SOURCES - page 1
1.
BIOGRAPHICAL MATERIAL -- in chronological order
ALCUIN (c735-804)
Phillip Drennon Thomas. Alcuin of York. DSB I, 104-105.
Robert Adamson. Alcuin, or Albinus. DNB, (I, 239-240), 20.
Andrew Fleming West. Alcuin and the Rise of the Christian Schools. (The Great Educators - III.) Heinemann, 1893. The only book on Alcuin that I found which deals with the
Propositiones.
Stephen Allott. Alcuin of York c. A.D. 732 to 804 -- his life and letters. William Sessions,
York, 1974.
FIBONACCI [LEONARDO PISANO] (c1170->1240)
See also the entries for Fibonacci in Common References.
Fibonacci. (1202 -- first paragraph); 1228 -- second paragraph, on p. 1. In this paragraph he
narrates almost everything we know about him. [In the second ed., he inserted a
dedication as the first paragraph.]
The paragraph ends with the notable sentence which I have used as a motto for
this work. "Si quid forte minus aut plus iusto vel necessario intermisi, mihi deprecor
indulgeatur, cum nemo sit qui vitio careat et in omnibus undique sit circumspectus." (If
I have perchance omitted anything more or less proper or necessary, I beg indulgence,
since there is no one who is blameless and utterly provident in all things. [Grimm's
translation.])
Richard E. Grimm. The autobiography of Leonardo Pisano. Fibonacci Quarterly 11:1 (Feb
1973) 99-104. He has collated six MSS of the autobiographical paragraph and presents
his critical version of it, with English translation and notes. Sigler, below, gives another
translation. I give Grimm's translation, omitting his notes.
After my father's appointment by his homeland as state official in the customs
house of Bugia for the Pisan merchants who thronged to it, he took charge; and, in view
of its future usefulness and convenience, had me in my boyhood come to him and there
wanted me to devote myself to and be instructed in the study of calculation for some
days. There, following my introduction, as a consequence of marvelous instruction in
the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to
me before all others, and for it I realized that all its aspects were studied in Egypt, Syria,
Greece, Sicily, and Provence, with their varying methods; and at these places thereafter,
while on business, I pursued my study in depth and learned the give-and-take of
disputation. But all this even, and the algorism, as well as the art of Pythagoras I
considered as almost a mistake in respect to the method of the Hindus. Therefore,
embracing more stringently that method of the Hindus, and taking stricter pains in its
study, while adding certain things from my own understanding and inserting also certain
things from the niceties of Euclid's geometric art, I have striven to compose this book in
its entirety as understandably as I could, dividing it into fifteen chapters. Almost
everything which I have introduced I have displayed with exact proof, in order that
those further seeking this knowledge, with its pre-eminent method, might be instructed,
and further, in order that the Latin people might not be discovered to be without it, as
they have been up to now. If I have perchance omitted anything more or less proper or
necessary, I beg indulgence, since there is no one who is blameless and utterly provident
in all things.
F. Bonaini. Memoria unica sincrona di Leonardo Fibonacci novamente scoperta. Giornale
Storico degli Archivi Toscani 1:4 (Oct-Dec 1857) 239-246. This reports the discovery
of a 1241 memorial of the Comune of Pisa, which I reproduce as it is not well known.
This grants Leonardo an annual honorarium of 20 pounds. In 1867, a plaque bearing
this inscription and an appropriate heading was placed in the atrium of the Archivio di
Stato in Pisa.
SOURCES - page 2
"Considerantes nostre civitatis et civium honorem atque profectum, qui eis, tam
per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi
Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis
quoties expedit, conferunter; ut eidem Leonardo, merito dilectionis et gratie, atque
scientie sue prerogativa, in recompensationem laboris sui quem substinet in audiendis et
consolidandis estimationibus et rationibus supradictis, a Comuni et camerariis publicis,
de Comuni et pro Comuni, mercede sive salario suo, annis singulis, libre xx denariorum
et amisceria consueta dari debeant (ipseque pisano Comuni et eius officialibus in
abbacatione de cetero more solito serviat), presenti constitutione firmamus."
A translation follows, but it can probably be improved. My thanks to Steph
Maury Gannon for many improvements over my initial version.
Considering the honour and progress of our city and its citizens that is brought to
them through both the knowledge and the diligent application of the discreet and wise
Maestro Leonardo Bigallo in the art of calculation for valuations and accounts for the
city and its officials and others, as often as necessary; we declare by this present decree
that there shall be given to the same Leonardo, from the Comune and on behalf of the
Comune, by reason of affection and gratitude, and for his excellence in science, in
recompense for the labour which he has done in auditing and consolidating the above
mentioned valuations and accounts for the Comune and the public bodies, as his wages
or salary, 20 pounds in money each year and his usual fees (the same Pisano shall
continue to render his usual services to the Comune and its officials in the art of
calculation etc.).
Bonaini also quotes a 1506 reference to Lionardo Fibonacci.
Mario Lazzarini. Leonardo Fibonacci Le sue Opere e la sua Famiglia. Bolletino di
Bibliografia e Storia delle Scienze Matematiche 6 (1903) 98-102 & 7 (1904) 1-7.
Traces the family to late 11C, saying Leonardo's father was Guglielmo and his
grandfather was probably Bonaccio. He estimates the birth date as c1170. He describes
a contract of 28 Aug 1226 in which Leonardo Bigollo, his father, Guglielmo, and his
brother, Bonaccingo, buy a piece of land from a relative. This land included a tower
and other buildings, outside the city, near S. Pietro in Vincoli. [G. Milanesi;
Documento inedito intorno a Leonardo Fibonacci; Rome, 1867 -- ??NYS]. Says
nothing is known of the 1202 ed of Liber Abbaci. Quotes the above memorial.
R. B. McClenon. Leonardo of Pisa and his liber quadratorum. AMM 26:1 (Jan 1919) 1-8.
Gino Loria. Leonardo Fibonacci. Gli Scienziati Italiana dall'inizio del medio evo ai nostri
giorni. Ed. by Aldo Mieli. (Dott. Attilio Nardecchia Editore, Rome, 1921;) Casa
Editrice Leonardo da Vinci, Rome, 1923. Vol. 1, pp. 4-12. This reproduces much of
the material in Lazzarini and the opening biographical paragraph of Liber Abaci.
Ettore Bortolotti. Article on Fibonacci in: Enciclopedia Italiana. G. Treccani, Rome, 1949
(reprint of 1932 ed.).
Charles King. Leonardo Fibonacci. Fibonacci Quarterly 1:4 (Dec 1963) 15-19.
Gino Arrighi, ed. Leonardo Fibonacci: La Practica di Geometria -- Volgarizzata da Cristofano
di Gherardo di Dino, cittadino pisano. Dal Codice 2186 della Biblioteca Riccardiana di
Firenze. Domus Galilaeana, Pisa, 1966. The Frontispiece is the mythical portrait of
Fibonacci, taken from I Benefattori dell'Umanità, vol. VI; Ducci, Florence, 1850.
(Smith, History II 214 says it is a "Modern engraving. The portrait is not based on
authentic sources".) P. 15 shows the plaque erected in the Archivio di Stato di Pisa in
1855 which reproduces the above memorial with an appropriate heading, but Arrighi
has no discussion of it. P. 19 is a photo of the statue in Pisa and p. 16 describes its
commissioning in 1859.
Joseph and Francis Gies. Leonard of Pisa and the New Mathematics of the Middle Ages.
Crowell, NY, 1969. This is a book for school students and contains a number of
dubious statements and several false statements.
Kurt Vogel. Fibonacci, Leonardo, or Leonardo of Pisa. DSB IV, 604-613.
A. F. Horadam. Eight hundred years young. Australian Mathematics Teacher 31 (1975)
123-134. Good survey of Fibonacci's life & work. Gives English of a few problems.
This is available on Kimberling's website - see below.
Ettore Picutti. Leonardo Pisano. Le Scienze 164 (Apr 1982) ??NYS. = Le Scienze,
SOURCES - page 3
Quaderni; 1984, pp. 30-39. (Le Scienze is a magazine; the Quaderni are collections of
articles into books.) Mostly concerned with the Liber Quadratorum, but surveys
Fibonacci's life and work. Says he was born around 1170. Includes photo of the plaque
in the Archivo di Stato di Pisa.
Leonardo Pisano Fibonacci. Liber quadratorum, 1225. Translated and edited by L. E. Sigler
as: The Book of Squares; Academic Press, NY, 1987. Introduction: A brief biography
of Leonardo Pisano (Fibonacci) [1170 - post 1240], pp. xv-xx. This is the best recent
biography, summarizing Picutti's article. Says he was born in 1170 and his father's
name was Guilielmo -- cf Loria above. Gives another translation of the biographical
paragraph of the Liber Abbaci.
A. F. Horadam & J. Lahr. Letter to the Editor. Fibonacci Quarterly 28:1 (Feb 1990) 90. The
authors volunteer to act as coordinators for work on the life and work of Fibonacci.
Addresses: A. F. Horadam, Mathematics etc., Univ. of New England, Armidale, New
South Wales, 2351, Australia; J. Lahr, 14 rue des Sept Arpents, L-1139 Luxembourg,
Luxembourg.
Thomas Koshy. Fibonacci and Lucas Numbers with Applications. Wiley-Interscience, Wiley,
2001. Claims to be 'the first attempt to compile a definitive history and authoritative
analysis' of the Fibonacci numbers, but the history is generally second-hand and marred
with a substantial number of errors, The mathematical work is extensive, covering
many topics not organised before, and is better done, but there are more errors than one
would like.
Laurence E. Sigler. Translation of Liber Abaci as: Fibonacci's Liber Abaci A Translation
into Modern English of Leonardo Pisano's Book of Calculation. Springer, 2002.
Clark Kimberling's site web includes biographical material on Fibonacci and other similar
number theorists. http://cedar.evansville.edu/~ck6/bstud/fibo.html .
Ron Knott has a huge website on Fibonacci numbers and their applications, with material on
many related topics, e.g. continued fractions, π, etc. with some history.
www.ee.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.html .
Luca PACIOLI (c1445-1517)
S. A. Jayawardene. Luca Pacioli. BDM 4, 1897-1900.
Bernardino Baldi (Catagallina) (1553-1617). Vita di Pacioli. (1589, first published in his
Cronica de Mathematici of 1707.) Reprinted in: Bollettino di bibliografia e di storia
delle scienze matematiche e fisiche 12 (1879) 421-427. ??NYS -- cited by Taylor,
p. 338.
Enrico Narducci. Intorno a due edizioni della "Summa de arithmetica" di Fra Luca Pacioli.
Rome, 1863. ??NYS -- cited by Riccardi [Biblioteca Matematica Italiana, 1952]
D. Ivano Ricci. Luca Pacioli, l'uomo e lo scienziato. San Sepolcro, 1940. ??NYS -- cited in
BDM.
R. Emmett Taylor. No Royal Road Luca Pacioli and His Times. Univ. of North Carolina
Press, Chapel Hill, 1942. BDM describes this as lively but unreliable.
Ettore Bortolotti. La Storia della Matematica nella Università di Bologna. Nicola Zanichelli
Editore, Bologna, 1947. Chap. I, § V, pp. 27-33: Luca Pacioli.
Margaret Daly Davis. Piero della Francesca's Mathematical Treatises The "Trattato d'abaco"
and "Libellus de quinque corporibus regularibus". Longo Editore, Ravenna, 1977. This
discusses Piero's reuse of his own material and Pacioli's reuse of Piero's material.
Fenella K. C. Rankin. The Arithmetic and Algebra of Luca Pacioli. PhD thesis, Univ. of
London, 1992 (copy at the Warburg Institute), ??NYR.
Enrico Giusti, ed. Descriptive booklet accompanying the 1994 facsimile of the Summa -- qv
in Common References.
Edward A. Fennell. Figures in Proportion: Art, Science and the Business Renaissance. The
contribution of Luca Pacioli to culture and commerce in the High Renaissance.
Catalogue for the exhibition, The Institute of Chartered Accountants in England and
Wales, London, 1994.
Claude-Gaspar BACHET de Méziriac (1581-1638)
C.-G. Collet & J. Itard. Un mathématicien humaniste -- Claude-Gaspar Bachet de Méziriac
(1581-1638). Revue d'Histoire des Sciences et leurs Applications 1 (1947) 26-50.
SOURCES - page 4
J. Itard. Avant-propos. IN: Bachet; Problemes; 1959 reprint, pp. v-viii. Based on the
previous article.
There is a Frontispiece portrait in the reprint.
Underwood Dudley. The first recreational mathematics book. JRM 3 (1970) 164-169. On
Bachet's Problemes.
William Schaaf. Bachet de Méziriac, Claude-Gaspar. DSB I, 367-368.
Jean LEURECHON (c1591-1670) and Henrik VAN ETTEN
A. Deblaye. Étude sur la récréation mathématique du P. Jean Leurechon, Jésuite. Mémoires
de la Société Philotechnique de Pont-à-Mousson 1 (1874) 171-183. [MUS #314.
Schaaf. Hall, OCB, pp. 86, 88 & 114, says the only known copy of this journal is at
Harvard, which has kindly supplied me with a photocopy of this article. Hall indicates
the article is in vol. II and says it is 12 pages, but only cites pp. 171 & 174.] This
simply assumes Leurechon is the author and gives a summary of his life. The essential
content is described by Hall.
G. Eneström. Girard Desargues und D.A.L.G. Biblioteca Mathematica (3) 14 (1914)
253-258. D.A.L.G. was an annotator of van Etten's book in c1630. Although D.A.L.G.
was used by Mydorge on one of his other books, it had been conjectured that this stood
for Des Argues Lyonnais Girard (or Géomètre). Eneström can find no real evidence for
this and feels that Mydorge is the most likely person.
Trevor H. Hall. Mathematicall Recreations. An Exercise in Seventeenth Century
Bibliography. Leeds Studies in Bibliography and Textual Criticism, No. 1. The
Bibliography Room, School of English, University of Leeds, 1969, 38pp. Pp. 18-38
discuss the question of authorship and Hall feels that van Etten probably was the author
and that there is very little evidence for Leurechon being the author. Much of the
mathematical content is in Bachet's Problemes and may have been copied from it or
some common source. [This booklet is reproduced as pp. 83-119 of Hall, OCB, with
the title page of the 1633 first English edition reproduced as plate 5, opp. p. 112. Some
changes have been made in the form of references since OCB is a big book, but the only
other substantial change is that he spells the name of the dedicatee of the book as
Verreyken rather than Verreycken.]
William Schaaf. Leurechon, Jean. DSB VIII, 271-272.
Jacques Voignier. Who was the author of "Recreation Mathematique" (1624)? The Perennial
Mystics #9 (1991) 5-48 (& 1-2 which are the cover and its reverse). [This journal is
edited and published by James Hagy, 2373 Arbeleda Lane, Northbrook, Illinois, 60062,
USA.] Presents some indirect evidence for Leurechon's authorship.
Jacques OZANAM (1640-1717)
On the flyleaf of J. E. Hofmann's copy of the 1696 edition of Ozanam's Recreations is a pencil
portrait labelled Ozanam -- the only one I know of. This copy is at the Institut für
Geschichte der Naturwissenschaft in Munich. Hofmann published the picture -- see
below.
Charles Hutton. A Mathematical and Philosophical Dictionary. 1795-1796. Vol. II, pp. 184185. ??NYS [Hall, OCB, p. 166.]
Charles Hutton. On the life and writings of Ozanam, the first author of these Mathematical
Recreations. Ozanam-Hutton. Vol. I. 1803: xiii-xv; 1814: ix-xi.
William L. Schaaf. Jacques Ozanam on mathematics .... MTr 50 (1957) 385-389. Mostly
based on Hutton. Includes a sketchy bibliography of Ozanam's works, generally
ignoring the Recreations.
Joseph Ehrenfried Hofmann. Leibniz und Ozanams Problem, drei Zahlen so zu bestimmen,
dass ihre Summe eine Quadratzahl und ihre Quadratsumme eine Biquadratzahl ergibt.
Studia Leibnitiana 1:2 (1969) 103-126. Outlines Ozanam's life, gives a bibliography of
his works and reproduces the above-mentioned drawing as a plate opp. p. 124. (My
thanks to Menso Folkerts for this information and a copy of Hofmann's article.)
William L. Schaaf. Ozanam, Jacques. DSB X, 263-265.
Jean Étienne MONTUCLA (1725-1799)
Charles Hutton. Some account of the life and writings of Montucla. Ozanam-Hutton. Vol. I.
SOURCES - page 5
1803: viii-xii; 1814: iv-viii.
Charles Hutton. A Philosophical and Mathematical Dictionary. 2nd ed. of the Dictionary
cited under Ozanam, 1815, Vol. II, pp. 63-64. ??NYS. According to Hall, OCB, p.
167, this is not in the 1795-1796 ed. and is a reworking of the previous item.
Lewis CARROLL (1832-1898)
Pseudonym of Charles Lutwidge Dodgson. There is so much written on Carroll that I
will only give references to his specifically recreational work and some basic references.
The Diaries of Lewis Carroll. Edited by Roger Lancelyn Green. (OUP, 1954); 2 vols,
Greenwood Publishers, Westport, Connecticut, 1971, HB.
Lewis Carroll's Diaries The private journals of Charles Lutwidge Dodgson (Lewis Carroll)
The first complete version of the nine surviving volumes with notes and annotations by
Edward Wakeling. Introduction by Roger Lancelyn-Green. The Lewis Carroll Society,
Publications Unit, Luton, Bedfordshire. [There were 13 journals, but 4 are lost.]
Vol. 1. Journal 2, Jan-Sep 1855. 1993, 158pp.
Vol. 2. Journal 4, Jan-Dec 1856. 1994, 158pp.
Vol. 3. Journal 5, Jan 1857 - Apr 1858. 1995, 199pp.
Vol. 4. Journal 8, May 1862 - Sep 1864 and a reconstruction of the four missing
years, 1858-1862. 1997, 399pp.
Vol. 5. Journal 9, Sep 1864 - Jan 1868, including the Russian Journal.
1999, 416pp.
Vol. 6. Journal 10, Apr 1868 - Dec 1876. 2001, 552pp.
Vol. 7. Journal 11, Jan 1877 - Jun 1883. 2003, 606pp.
The Letters of Lewis Carroll. Edited by Morton N. Cohen with the assistance of Roger
Lancelyn Green. Volume One ca.1837 - 1885; Volume Two 1886 - 1898. Macmillan
London, 1979.
Stuart Dodgson Collingwood. The Life and Letters of Lewis Carroll. T. Fisher Unwin,
London, 1898.
Stuart Dodgson Collingwood, ed. The Lewis Carroll Picture Book. T. Fisher Unwin,
London, 1899. = Diversions and Digressions of Lewis Carroll, Dover, 1961. = The
Unknown Lewis Carroll, Dover, 1961(?). Reprint, in reduced format, Collins, c1910.
The pagination of the main text is the same in the 1899 and in both Dover reprints, but
is quite different than the Collins. Cited as: Carroll-Collingwood, qv in Common
References.
R. B. Braithwaite. Lewis Carroll as logician. MG 16 (No. 219) (Jul 1932) 174-178. He notes
that Carroll assumed that a universal statement implied the existence of an object
satisfying the antecedent, e.g. 'all unicorns are blue' would imply the existence of
unicorns, contrary to modern convention.
Derek Hudson. Lewis Carroll -- An Illustrated Biography. Constable, 1954; new illustrated
ed., 1976.
Warren Weaver. Lewis Carroll: Mathematician. SA 194:4 (Apr 1956) 116-128. + Letters
and response. SA 194:6 (Jun 1956) 19-22.
Martin Gardner. The Annotated Alice. C. N. Potter, NY, 1960. Penguin, 1965; 2nd ed.,
1971. Revised as: More Annotated Alice, 1990, qv.
Martin Gardner. The Annotated Snark. Bramhall House, 1962. Penguin, 1967; revised,
1973 & 1974.
John Fisher. The Magic of Lewis Carroll. Nelson, 1973. Penguin, 1975.
Morton N. Cohen, ed. The Selected Letters of Lewis Carroll. Papermac (Macmillan), 1982.
Martin Gardner. More Annotated Alice. [Extension of The Annotated Alice.] Random
House, 1990.
Edward Wakeling. Lewis Carroll's Games and Puzzles. Dover and the Lewis Carroll
Birthplace Trust, 1992. Cited as Carroll-Wakeling, qv in Common References.
Francine F. Abeles, ed. The Pamphlets of Lewis Carroll -- Vol. 2: The Mathematical
Pamphlets of Charles Lutwidge Dodgson and Related Pieces. Lewis Carroll Society of
North America, distributed by University Press of Virginia, Charlottesville, 1994.
Edward Wakeling. Rediscovered Lewis Carroll Puzzles. Dover, 1995. Cited as
Carroll-Wakeling II, qv in Common References.
Martin Gardner. The Universe in a Handkerchief. Lewis Carroll's Mathematical Recreations,
SOURCES - page 6
Games, Puzzles and Word Plays. Copernicus (Springer, NY), 1996. Cited as
Carroll-Gardner, qv in Common References.
Martin Gardner. The Annotated Alice: The Definitive Edition. 1999. [A combined version
of The Annotated Alice and More Annotated Alice.]
Professor Louis HOFFMANN (1839-1919)
Pseudonym of Angelo John Lewis.
Joseph Foster. Men-at-the-Bar: A biographical Hand-List of the Members of the Various Inns
of Court, including Her Majesty's Judges, etc. 2nd ed, the author, 1885. P. 277 is the
entry for Lewis. Born in London, eldest son of John Lewis. Graduated from Wadham
College, Oxford. Entered Lincoln's Inn as a student in 1858, called to the bar there in
1861. Married Mary Ann Avery in 1864. Author of Manual of Indian Penal Code
and Manual of Indian Civil Procedure. Address: 12 Crescent Place, Mornington
Crescent, London, NW. (My thanks to the Library of Lincoln's Inn for this
information.)
Anonymous. Professor Hoffmann. Mahatma 4:1 (Jul 1900) 377-378. A brief note, with
photograph, stating that he is Mr. Angelo Lewis, M.A. and Barrister-at-Law.
Will Goldston. Will Goldston's Who's Who in Magic. My version is included in a
compendium called: Tricks that Mystify; Will Goldston, London, nd [1934-NUC]. Pp.
106-107. Says he was a barrister, retired to Hastings about 1903 and died in 1917.
Who Was Who, 1916-1928, p. 627. This says he attended North London Collegiate School
and that he only practised law until 1876. He was on the staff of the Saturday Review
and a contributor to many journals. Won the £100 prize offered by Youth's Companion
(Boston) for best short story for boys. Lists 36 books by him and 9 card games he
invented. Address: Manningford, Upper Bolebrooke Road, Bexhill-on-Sea. (My
thanks to the Library of Lincoln's Inn for this information.)
J. B. Findlay & Thomas A. Sawyer. Professor Hoffmann: A Study. Published by Thomas A.
Sawyer, Tustin, California, 1977. A short book, 12 + 67 pp, with two portraits (one
from Mahatma) and 27pp of bibliography. He was born at 3 Crescent Place,
Mornington Crescent, London. He was a barrister and wrote two books on Indian law.
Charles Reynolds. Introduction -- to the reprint of Hoffmann's Modern Magic, Dover, 1978,
pp. v-xiv. This says Lewis was a barrister, which is mentioned in another reprint of a
Hoffmann book and in S. H. Sharpe's translation of Ponsin on Conjuring.
Edward Hordern. Foreword to this edition. In: Hoffmann's Puzzles Old and New (see under
Common References), 1988 reprint, pp. v-vi. This says he was the Reverend Lewis, but
this is corrected in Hoffmann-Hordern to saying he was a barrister.
Hoffmann-Hordern, p. viii, is a version of the photograph in Mahatma.
Hall, OCB, p. 189, gives Hoffmann's address as Ireton Lodge, Cromwell Ave., N. -presumably the Cromwell Ave. in Highgate.
Toole Stott 386 gives a little information about Hoffmann and Modern Magic, including an
address in Mornington Crescent in 1877.
No DNB or DSB entry -- I have suggested a DNB entry.
Sam LOYD (1841-1911) and Sam LOYD JR. (1873-1934)
[W. R. Henry.] Samuel Loyd. [Biography.] Dubuque Chess Journal, No. 66 (Aug-Sep 1875)
361-365. ??NX -- o/o (11 Jul 91).
Loyd. US Design 4793 -- Design for Puzzle-Blocks. 11 April 1871. These are solid pieces,
but unfortunately the drawing did not come with this, so I am not clear what they are.
??Need drawing -- o/o (11 Jul 91).
Anonymous & Sam Loyd. Loyd's puzzles (Introductory column). Brooklyn Daily Eagle (22
Mar 1896) 23. Says he lives at 153 Halsey St., Brooklyn.
L. D. Broughton Jr. Samuel Loyd. [A Biography.] Lasker's Chess Magazine 1:2 (Dec 1904)
83-85. About his chess problems with a mention of some of his puzzles.
G. G. Bain. The prince of puzzle-makers. An interview with Sam Loyd. Strand Magazine 34
(No. 204) (Dec 1907) 771-777. Solutions of Sam Loyd's puzzles. Ibid. 35 (No. 205)
(Jan 1908) 110.
Walter Prichard Eaton. My fifty years in puzzleland -- Sam Loyd and his ten thousand
brain-teasers. The Delineator (New York) (April 1911) 274 & 328. Drawn portrait of
SOURCES - page 7
Loyd, age 69.
Anon. Puzzle inventor dead. New-York Daily Tribune (12 Apr 1911) 7. Says he died at his
house, 153 Halsey St. "He declared no one had ever succeeded in solving [the
"Disappearing Chinaman"]." Says he is survived by a son and two daughters (!! -- has
anyone ever tracked the daughters and their descendents??).
Anon. Sam Loyd, puzzle man, dies. New York Times (12 Apr 1911) 13. Says he was for
some time editor of The Sanitary Engineer and a shrewd operator on Wall Street.
Anon. Sam Loyd. SA (22 Apr 1911) 40-41?? Says he was for some years chess editor of SA
and was puzzle editor of Woman's Home Companion when he died.
W. P. Eaton. Sam Loyd. The American Magazine 72 (May 1911) 50, 51, 53. Abridged
version of Eaton's earlier article. Photo of Loyd on p. 50.
P. J. Doyle. Letter to the Chess column. The Sunday Call [Newark, NJ] (21 May 1911),
section III, p. 10.
A. C. White. Sam Loyd and His Chess Problems. Whitehead and Miller, Leeds, UK, 1913;
corrected, Dover, 1962.
Alain C. White. Supplement to Sam Loyd and His Chess Problems. Good Companion Chess
Problem Club, Philadelphia, vol. I, nos. 11-12 (Aug 1914), 12pp. This is mostly
corrections of the chess problems, but adds a few family details with a picture of the
Loyd Homestead and Grist Mill in Moylan, Pennsylvania.
Alain C. White. Reminiscences of Sam Loyd's family. The Problem [Pittsburgh]
(28 Mar 1914) 2, 3, 6, 7.
Louis C. Karpinski. Loyd, Samuel. Dictionary of American Biography, Scribner's, NY,
vol. XI, 1933, pp. 479-480.
Loyd Jr. SLAHP. 1928. Preface gives some details of his life, making little mention of his
father, "who was a famous mathematician and chess player". He claims to have created
over 10,000 puzzles. There are some vague biographical details on pp. 1-22, e.g. 'Father
conducted a printing establishment.' 'My "Missing Chinaman Puzzle"'. (It may have
been some such assertion that led me to estimate his birthdate as 1865, but I now see it
is well known to be 1873.)
Anonymous. Sam Loyd dead; puzzle creator. New York Times (25 Feb 1934). Obituary of
Sam Loyd Jr. Says he resided at 153 Halsey St., Brooklyn -- the same address as his
father -- see the Brooklyn Daily Eagle article of 1896, above. He worked from a studio
at 246 Fulton St., Brooklyn. It says Jr. invented 'How Old is Ann?'.
Clark Kinnaird. Encyclopedia of Puzzles and Pastimes. Grosset & Dunlap, NY, 1946.
Pp. 263-267: Sam Loyd. Asserts that Loyd Jr. invented 'How Old is Ann?'
Gardner. Sam Loyd: America's greatest puzzlist. SA (Aug 1957) c= First Book, Chap. 9.
Gardner. Advertising premiums. SA (Nov 1971) c= Wheels, chap. 12.
Will Shortz is working on a biography.
No DSB entry.
François Anatole Édouard LUCAS (1842-1891)
Jeux Scientifiques de Ed. Lucas. Advertisement by Chambon & Baye (14 rue Etienne-Marcel,
Paris) for the 1re Serie of six games. Cosmos. Revue des Sciences et Leurs
Applications 39 (NS No. 254) (7 Dec 1889) no page number on my photocopy.
B. Bailly [name not given, but supplied by Hinz]. Article on Lucas's puzzles. Cosmos.
Revue des Sciences et Leurs Applications. NS, 39 (No. 259) (11 Jan 1890) 156-159.
NEED 156-157.
Nécrologie: Édouard Lucas. La Nature 19 (1891) II, 302.
Obituary notice: "La Nature announces the death of Prof. Edouard Lucas ...." Nature 44
(15 Oct 1891) 574-575.
Duncan Harkin. On the mathematical work of François-Édouard-Anatole Lucas.
L'Enseignement Math. (2) 3 (1957) 276-288. Pp. 282-288 is a bibliography of 184
items. I have found many Lucas publication not listed here and have started a new
Bibliography -- see below.
P. J. Campbell. Lucas' solution to the non-attacking rooks problem. JRM 9 (1976/77)
195-200. Gives life of Lucas.
A photo of Lucas is available from Bibliothèque Nationale, Service Photographique, 58 rue
Richelieu, F-75084 Paris Cedex 02, France. Quote Cote du Document Ln27 . 43345 and
Cote du Cliche 83 A 51772. (??*) I have obtained a copy, about 55 x 85 mm, with the
photo in an oval surround. It looks like a carte-de-visite, but has Édouard LUCAS
SOURCES - page 8
(1842-1891). -- Phot. Zagel. underneath. (Thanks to H. W. Lenstra for the
information.)
Norman T. Gridgeman. Lucas, François-Édouard-Anatole. DSB VIII, 531-532.
Susanna S. Epp. Discrete Mathematics with Applications. Wadsworth, Belmont, Calif.,
1990, p. 477 gives a small photo of Lucas which looks nothing like the photo from the
BN. I have since received a note from Epp via Paul Campbell that a wrong photo was
used in the first edition, but this was corrected in later editions.
Alain Zalmanski. Edouard Lucas Quand l'arithmétique devient amusante. Jouer Jeux
Mathématiques 3 (Jul/Sep 1991) 5. Brief notice of his life and work.
Andreas M. Hinz. Pascal's triangle and the Tower of Hanoi. AMM 99 (1992) 538-544.
Sketches Lucas' life and work, giving details that are not in the above items.
David Singmaster. The publications of Édouard Lucas. Draft version, 14pp, 1998. I
discovered many items in Dickson's History of the Theory of Numbers and elsewhere
which are not given by Harkin (cf above). This has 248 items, though many of these are
multiple items so the actual count is perhaps 275. However, Dickson does not give
article titles, and may not give the pages of the entire article, so the same article may be
cited more than once, at different pages. I hope to fill in the missing information at
some time.
Hermann Cäsar Hannibal SCHUBERT (1848-1911)
Acta Mathematica 1882-1912. Table Générale des Tomes 1-35. 1913. P. 169. Portrait of
Schubert.
Werner Burau. Schubert, Hermann Cäsar Hannibal. DSB XII, 227-229.
Walter William Rouse BALL (1850-1925)
Anon. Obituary: Mr. Rouse Ball. The Times (6 Apr 1925) 16.
Anon. Funeral notice: Mr. W. W. R. Ball. The Times (9 Apr 1925) 13.
(Lord) Phillimore. Letter: Mr. Rouse Ball. The Times (9 Apr 1925) 15.
"An old pupil". The late Mr. Rouse Ball. The Times (13 Apr 1925) 12.
J. J. Thomson. W. W. Rouse Ball. The Cambridge Review (24 Apr 1925) 341-342.
Anon. Obituary of W. W. Rouse Ball. Nature 115 (23 May 1925) 808-809.
Anon. The late Mr. W. W. Rouse Ball. The Trinity Magazine (Jun 1925) 53-54.
Anon. Entry in Who's Who, 1925, p. 127.
Anon. Wills and bequests: Mr. Walter William Rouse Ball. The Times (7 Sep 1925) 15.
E. T. Whittaker. Obituary. W. W. Rouse Ball. Math. Gaz. 12 (No. 178) (Oct 1925) 449-454,
with photo opp. p. 449.
F. Cajori. Walter William Rouse Ball. Isis 8 (1926) 321-324. Photo on plate 15, opp. p. 321.
Copy of Ball's 1924 Xmas card on p. 324.
J. A. Venn. Alumni Cantabrigienses. Part II: From 1752 to 1900. Vol. I, p. 136. CUP,
1940.
David Singmaster. Walter William Rouse Ball (1850-1925). 6pp handout for 1st UK
Meeting on the History of Recreational Mathematics, 24 Oct 1992. Plus extended
biographical (6pp) and bibliographical (8pp) notes which repeat some of the material in
the handout.
No DNB or DSB entry -- however I have offered to write a DNB entry. I have since seen the
proposed list of names for the next edition and Ball is already on it.
Henry Ernest DUDENEY (1857-1930)
Anon. & Dudeney. A chat with the puzzle king. The Captain 2 (Dec? 1899) 314-320, with
photo. Partly an interview. Includes photos of Littlewick Meadow.
Anon. Solutions to "Sphinx's puzzles". The Captain 2:6 (Mar 1900) 598-599 & 3:1
(Apr 1900) 89.
Anon. Master of the breakfast table problem. Daily Mail (1 Feb 1905) 7. An interview with
Dudeney in which he gives the better version of his spider and fly problem.
Fenn Sherie. The Puzzle King: An Interview with Henry E. Dudeney. Strand Magazine 71
(Apr 1926) 398-4O4.
Alice Dudeney. Preface to PCP, dated Dec 1931, pp. vii-x. The date of his death is
erroneously given as 1931.
SOURCES - page 9
Gardner. Henry Ernest Dudeney: England's greatest puzzlist. SA (Jun 1958) c= Second
Book, chap. 3.
Angela Newing. The Life and Work of H. E. Dudeney. MS 21 (1988/89) 37-44.
Angela Newing is working on a biography.
No DNB or DSB entry. I have suggested a DNB entry.
Wilhelm Ernst Martin Georg AHRENS (1872-1927)
Wilhelm Lorey. Wilhelm Ahrens zum Gedächtnis. Archiv für Geschichte der Mathematik,
der Naturwissenschaften und der Technik 10 (1927/28) 328-333. Photo on p. 328.
O. Staude. Dem Andenken an Dr. Wilhelm Ahrens. Jahresbericht DMV 37 (1928) 286-287.
No DSB entry.
Yakov Isidorovich PERELMAN [Я . И . П е р е л м а н ] (1882-1942)
Perelman. FMP. 1984. P. 2 (opp. TP) is a sketch of his life and the history of the book.
There is a small drawing of Perelman at the top of the page.
Patricio Barros. Website -- Yakov I. Perelman [in Spanish]:
www.geocities.com/yakov_perelman/index.html. This includes a four page biography,
in collaboration with Antonio Bravo, and two photos.
Hubert PHILLIPS (1891-1964)
Hubert Phillips. Journey to Nowhere. A Discursive Autobiography. Macgibbon & Kee,
London, 1960. ??NYR
No DNB entry -- I have suggested one.
2.
GENERAL PUZZLE COLLECTIONS AND SURVEYS
H. E. Dudeney. Great puzzle crazes. London Magazine 13?? (Nov 1904) 478-482. Fifteen
Puzzle. Pigs in Clover, Answers, Pick-me-up (spiral ramp) and other dexterity puzzles.
Get Off the Earth. Conjurer's Medal (ring maze). Chinese Rings. Chinese Cross (six
piece burr). Puzzle rings. Solitaire. The Mathematician's Puzzle (square, circle,
triangle). Imperial Scale. Heart and Balls.
H. E. Dudeney. Puzzles from games. Strand Magazine 35 (No. 207) (Mar 1908) 339-344.
Solutions. Ibid. 35 (No. 208) (Apr 1908) 455-458.
H. E. Dudeney. Some much-discussed puzzles. Strand Magazine 35 (No. 209) (May 1908)
580-584. Solutions. Ibid. 35 (No. 210) (Jun 1908) 696.
H. E. Dudeney. The world's best puzzles. Strand Magazine 36 (No. 216) (Dec 1908)
779-787. Solutions. Ibid. 37 (No. 217) (Jan 1909) 113-116.
H. E. Dudeney. The psychology of puzzle crazes. The Nineteenth Century 100:6 (Dec 1926)
868-879. Repeats much of his 1904 article.
Sam Loyd Jr. Are you good at solving puzzles? The American Magazine (Sep 1931) 61-63,
133-137.
Orville A. Sullivan. Problems involving unusual situations. SM 9 (1943) 114-118 &
13 (1947) 102-104.
3.
GENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIAL
I have tried to divide this material into historical and bibliographical parts, but the two
overlap considerably.
3.A. GENERAL HISTORICAL MATERIAL
Raffaella Franci. Giochi matematici in trattati d'abaco del medioevo e del rinascimento. Atti
del Convegno Nazionale sui Giochi Creative, Siena, 11-14 Jun 1981. Tipografia
Senese for GIOCREA (Società Italiana Giochi Creativi), 1981. Pp. 18-43. Describes
and quotes many typical problems. 17 references, several previously unknown to me.
Heinrich Hermelink. Arabische Unterhaltungsmathematik als Spiegel Jahrtausendealter
SOURCES - page 10
Kulturbeziehungen zwischen Ost und West. Janus 65 (1978) 105-117, with English
summary. An English translation appeared as: Arabic recreational mathematics as a
mirror of age-old cultural relations between Eastern and Western civilizations; in:
Ahmad Y. Al-Hassan, Ghada Karmi & Nizar Namnum, eds.; Proceedings of the First
International Symposium for the History of Arabic Science, April 1976 -- Vol. Two:
Papers in European Languages; Institute for the History of Arabic Science, Aleppo,
1978, pp. 44-52. (There are a few translation and typographical errors, which make it
clear that the English version is a translation of the German.)
D. E. Smith. On the origin of certain typical problems. AMM 24 (1917) 64-71. (This is
mostly contained in his History, vol. II, pp. 536-548.)
3.B. BIBLIOGRAPHICAL MATERIAL
Many of the items cited in the Common References have extensive bibliographies. In
particular: BLC; BMC; BNC; DNB; DSB; Halwas; NUC; Schaaf;
Smith & De Morgan: Rara; Suter are basic bibliographical sources. Datta & Singh;
Dickson; Heath: HGM; Murray; Sanford: H&S & Short History; Smith: History &
Source Book; Struik; Tropfke are histories with extensive bibliographical references. AR;
BR are editions of early texts with substantial bibliographical material. Ahrens: MUS; Ball:
MRE; Berlekamp, Conway & Guy: Winning Ways; Gardner; Lucas: RM are recreational
books with some useful bibliographical material. Of these, the material in Ahrens is by far the
most useful. The magic bibliographies of Christopher, Clarke & Blind, Hall, Heyl, Price (see
HPL), Toole Stott and Volkmann & Tummers have considerable overlap with the present
material, particularly for older books, though Hall, Heyl and Toole Stott restrict themselves to
English material, while Volkmann & Tummers only considers German. Santi is also very
useful. Below I give some additional bibliographical material which may be useful, arranged
in author order.
Anonymous. Mathematical bibliography. SSM 48 (1948) 757-760. Covers recreations.
Wilhelm Ahrens. Mathematische Spiele. Section I G 1 of Encyklopadie der Math. Wiss.,
Vol. I, part 2, Teubner, Leipzig, 1900-1904, pp. 1080-1093.
Raymond Clare Archibald. Notes on some minor English mathematical serials. MG 14
(1928-29) 379-400.
Elliott M. Avedon & Brian Sutton-Smith. The Study of Games. (Wiley, NY, 1971);
Krieger, Huntington, NY, 1979.
Anthony S. M. Dickins. A Catalogue of Fairy Chess Books and Opuscules Donated to
Cambridge University Library, 1972-1973, by Anthony Dickins M.A. Third ed.,
Q Press, Kew Gardens, UK, 1983.
Underwood Dudley. An annotated list of recreational mathematics books. JRM 2:1
(Jan 1969) 13-20. 61 titles, in English and in print at the time.
Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games and Some Related
Material. There have been several versions with slightly varying titles. The most recent
printed version is: 400 items, 28 pp., including 4 pp of text, Sep 1990. Technical
Report CS90-23, Weizmann Institute of Science, Rehovot, Israel. = Proc. Symp. Appl.
Math. 43 (1991) 191-226. Fraenkel has since produced Update 1 to this which lists 430
items on 31pp, Aug 1992; and Update 2, 480 items on 33pp, with 5 pp of text,
accidentally dated Aug 1992 at the top but produced in Feb 1994. On 22 Nov 1994, it
became a dynamic survey on the Electronic J. Combinatorics and can be accessed from:
http://ejc.math.gatech.edu:8080/journal/surveys/index.html.
It can also be accessed via anonymous ftp from ftp.wisdom.weizmann.ac.il. After
logging in, do cd pub/fraenkel and then get one of the following three compressed
files: games.tex.z; games.dvi.z; games.ps.z.
Martin P. Gaffney & Lynn Arthur Steen. Annotated Bibliography of Expository Writing in
the Mathematical Sciences. MAA, 1976.
JoAnne S. Growney. Mathematics and the arts -- A bibliography. Humanistic Mathematics
Network Journal 8 (1993) 22-36. General references. Aesthetic standards for
mathematics and other arts. Biographies/autobiographies of mathematicians.
Mathematics and display of information (including mapmaking). Mathematics and
humor. Mathematics and literature (fiction and fantasy). Mathematics and music.
Mathematics and poetry. Mathematics and the visual arts.
SOURCES - page 11
JoAnne S. Growney. Mathematics in Literature and Poetry. Humanistic Mathematics
Network Journal 10 (Aug 1994) 25-30. Short survey. 3 pages of annotated references
to 29 authors, some of several books.
R. C. Gupta. A bibliography of selected book [sic] on history of mathematics. The
Mathematics Education 23 (1989) 21-29.
Trevor H. Hall. Mathematicall Recreations. Op. cit. in 1. This is primarily concerned with
the history of the book by van Etten. [This booklet is revised as pp. 83-119 of Hall,
OCB -- see Section 1.]
Catherine Perry Hargrave. A History of Playing Cards and a Bibliography of Cards and
Gaming. (Houghton Mifflin, Boston, 1930); Dover, 1966.
Susan Hill. Catalogue of the Turner Collection of the History of Mathematics Held in the
Library of the University of Keele. University Library, Keele, 1982. (Sadly this
collection was secretly sold by Keele University in 1998 and has now been dispersed.)
Honeyman Collection -- see: Sotheby's.
Horblit Collection -- see: Sotheby's and H. P. Kraus.
Else Høyrup. Books about Mathematics. Roskilde Univ. Center, PO Box 260, DK-4000,
Roskilde, Denmark, 1979.
D. O. Koehler. Mathematics and literature. MM 55 (1982) 81-95. 64 references. See Utz for
some further material.
H. P. Kraus (16 East 46th Street, New York, 10017). The History of Science including
Navigation.
Catalogue 168. A First Selection of Books from the Library of Harrison D. Horblit. Nd
[c1976].
Catalogue 169. A Further Selection of Books, 1641-1700 (Wing Period) from the
Library of Harrison D. Horblit. Nd [c1976].
Catalogue 171. Another Selection of Books from the Library of Harrison D. Horblit.
Nd [c1976].
These are the continuations of the catalogues issued by Sotheby's, qv.
John S. Lew. Mathematical references in literature. Humanistic Mathematics Network
Journal 7 (1992) 26-47.
Antonius van der Linde. Das erst Jartausend [sic] der Schachlitteratur -- (850-1880). (1880);
Facsimile reprint by Caissa Limited Editions, Yorklyn, Delaware, 1979, HB.
Andy Liu. Appendix III: A selected bibliography on popular mathematics. Delta-k 27:3
(Apr 1989) -- Special issue: Mathematics for Gifted Students, 55-83.
Édouard Lucas. Récréations mathématiques, vol 1 (i.e. RM1), pp. 237-248 is an Index
Bibliographique.
Felix Müller. Führer durch die mathematische Literature mit besonderer Berücksichtigung
der historisch wichtigen Schriften. Abhandlungen zur Geschichte der Mathematik 27
(1903).
Charles W. Newhall. "Recreations" in secondary mathematics. SSM 15 (1915) 277-293.
Mathematical Association. 259 London Road, Leicester, LE2 3BE.
Catalogue of Books and Pamphlets in the Library. No details, [c1912], 19pp, bound in at end
of Mathematical Gazette, vol. 6 (1911-1912).
A First List of Books & Pamphlets in the Library of the Mathematical Association -Books and Pamphlets acquired before 1924. Bell, London, 1926.
A Second List of Books & Pamphlets in the Library of the Mathematical Association -Books and Pamphlets acquired during 1924 and 1925. Bell, London, 1929.
A Third List of Books & Pamphlets in the Library of the Mathematical Association -Books and Pamphlets added from 1926 to 1929. Bell, London, 1930.
A Fourth List of Books & Pamphlets in the Library of the Mathematical Association -Books and Pamphlets added from 1930 to 1935. Bell, London, 1936.
Lists 1-4 edited by E. H. Neville.
Books and Periodicals in the Library of the Mathematical Association. Ed. by R. L.
Goodstein. MA, 1962. Includes the four previous lists and additions through
1961.
SEE ALSO: Riley; Rollett; F. R. Watson.
Stanley Rabinowitz. Index to Mathematical Problems 1980-1984. MathPro Press, Westford,
Massachusetts, 1992.
Cecil B. Read & James K. Bidwell.
Selected articles dealing with the history of elementary Mathematics. SSM 76 (1976)
477-483.
SOURCES - page 12
Periodical articles dealing with the history of advanced mathematics -- Parts I & II.
SSM 76 (1976) 581-598 & 687-703.
Rudolf H. Rheinhardt. Bibliography on Whist and Playing Cards. From: Whist Scores and
Card-table Talk, Chicago, 1887. Reprinted by L. & P. Parris, Llandrindod Wells, nd
[1980s].
Pietro Riccardi. Biblioteca Matematica Italiana dalla Origine della Stampa ai Primi Anni del
Secolo XIX. G. G. Görlich, Milan, 1952, 2 vols. This work appeared in several parts
and supplements in the late 19C and early 20C, mostly published by the Società
Tipografica Modense, Modena, 1878-1893. Because it appeared in parts, the contents
of early copies are variable and even the reprints may vary. The contents of this set are
as follows.
I.
20pp prelims + Col. 1 - 656 (Abaco - Kirchoffer). [= original Vol. I.]
Col. 1 - 676 (La Cometa - Zuzzeri) + 2pp correzioni. [= original Vol. II.]
II. 4pp titles and reverses. Correzioni ed Aggiunte. [= original Appendice.]
Serie I.a Col. 1 - 78 + 1½pp Continuazione delle Correzioni (note that these
have Pag. when they mean Col.).
Serie II.a. Col. 81 - 156.
Serie III.a. Col. 157 - 192 + Aggiunte al Catalogo delle Opere di sovente citate,
col. 193-194 + 1p Continuazione delle Correzioni (note that these have
Pag. when they mean Col.).
Serie IV.a. Col. 197 - 208 + Seconda Aggiunta al Catalogo delle Opere più di
sovente citate, col. 209 - 212 + Continuazione delle Correzioni in
col. 211-212.
Serie V.a. Col. 1 - 180.
Serie VI.a. Col. 179 - 200.
Serie V & VI must have been published as one volume as Serie V ends
halfway down a page and then Serie VI begins on the same page.
Serie VII.a. 2pp introductory note by Ettore Bortolotti in 1928 saying that this
material was left as a manuscript by Riccardi and never previously
published + Col. 1 - 106.
Indice Alfabetico, of authors, covering the original material and all seven Series
of Correzioni ed Aggiunte, in 34 unnumbered columns.
Parte Seconda. Classificazione per materie delle opere nella Parte I. 18pp
(including a chronological table) + subject index, pp. 1 - 294.
Catalogo Delle opere più di sovente citate, col. 1 - 54.
[I have seen an early version which had the following parts: Vol. I, 1893, col.
1-656; Vol. II, 1873, col. 1-676; Appendice, 1878-1880-1893, col. 1-228. Appendice,
nd, col. 1-212. Serie V, col. 1-228. Parte 2, Vol. 1, 1880, pp. 1-294. Renner Katalog
87 describes it as 5 in 2 vols.]
A. W. Riley. School Library Mathematics List -- Supplement No. 1. MA, 1973.
SEE ALSO: Rollett.
Tom Rodgers. Catalog of his collection of books on recreational mathematics, etc. The
author, Atlanta, May 1991, 40pp.
Leo F. Rogers. Finding Out in the History of Mathematics. Produced by the author, London,
c1985, 52pp.
A. P. Rollett. School Library Mathematics List. Bell, London, for MA, 1966.
SEE ALSO: Riley.
Charles L. Rulfs. Origins of some conjuring works. Magicol 24 (May 1971) 3-5.
José A. Sánchez Pérez. Las Matematicas en la Biblioteca del Escorial. Imprenta de
Estanislao Maestre, Madrid, 1929.
William L. Schaaf.
List of works on recreational mathematics. SM 10 (1944) 193-200.
PLUS: A. Gloden; Additions to Schaaf's "List of works on mathematical
recreations"; SM 13 (1947) 127.
A Bibliography of Recreational Mathematics. Op. cit. in Common References, 4 vols.,
1955-1978. In these volumes he gives several lists of relevant books.
Books for the periods 1900-1925 and 1925-c1956 are given as Sections 1.1
(pp. 2-3) and 1.2 (pp. 4-12) in Vol. 1.
Chapter 9, pp. 144-148, of Vol. 1, is a Supplement, generally covering
c1954-c1962, but with some older items.
In Vol. 2, 1970, the Appendix, pp. 181-191, extends to c1969, including
SOURCES - page 13
some older items and repeating a few from the Supplement of Vol. 1.
Appendix A of Vol. 3, 1973, pp. 111-113, adds some more items up
through 1972.
Appendix A, pp. 134-137, of Vol. 4, 1978, extends up through 1977.
The following VESTPOCKET BIBLIOGRAPHIES are extensions of the material
in his Bibliographies.
No. 1:
Pythagoras and rational triangles; Geoboards and lattices. JRM 16:2
(1983-84) 81-88.
No. 2:
Combinatorics; Gambling and sports. JRM 16:3 (1983-84) 170-181.
No. 3:
Tessellations and polyominoes; Art and music. JRM 16:4 (1983-84)
268-280.
No. 4:
Recreational miscellany. JRM 17:1 (1984-85) 22-31.
No. 5:
Polyhedra; Topology; Map coloring. JRM 17:2 (1984-85) 95-105.
No. 6:
Sundry algebraic notes. JRM 17:3 (1984-85) 195-203.
No. 7:
Sundry geometric notes. JRM 18:1 (1985-86) 36-44.
No. 8:
Probability; Gambling. JRM 18:2 (1985-86) 101-109.
No. 9:
Games and puzzles. JRM 18:3 (1985-86) 161-167.
No. 10:
Recreational mathematics; Logical puzzles; Expository mathematics.
JRM 18:4 (1985-86) 241-246.
No. 11:
Logic, Artificial intelligence, and Mathematical foundations. JRM 19:1
(1987) 3-9.
No. 12:
Magic squares and cubes; Latin squares; Mystic arrays and Number
patterns. JRM 19:2 (1987) 81-86.
The High School Mathematics Library. NCTM, (1960, 1963, 1967, 1970, 1973); 6th
ed., 1976; 7th ed., 1982; 8th ed., 1987.
SEE ALSO: Wheeler; Wheeler & Hardgrove.
Early Books on Magic Squares. JRM 16:1 (1983-84) 1-6.
William L. Schaaf & David Singmaster. Books on Recreational Mathematics. A
Supplement to the Lists in William L. Schaaf's A Bibliography of Recreational
Mathematics. Collected by William L. Schaaf; typed and annotated by David
Singmaster. School of Computing, Information Systems and Mathematics, South Bank
University, London, SE1 0AA. 18pp, Dec 1992 and revised several times afterwards.
Peter Schreiber.
Mathematik und belletristik [1.] & 2. Teil. Mitteilungen der Mathematischen
Gesellschaft der Deutschen Demokratischer Republik. (1986), no. 4, 57-71 &
(1988), no. 1-2, 55-61. Good on German works relating mathematics and arts.
Mathematiker als Memoirenschreiber. Alpha (Berlin) (1991), no. 4, no page numbers
on copy received from author. Extends previous work.
S. N. Sen. Scientific works in Sanskrit, translated into foreign languages and vice-versa in the
18th and 19th century A.D. Indian J. History of Science 7 (1972) 44-70.
Will Shortz. Puzzleana [catalogue of his puzzle books]. Produced by the author. 14 editions
have appeared. The latest is: May 1992, 88pp with 1175 entries in 26 categories, with
indexes of authors and anonymous titles. Some entries cover multiple items. In Jan
1995, he produced a 19pp Supplement extending to a total of 1451 entries.
David Singmaster.
The Bibliography of Some Recreational Mathematics Books. School of Computing,
Information Systems
and Mathematics, South Bank Univ.
13 Nov 1994, 39pp. Technical Report SBU-CISM-94-09.
2nd ed., Aug 1995, 41pp. Technical Report SBU-CISM-95-08.
3rd ed., Jun 1996, 42pp. Technical Report SBU-CISM-96-12.
4th ed., Jun 1998, 44pp. Technical Report SBU-CISM-98-02.
(Current version is 61pp.)
Books on Recreational Mathematics. School of Computing, Information Systems and
Mathematics, South Bank Univ., until 1996.
21 Jan 1991. Approx. 2951 items on 120pp, ringbound.
30 Jan 1992. Approx. 3314 items on 138pp, ringbound.
10 Jan 1993. Approx. 3606 items on 95pp, ringbound.
10 Dec 1994. Approx. 4303 items plus 67 Old Books on 110pp. Technical
SOURCES - page 14
Report SBU-CISM-94-11.
10 Oct 1996. Approx. 4842 items plus 84 Old Books on 127pp. Technical
Report SBU-CISM-96-17.
24 May 1999. Approx. 6015 items plus 133 Old Books on 166pp. Technical
Report SBU-CISM-99-14.
26 Feb 2002. Approx. 7185 items plus 192 Old Books plus Supplement of
Calculating Devices, on 220pp. thermal bound.
22 Nov 2003. Approx. 7811 items plus 202 Old Books plus Supplement of
Calculating Devices, on 244pp. thermal bound.
Index to Martin Gardner's Columns and Cross Reference to His Books. (Oct 1993.)
Slightly revised as: Technical Report SBU-CISM-95-09; School of Computing,
Information Systems, and Mathematics; South Bank University, London, Aug
1995, 22pp. (Current version is 23pp and Don Knuth has sent 9pp of additional
material and I will combine these at some time.)
Harold Adrian Smith. Dick and Fitzgerald Publishers. Books at Brown 34 (1987) 108-114.
Sotheby's [Sotheby Parke Bernet].
Catalogue of the J. B. Findlay Collection Books and Periodicals on Conjuring and the
Allied Arts. Part I: A-O 5-6 Jul 1979. Part II: P-Z plus: Mimeographed Books
and Instructions; Flick Books Catalogues of Apparatus and Tricks Autograph
Letters, Manuscripts, and Typescripts 4-5 Oct 1979. Part III: Posters and
Playbills 3-4 Jul 1980. Each with estimates and results lists.
The Celebrated Library of Harrison D. Horblit Esq. Early Science Navigation &
Travel Including Americana with a few medical books. Part I A - C 10/11 Jun
1974. Part II D - G 11 Nov 1974. HB. The sale was then cancelled and the
library was sold to E. P. Kraus, qv, who issued three further catalogues, c1976.
The Honeyman Collection of Scientific Books and Manuscripts. Seven volumes, each
with estimates and results booklets.
Part I: Printed Books A-B, 30-31 Oct 1978.
Part II: Printed Books C-E, 30 Apr - 1 May 1979.
Part III: Manuscripts and Autograph Letters of the 12th to the 20th Centuries.
Part IV: Printed Books F-J, 5-6 Nov 1979.
Part V: Printed Books K-M, 12-13 May 1980.
Part VI: Printed Books N-Sa, 10-11 Nov 1980.
Part VII: Printed Books Sc-Z and Addenda, 19-20 May 1981.
Lynn A. Steen, ed.
Library Recommendations for Undergraduate Mathematics. MAA Reports No. 4,
1992.
Two-Year College Mathematics Library Recommendations. MAA Reports No. 5,
1992.
Strens/Guy Collection. Author/Title Listing. Univ. of Calgary. Preliminary Catalogue,
319 pp., July 1986. [The original has a lot of blank space. I have a computer version
which is reduced to 67pp.]
Eva Germaine Rimington Taylor. The Mathematical Practitioners of Tudor & Stuart England
1485-1714. CUP for the Institute of Navigation, 1970.
Eva Germaine Rimington Taylor. The Mathematical Practitioners of Hanoverian England
1714-1840. CUP for the Institute of Navigation, 1966.
PLUS: Kate Bostock, Susan Hurt & Michael Hart; An Index to the Mathematical
Practitioners of Hanoverian England 1714-1840; Harriet Wynter Ltd., London, 1980.
W. R. Utz. Letter: Mathematics in literature. MM 55 (1982) 249-250. Utz has sent his 3pp
original more detailed version along with 4pp of further citations. This extends
Koehler's article.
George Walker. The Art of Chess-Play: A New Treatise on the Game of Chess. 4th ed.,
Sherwood, Gilbert & Piper, London, 1846. Appendix: Bibliographical Catalogue of
the chief printed books, writers, and miscellaneous articles on chess, up to the present
time, pp. 339-375.
Frank R. [Joe] Watson, ed. Booklists. MA.
Puzzles, Problems, Games and Mathematical Recreations. 16pp, 1980.
Selections from the Recommended Books. 18pp, 1980.
Full List of Recommended Books. 105pp, 1984.
Margariete Montague Wheeler. Mathematics Library -- Elementary and Junior High School.
5th ed., NCTM, 1986.
SOURCES - page 15
SEE ALSO: Schaaf; Wheeler & Hardgrove.
Margariete Montague Wheeler & Clarence Ethel Hardgrove. Mathematics Library -Elementary and Junior High School. NCTM, (1960; 1968; 1973); 4th ed., 1978.
SEE ALSO: Schaaf; Wheeler.
Ernst Wölffing. Mathematischer Bücherschatz. Systematisches Verzeichnis der wichtigsten
deutschen und ausländischen Lehrbücher und Monographien des 19. Jahrhunderts auf
dem Gebiete der mathematischen Wissenschaften. I: Reine Mathematik; (II:
Angewandte Mathematik never appeared). AGM 16, part I (1903).
SOURCES - page 16
4.
MATHEMATICAL GAMES
Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games and Some Related
Material. Op. cit. in 3.B.
4.A. GENERAL THEORY AND NIM-LIKE GAMES
Conway's extension of this theory is well described in Winning Ways and later work is
listed in Fraenkel's Bibliography -- see section 3.B & 4 -- so I will not cover such material
here.
4.A.1.ONE PILE GAME
See MUS I 145-147.
(a, b) denotes the game where one can take 1, 2, ..., or a away from one pile, starting
with b in the pile, with the last player winning. The version (10, 100) is sometimes called
Piquet des Cavaliers or Piquet à Cheval, a name which initially perplexed me. Piquet is one
of the older card games, being well known to Rabelais (1534) and was known in the 16C as
Cent (or Saunt or Saint) because of its goal of 100 points. See: David Parlett; (Oxford Guide
to Card Games, 1990 =) A History of Card Games; OUP, 1991, pp. 24 & 175-181. The
connection with horses undoubtedly indicates that (10, 100) was viewed as a game which
could be played without cards, while riding -- see Les Amusemens, Decremps.
INDEX
Dudeney, Stong
Mittenzwey, Hoffmann, Mr. X, Dudeney, Blyth,
Fourrey,
Blyth, Hummerston,
Mittenzwey,
Pacioli, Leske, Mittenzwey, Ducret,
Baker,
Ball-FitzPatrick,
Rational Recreations
Hummerston,
Mittenzwey,
Sprague,
Mittenzwey,
Decremps,
Fourrey,
Bachet, Carroll,
Bachet, Ozanam, Alberti
Bachet, Henrion, Ozanam, Alberti, Les Amusemens, Hooper, Decremps,
Badcock, Jackson, Rational Recreations, Manuel des Sorciers,
Boy's Own Book, Nuts to Crack, Young Man's Book, Carroll,
Magician's Own Book, Book of 500 Puzzles, Secret Out,
Boy's Own Conjuring Book, Vinot, Riecke, Fourrey, Ducret, Devant,
(10, 120)
Bachet,
(12, 134)
Decremps,
General case: Bachet, Ozanam, Alberti, Decremps, Boy's Own Book,
Young Man's Book, Vinot, Mittenzwey, (others ?? check)
Versions with limited numbers of each value or using a die -- see 4.A.1.a.
Version where an odd number in total has to be taken: Dudeney, Grossman & Kramer,
Sprague.
Versions with last player losing: Mittenzwey,
( 3, 13)
( 3, 15)
( 3, 17)
( 3, 21)
( 4, 15)
( 6, 30)
( 6, 31)
( 6, 50)
( 6, 52)
( 6, 57)
( 7, 40)
( 7, 41)
( 7, 45)
( 7, 50)
( 7, 60)
( 8, 100)
( 9, 100)
(10, 100)
Pacioli. De Viribus. c1500. Ff. 73v - 76v. XXXIIII effecto afinire qualunch' numero na'ze al
compagno anon prendere piu de un termi(n)ato .n. (34th effect to finish whatever
number is before the company, not taking more than a limiting number) = Peirani
109-112. Phrases it as an addition problem. Considers (6, 30) and the general
problem.
SOURCES - page 17
David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A
History of Card Games. Penguin, 1991, pp. 174-175. "Early references to 'les luettes',
said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503,
and by Gargantua in 1534, seem to suggest a game of the Nim family (removing
numbers of objects from rows and columns)."
Cardan. Practica Arithmetice. 1539. Chap. 61, section 18, ff. T.iiii.v - T.v.r (p. 113). "Ludi
mentales". One has 1, 3, 6 and the other has 2, 4, 5; or one has 1, 3, 5, 8, 9 and the
other has 2, 4, 6, 7, 10; one one wants to make 100. "Sunt magnæ inventionis, & ego
inveni æquitando & sine aliquo auxilio cum socio potes ludere & memorium exercere
...."
Baker. Well Spring of Sciences. 1562? Prob. 5: To play at 31 with Numbers, 1670:
pp. 353-354. ??NX. (6, 31).
Bachet. Problemes. 1612. Prob. XIX: 1612, 99-103. Prob. XXII, 1624: 170-173;
1884: 115-117. Phrases it as an addition problem. First considers (10, 100), then
(10, 120), (8, 100), (9, 100), and the general case. Labosne omits the demonstration.
Dennis Henrion. Nottes to van Etten. 1630. Pp. 19-20. (10, 100) as an addition problem,
citing Bachet.
Ozanam. 1694. Prob. 21, 1696: 71-72; 1708: 63-64. Prob. 25, 1725: 182-184. Prob. 14,
1778: 162-164; 1803: 163-164; 1814: 143-145. Prob. 13, 1840: 73-74. Phrases it as
an addition problem. Considers (10, 100) and (9, 100) and remarks on the general
case.
Alberti. 1747. Due persone essendo convenuto ..., pp. 105-108 (66-67). This is a slight
recasting of Ozanam.
Les Amusemens. 1749. Prob. 10, p. 130: Le Piquet des Cavaliers. (10, 100) in additive
form. "Deux amis voyagent à cheval, l'un propose à l'autre un cent de Piquet sans
carte."
William Hooper. Rational Recreations, In which the Principles of Numbers and Natural
Philosophy Are clearly and copiously elucidated, by a series of Easy, Entertaining,
Interesting Experiments. Among which are All those commonly performed with the
cards. [Taken from my 2nd ed.] 4 vols., L. Davis et al., London, 1774; 2nd ed.,
corrected, L. Davis et al., London, 1783-1782 (vol. 1 says 1783, the others say 1782;
BMC gives 1783-82); 3rd ed., corrected, 1787; 4th ed., corrected, B. Law et al.,
London, 1794. [Hall, BCB 180-184 & Toole Stott 389-392. Hall says the first four
eds. have identical pagination. I have not seen any difference in the first four editions,
except as noted in Section 6.P.2. Hall, OCB, p. 155. Heyl 177 notes the different
datings of the 2nd ed, Hall, BCB 184 and Toole Stott 393 is a 2 vol. 4th ed., corrected,
London, 1802. Toole Stott 394 is a 2 vol. ed. from Perth, 1801. I have a note that there
was an 1816 ed, but I have no details. Since all relevant material seems the same in all
volumes, I will cite this as 1774.] Vol. 1, recreation VIII: The magical century.
(10, 100) in additive form. Mentions other versions and the general rule.
I don't see any connection between this and Rational Recreations, 1824.
Henri Decremps. Codicile de Jérôme Sharp, Professeur de Physique amusante; Où l'on
trouve parmi plusieurs Tours dont il n'est point parlé dans son Testament, diverses
récréations relatives aux Sciences & Beaux-Arts; Pour servir de troisième suite À La
Magie Blanche Dévoilée. Lesclapart, Paris, 1788. Chap. XXVII, pp. 177-184:
Principes mathématiques sur le piquet à cheval, ou l'art de gagner son diner en se
promenant. Does (10, 100) in additive form, then discusses the general method,
illustrating with (7, 50) and (12, 134).
Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 33-34, no. 48: A
curious recreation with a hundred numbers, usually called the magical century.
(10, 100) as an additive problem where each person starts with 50 counters.
Discusses general case, but doesn't notice that the limitation to 50 counters each
considerably changes the game!
Jackson. Rational Amusement. 1821. Arithmetical Puzzles, no. 47, pp. 11 & 64. Additive
form of (10, 100).
Rational Recreations. 1824. Exercise 12(?), pp. 57-58. As in Badcock. Then says it can be
generalised and gives (6, 52).
Manuel des Sorciers. 1825. Pp. 57-58, art. 30: Le piquet sans cartes. ??NX (10, 100) done
subtractively.
The Boy's Own Book.
The certain game. 1828: 177; 1828-2: 236; 1829 (US): 104; 1855: 386-387;
SOURCES - page 18
1868: 427.
The magical century. 1828: 180; 1828-2: 236-237; 1829 (US): 104-105;
1855: 391-392.
Both are additive phrasings of (10, 100). The latter mentions using other
numbers and how to win then.
Nuts to Crack V (1836), no. 70. An arithmetical problem. (10, 100).
Young Man's Book. 1839. Pp. 294-295. A curious Recreation with a Hundred Numbers,
usually called the Magical Century. Almost identical to Boy's Own Book.
Lewis Carroll.
Diary entry for 5 Feb 1856. In Carroll-Gardner, pp. 42-43. (10, 100). Wakeling's note
in the Diaries indicates he is not familiar with this game.
Diary entry for 24 Oct 1872. Says he has written out the rules for Arithmetical Croquet,
a game he recently invented. Roger Lancelyn Green's abridged version of the
Diaries, 1954, prints a MS version dated 22 Apr 1889. Carroll-Wakeling, prob.
38, pp. 52-53 and Carroll-Gardner, pp. 39 & 42 reprint this, but Gardner has a
misprinted date of 1899. Basically (8, 100), but passing the values 10, 20, ...,
requires special moves and one may have to go backward. Also, when a move is
made, some moves are then barred for the next player. Overall, the rules are
typically Carrollian-baroque.
Magician's Own Book. 1857.
The certain game, p. 243. As in Boy's Own Book.
The magical century, pp. 244-245. As in Boy's Own Book.
Book of 500 Puzzles. 1859.
The certain game, p. 57. As in Boy's Own Book.
The magical century, pp. 58-59. As in Boy's Own Book.
The Secret Out. 1859. Piquet on horseback, pp. 397-398 (UK: 130-131) -- additive (10, 100)
unclearly explained.
Boy's Own Conjuring Book. 1860.
The certain game, pp. 213-214. As in Boy's Own Book.
Magical century, pp. 215. As in Boy's Own Book.
Vinot. 1860. Art. XI: Un cent de piquet sans cartes, pp. 19-20. (10. 100). Says the idea can
be generalised, giving (7, 52) as an example.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 563-III, pp. 247: Wer von 30
Rechenpfennigen den letzen wegnimmt, hat gewonnen. (6, 30).
F. J. P. Riecke. Mathematische Unterhaltungen. 3 vols., Karl Aue, Stuttgart, 1867, 1868 &
1873; reprint in one vol., Sändig, Wiesbaden, 1973. Vol. 3, art 22.2, p. 44. Additive
form of (10, 100).
Mittenzwey. 1880. Probs. 286-287, pp. 52 & 101-102; 1895?: 315-317, pp. 56 & 103-104;
1917: 315-317, pp. 51 & 98.
(6, 30), last player wins.
(4, 15), last player loses, the solution discusses other cases: (7, 40), (7, 45) and indicates
the general solution.
(added in 1895?) (3, 15), last player loses.
Hoffmann. 1893. Chap VII, no. 19: The fifteen matches puzzle, pp. 292 & 300-301
= Hoffmann-Hordern, p. 197. (3, 15). c= Benson, 1904, The fifteen match puzzle,
pp. 241-242.
Ball-FitzPatrick. 1st ed., 1898. Deuxième exemple, pp. 29-30. (6, 50).
E. Fourrey. Récréations Arithmétiques. (Nony, Paris, 1899; 2nd ed., 1901); 3rd ed., Vuibert
& Nony, Paris, 1904; (4th ed., 1907); 8th ed., Librairie Vuibert, Paris, 1947. [The 3rd
and 8th eds are identical except for the title page, so presumably are identical to the 1st
ed.] Sections 65-66: Le jeu du piquet à cheval, pp. 48-49. Additive forms of (10, 100)
and (7, 60). Then gives subtractive form for a pile of matches for (3, 17).
Étienne Ducret. Récréations Mathématiques. Garnier Frères, Paris, nd [not in BN, but a
similar book, nouv. ed., is 1892]. Pp. 102-104: Le piquet à cheval. Additive version of
(10, 100) with some explanation of the use of the term piquet. Discusses (6, 30).
Mr. X [possibly J. K. Benson -- see entry for Benson in Abbreviations]. His Pages. The
Royal Magazine 9:3 (Jan 1903) 298-299. A good game for two. (3, 15) as a subtraction
game.
David Devant. Tricks for Everyone. Clever Conjuring with Everyday Objects. C. Arthur
Pearson, London, 1910. A counting race, pp. 52-53. (10, 100).
Dudeney. AM. 1917. Prob. 392: The pebble game, pp. 117 & 240. (3, 15) & (3, 13) with
SOURCES - page 19
the object being to take an odd number in total. For 15, first player wins; for 13,
second player wins. (Barnard (50 Telegraph ..., 1985) gives the case (3, 13).)
Blyth. Match-Stick Magic. 1921.
Fifteen matchstick game, pp. 87-88. (3, 15).
Majority matchstick game, p. 88. (3, 21).
Hummerston. Fun, Mirth & Mystery. 1924.
Two second-sight tricks (no. 2), p. 84. (6, 57), last player losing.
A match mystery, p. 99. (3, 21), last player losing.
H. D. Grossman & David Kramer. A new match-game. AMM 52 (1945) 441-443. Cites
Dudeney and says Games Digest (April 1938) also gave a version, but without solution.
Gives a general solution whether one wants to take an odd total or an even total.
C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. How to design a
"Pircuit" or Puzzle circuit, pp. 388-394. On pp. 388-391, Harry Rudloe describes a
relay circuit for playing the subtractive form of (3, 13), which he calls the "battle of
numbers" game.
Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by
T. H. O'Beirne as: Recreations in Mathematics, Blackie, London, 1963. Problem 24:
"Ungerade" gewinnt, pp. 16 & 44-45. (= 'Odd' is the winner, pp. 18 & 53-55.) (7, 41)
with the winner being the one who takes an odd number in total. Solves (7, b) and
states the structure for (a, b).
I also have some other recent references to this problem. Lewis (1983) gives a
general solution which seems to be wrong.
4.A.1.a.
THE 31 GAME
Numerical variations: Badcock, Gibson, McKay.
Die versions: Secret Out (UK), Loyd, Mott-Smith, Murphy.
Baker. Well Spring of Sciences. 1562? Prob. 5: To play at 31 with Numbers, 1670:
pp. 353-354. ??NX. (6, 31). ??CHECK if this has the limited use of numbers.
John Fisher. Never Give a Sucker an Even Break. (1976); Sphere Books, London, 1978.
Thirty-one, pp. 102-104. (6, 31) additively, but played with just 4 of each value, the 24
cards of ranks 1 -- 6, and the first to exceed 31 loses. He says it is played extensively
in Australia and often referred to as "The Australian Gambling Game of 31". Cites the
19C gambling expert Jonathan Harrington Green who says it was invented by Charles
James Fox (1749-1806). Gives some analysis.
Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 33-34, no. 48: A
curious recreation with a hundred numbers, usually called the magical century.
(10, 100) as an additive problem where each person starts with 50 counters.
Discusses general case, but doesn't notice that the limitation to 50 counters each
considerably changes the game!
Nuts to Crack V (1836), no. 71. (6, 31) additively, with four of each value. "Set down on a
slate, four rows of figures, thus:-- ... You agree to rub out one figure alternately, to see
who shall first make the number thirty-one."
Magician's Own Book. 1857. Art. 31: The trick of thirty-one, pp. 70-71. (6, 31) additively,
but played with just 4 of each value -- e.g. the 24 cards of ranks 1 -- 6. The author
advises you not to play it for money with "sporting men" and says it it due to Mr. Fox.
Cf Fisher. = Boy's Own Conjuring Book; 1860; Art. 29: The trick of thirty-one,
pp. 78-79. = The Secret Out; 1859, pp. 65-66, which adds a footnote that the trick is
taken from the book One Hundred Gambler Tricks with Cards by J. H. Green, reformed
gambler, published by Dick & Fitzgerald.
The Secret Out (UK), c1860. To throw thirty-one with a die before your antagonist, p. 7.
This is incomprehensible, but is probably the version discussed by Mott-Smith.
Edward S. Sackett. US Patent 275,526 -- Game. Filed: 9 Dec 1882; patented: 10 Apr 1883.
1p + 1p diagrams. Frame of six rows holding four blocks which can be slid from one
side to the other to play the 31 game, though other numbers of rows, blocks and goal
may be used. Gives an example of a play, but doesn't go into the strategy at all.
Larry Freeman. Yesterday's Games. Taken from "an 1880 text" of games. (American edition
by H. Chadwick.) Century House, Watkins Glen, NY, 1970. P. 107: Thirty-one.
(6, 31) with 4 of each value -- as in Magician's Own Book.
Algernon Bray. Letter: "31" game. Knowledge 3 (4 May 1883) 268, item 806. "... has lately
SOURCES - page 20
made its appearance in New York, ...." Seems to have no idea as how to win.
Loyd. Problem 38: The twenty-five up puzzle. Tit-Bits 32 (12 Jun & 3 Jul 1897) 193 &
258. = Cyclopedia. 1914. The dice game, pp. 243 & 372. = SLAHP: How games
originate, pp. 73 & 114. The first play is arbitrary. The second play is by throwing a
die. Further values are obtained by rolling the die by a quarter turn.
Ball-FitzPatrick. 1st ed., 1898. Généralization récente de cette question, pp. 30-31. (6, 50)
with each number usable at most 3 times. Some analysis.
Ball. MRE, 4th ed., 1905, p. 20. Some analysis of (6, 50) where each player can play a
value at most 3 times -- as in Ball-FitzPatrick, but with the additional sentence: "I have
never seen this extension described in print ...." He also mentions playing with values
limited to two times. In the 5th ed., 1911, pp. 19-21, he elaborates his analysis.
Dudeney. CP. 1907. Prob. 79: The thirty-one game, pp. 125-127 & 224. Says it used to be
popular with card-sharpers at racecourses, etc. States the first player can win if he starts
with 1, 2 or 5, but the analysis of cases 1 and 2 is complicated. This occurs as No. 459:
The thirty-one puzzle, Weekly Dispatch (17 Aug 1902) 13 & (31 Aug 1902) 13, but he
leaves the case of opening move 2 to the reader, but I don't see the answer given in the
next few columns.
Devant. Tricks for Everyone. Op. cit. in 4.A.1. 1910. The thirty-one trick, pp. 53-54. Says
to get to 3, 10, 17, 24.
Hummerston. Fun, Mirth & Mystery. 1924. Thirty-one -- a game of skill, pp. 95-96. This
uses a layout of four copies of the numbers 1, 2, 3, 4, 5, 6 with one copy of 20 in a
5 x 5 square with the 20 in the centre. Says to get to 3, 10, 17, 24, but that this will
lose to an experienced player.
Loyd Jr. SLAHP. 1928. The "31 Puzzle Game", pp. 3 & 87. Loyd Jr says that as a boy, he
often had to play it against all comers with a $50 prize to anyone who could beat 'Loyd's
boy'. This is the game that Loyd Sr called 'Blind Luck', but I haven't found it in the
Cyclopedia. States the first player wins with 1, 2 or 5, but only sketches the case for
opening with 5. I have seen an example of Blind Luck -- it has four each of the
numbers 1 - 6 arranged around a frame containing a horseshoe with 13 in it.
McKay. Party Night. 1940. The 21 race, pp. 166. Using the numbers 1, 2, 3, 4, at most
four times, achieve 21. Says to get 1, 6, 11, 16. He doesn't realise that the sucker can
be mislead into playing first with a 1 and losing! Says that with 1, ..., 5 at most four
times, one wants to achieve 26 and that with 1, ..., 6 at most four times, one wants to
achieve 31. Gives just the key numbers each time.
Geoffrey Mott-Smith. Mathematical Puzzles for Beginners and Enthusiasts. (Blakiston,
1946); revised 2nd ed., Dover, 1954.
Prob. 179: The thirty-one game, pp. 117-119 & 231-232. As in Dudeney.
Prob. 180: Thirty-one with dice, p. 119 & 232-233. Throw a die, then make quarter
turns to produce a total of 31. Analysis based on digital roots (i.e. remainders
(mod 9)). First player wins if the die comes up 4, otherwise the second player
can win. He doesn't treat any other totals.
"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and Sciences, Croydon,
1947. "Trente et un", pp. 56-57. Says he doesn't know any name for this. Get 31 using
4 each of the cards A, 2, ..., 6. Says first player loses easily if he starts with 4, 5, 6
(not true according to Dudeney) and that gamblers dupe the sucker by starting with 3
and winning enough that the sucker thinks he can win by starting with 3. But if he
starts with a 1 or 2, then the second player must play low and hope for a break.
Walter B. Gibson. Fell's Guide to Papercraft Tricks, Games and Puzzles. Frederick Fell, NY,
1963. Pp. 54-55: First to fifty. First describes (50, 6), but then adds a version with
slips of paper: eight marked 1 and seven marked with 2, 3, 4, 5, 6 and you secretly
extract a 6 slip when the other player starts.
Harold Newman. The 31 Game. JRM 23:3 (1991) 205-209. Extended analysis. Confirms
Dudeney. Only cites Dudeney & Mott-Smith.
Bernard Murphy. The rotating die game. Plus 27 (Summer 1994) 14-16. Analyses the die
version as described by Mott-Smith and finds the set, S(n), of winning moves for
achieving a count of n by the first player, is periodic with period 9 from n = 8, i.e.
S(n+9) = S(n) for n  8. There is no first player winning move if and only if n is a
multiple of 9. [I have confirmed this independently.]
Ken de Courcy. The Australian Gambling Game of 31. Supreme Magic Publication,
Bideford, Devon, nd [1980s?]. Brief description of the game and some indications of
how to win. He then plays the game with face-down cards! However, he insures that
SOURCES - page 21
the cards by him are one of of each rank and he knows where they are.
4.A.2.
SYMMETRY ARGUMENTS
Loyd?? Problem 43: The daisy game. Tit-Bits 32 (17 Jul & 7 Aug 1897) 291 & 349.
(= Cyclopedia. 1914. A daisy puzzle game, pp. 85 & 350. c= MPSL2, prob. 57, pp.
40-41 & 140. c= SLAHP: The daisy game, pp. 42 & 99.) Circular version of Kayles
with 13 objects. Solution uses a symmetry argument -- but the Tit-Bits solution was
written by Dudeney.
Dudeney. Problem 500: The cigar puzzle. Weekly Dispatch (7 Jun, 21 Jun, 5 Jul, 1903) all
p. 16. (= AM, prob. 398, pp. 119, 242.) Symmetry in placement game, using cigars on
a table.
Loyd. Cyclopedia. 1914. The great Columbus problem, pp. 169 & 361. (= MPSL1, prob.
65, pp. 62 & 144. = SLAHP: When men laid eggs, pp. 75 & 115.) Placing eggs on a
table.
Maurice Kraitchik. La Mathématique des Jeux. Stevens, Bruxelles, 1930. Section XII,
prob. 1, p. 296. (= Mathematical Recreations; Allen & Unwin, London, 1943;
Problem 1, pp. 13-14.) Child plays black and white against two chess players and
guarantees to win one game. [MJ cites L'Echiquier (1925) 84, 151.]
CAUTION. The 2nd edition of Math. des Jeux, 1953, is a translation of
Mathematical Recreations and hence omits much of the earlier edition.
Leopold. At Ease! 1943. Chess wizardry in two minutes, pp. 105-106. Same as Kraitchik.
4.A.3.KAYLES
This has objects in a line or a circle and one can remove one object or two adjacent
objects (or more adjacent objects in a generalized version of the game). This derives from
earlier games with an array of pins at which one throws a ball or stick.
Murray 442 cites Act 17 of Edward IV, c.3 (1477): "Diversez novelx ymagines jeuez
appellez Cloishe Kayles ..." This outlawed such games. A 14C picture is given in [J. A. R.
Pimlott; Recreations; Studio Vista, 1968, plate 9, from BM Royal MS 10 E IV f.99] showing
a 3 x 3 array of pins. A version is shown in Pieter Bruegel's painting "Children's Games" of
1560 with balls being thrown at a row of pins by a wall, in the back right of the scene.
Versions of the game are given in the works of Strutt and Gomme cited in 4.B.1. Gomme II
115-116 discusses it under Roly-poly, citing Strutt and some other sources. Strutt 270-271 (=
Strutt-Cox 219-220) calls it "Kayles, written also cayles and keiles, derived from the French
word quilles". He has redrawings of two 14C engravings (neither that in Pimlott) showing
lines of pins at which one throws a stick (= plate opp. 220 in Strutt-Cox). He also says Closh
or Cloish seems to be the same game and cites prohibitions of it in c1478 et seq. Loggats was
analogous and was prohibited under Henry VIII and is mentioned in Hamlet.
14C MS in the British Museum, Royal Library, No. 2, B. vii. Reproduced in Strutt, p. 271.
Shows a monk(?) standing by a line of eight conical pins and another monk(?) throwing
a stick at the pins.
Anonymous. Games of the 16th Century. The Rockliff New Project Series. Devised by
Arthur B. Allen. The Spacious Days of Queen Elizabeth. Background Book No. 5.
Rockliff Publishing, London, ©1950, 4th ptg. The Background Books seem to be
consecutively paginated as this booklet is paginated 129-152. Pp. 133-134 describes
loggats, quoting Hamlet and an unknown poet of 1611. P. 137 is a photograph of the
above 14C illustration. The caption is "Skittles, or "Kayals", and Throwing a Whirling
Stick".
van Etten. 1624. Prob. 72 (misnumbered 58) (65), pp 68-69 (97-98): Du jeu des quilles (Of
the play at Keyles or Nine-Pins). Describes the game as a kind of ninepins.
Loyd. Problem 43: The daisy game. Tit-Bits 32 (17 Jul & 7 Aug 1897) 291 & 349.
(= Cyclopedia. 1914. A daisy puzzle game, pp. 85 & 350. c= MPSL2, prob. 57,
pp. 40-41 & 140. c= SLAHP: The daisy game, pp. 42 & 99.) Circular version of
Kayles with 13 objects. See also 4.A.2.
Dudeney. Sharpshooters puzzle. Problem 430. Weekly Dispatch (26 Jan, 9 Feb, 1902) both
p. 13. Simple version of Kayles.
Ball. MRE, 4th ed., 1905, pp. 19-20. Cites Loyd in Tit-Bits. Gives the general version:
place p counters in a circle and one can take not more than m adjacent ones.
SOURCES - page 22
Dudeney. CP. 1907. Prob. 73: The game of Kayles, pp. 118-119 & 220. Kayles with 13
objects.
Loyd. Cyclopedia. 1914. Rip van Winkle puzzle, pp. 232 & 369-370. (c= MPSL2, prob. 6,
pp. 5 & 122.) Linear version with 13 pins and the second knocked down. Gardner
asserts that Dudeney invented Kayles, but it seems to be an abstraction from the old
form of the game.
Rohrbough. Puzzle Craft, later version, 1940s?. Daisy Game, p. 22. Kayles with 13 petals of
a daisy.
Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965.
Prob. 45, pp. 48 & 95. Circular kayles with five objects.
Doubleday - 2. 1971. Take your pick, pp. 63-65. This is Kayles with a row of 10, but he says
the first player can only take one.
4.A.4.NIM
Nim is the game with a number of piles and a player can take any number from one of
the piles. Normally the last one to play wins.
David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A
History of Card Games. Penguin, 1991. Pp. 174-175. "Early references to 'les luettes',
said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503,
and by Gargantua in 1534, seem to suggest a game of the Nim family (removing
numbers of objects from rows and columns)."
Charles L. Bouton. Nim: a game with a complete mathematical theory. Annals of Math. (2) 3
(1901/02) 35-39. He says Nim is played at American colleges and "has been called
Fan-Tan, but as it is not the Chinese game of that name, the name in the title is
proposed for it." He says Paul E. More showed him the misère (= last player loses)
version in 1899, so it seems that Bouton did not actually invent the game himself.
Ahrens. "Nim", ein amerikanisches Spiel mit mathematischer Theorie.
Naturwissenschaftliche Wochenschrift 17:22 (2 Mar 1902) 258-260. He says that
Bouton has admitted that he had confused Nim and Fan-Tan. Fan-Tan is a Chinese
game where you bet on the number of counters (mod 4) in someone's hand. Parker,
Ancient Ceylon, op. cit. in 4.B.1, pp. 570-571, describes a similar game, based on odd
and even, as popular in Ceylon and "certainly one of the earliest of all games".
For more about Fan-Tan, see the following.
Stewart Culin. Chess and playing cards. Catalogue of games and implements for divination
exhibited by the United States National Museum in connection with the Department of
Archæology and Paleontology of the University of Pennsylvania at the Cotton States
and International Exposition, Atlanta, Georgia, 1895. IN: Report of the U. S. National
Museum, year ending June 30, 1896. Government Printing Office, Washington, 1898,
HB, pp. 665-942. [There is a reprint by Ayer Co., Salem, Mass., c1990.] Fan-Tan
(= Fán t‘án = repeatedly spreading out) is described on pp. 891 & 896, with discussion
of related games on pp. 889-902.
Alan S. C. Ross. Note 2334: The name of the game of Nim. MG 37 (No. 320) (May 1953)
119-120. Conjectures Bouton formed the word 'nim' from the German 'nimm'. Gives
some discussion of Fan-Tan and quotes MUS I 72.
J. L. Walsh. Letter: The name of the game of Nim. MG 37 (No. 322) (Dec 1953) 290.
Relates that Bouton said that he had chosen the word from the German 'nimm' and
dropped one 'm'.
W. A. Wythoff. A modification of the game of Nim. Nieuw Archief voor Wiskunde
(Groningen) (2) 7 (1907) 199-202. He considers a Nim game with two piles allows the
extra move of taking the same amount from both piles. [Is there a version with more
piles where one can take any number from one pile or equal amounts from two piles??
See Barnard, below for a three pile version.]
Ahrens. MUS I. 1910. III.3.VII: Nim, pp. 72-88. Notes that Nim is not the same as Fan-Tan,
has been known in Germany for decades and is played in China. Gives a thorough
discussion of the theory of Nim and of an equivalent game and of Wythoff's game.
E. H. Moore. A generalization of the game called Nim. Annals of Math. (2) 11 (1910)
93-94. He considers a Nim game with n piles and one is allowed to take any number
from at most k piles.
Ball. MRE, 5th ed., 1911, p. 21. Sketches the game of Nim and its theory.
SOURCES - page 23
A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton
& Co., London, nd [1927 -- BMC]. No. 13: The last match, pp. 10-11. Thirty matches
divided at random into three heaps. Last player loses. Explanation of how to win is
rather cryptic: "you must try and take away ... sufficient ... to leave the matches in the
two or three heaps remaining, paired in ones, twos, fours, etc., in respect of each other."
Loyd Jr. SLAHP. 1928. A tricky game, pp. 47 & 102. Nim (3, 4, 8).
Emanuel Lasker. Brettspiele der Völker. 1931. See comments in 4.A.5. Jörg Bewersdorff
[email of 6 Jun 1999] says that Lasker considered a three person Nim and found an
equilibrium for it -- see: Jörg Bewersdorff; Glück, Logik und Bluff Mathematik im
Spiel -- Methoden, Ergebnisse und Grenzen; Vieweg, 1998, Section 2.3 Ein Spiel zu
dritt, pp. 110-115.
Lynn Rohrbough, ed. Fun in Small Spaces. Handy Series, Kit Q, Cooperative Recreation
Service, Delaware, Ohio, nd [c1935]. Take Last, p. 10. Last player loses Nim (3, 5, 7).
Rohrbough. Puzzle Craft. 1932.
Japanese Corn Game, p. 6 (= p. 6 of 1940s?). Last player loses Nim (1, 2, 3, 4, 5).
Japanese Corn Game, p. 23. Last player loses Nim (3, 5, 7).
René de Possel. Sur la Théorie Mathématique des Jeux de Hasard et de Réflexion. Actualités
Scientifiques et Industrielles 436. Hermann, Paris, 1936. Gives the theory of Nim and
also the misère version.
Depew. Cokesbury Game Book. 1939. Make him take it, pp. 187-188. Nim (3, 4, 5), last
player loses.
Edward U. Condon, Gereld L. Tawney & Willard A. Derr. US Patent 2,215,544 -- Machine to
Play Game of Nim. Filed: 26 Apr 1940; patented: 24 Sep 1940. 10pp + 11pp
diagrams.
E. U. Condon. The Nimatron. AMM 49 (1942) 330-332. Has photo of the machine.
Benedict Nixon & Len Johnson. Letters to the Notes & Queries Column. The Guardian
(4 Dec 1989) 27. Reprinted in: Notes & Queries, Vol. 1; Fourth Estate, London, 1990,
pp. 14-15. These describe the Ferranti Nimrod machine for playing Nim at the Festival
of Britain, 1951. Johnson says it played Nim (3, 5, 6) with a maximum move of 3.
The Catalogue of the Exhibition of Science shows this as taking place in the Science
Museum.
H. S. M. Coxeter. The golden section, phyllotaxis, and Wythoff's game. SM 19 (1953)
135-143. Sketches history and interconnections.
H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Chap. 11: The golden section and
phyllotaxis, pp. 160-172. Extends his 1953 material.
A. P. Domoryad. Mathematical Games and Pastimes. (Moscow, 1961). Translated by Halina
Moss. Pergamon, Oxford, 1963. Chap. 10: Games with piles of objects, pp. 61-70. On
p. 62, he asserts that Wythoff's game is 'the Chinese national game tsyanshidzi ("picking
stones")'. However M.-K. Siu cannot recognise such a Chinese game, unless it refers to
a form of jacks, which has no obvious connection with Wythoff's game or other Nim
games. He says there is a Chinese character, 'nian', which is pronounced 'nim' in
Cantonese and means to pick up or take things.
N. L. Haddock. Note 2973: A note on the game of Nim. MG 45 (No. 353) (Oct 1961)
245-246. Wonders if the game of Nim is related to Mancala games.
T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Section on misère version of Wythoff's
game, p. 133. Richard Guy (letter of 27 Feb 1985) says this is one of O'Beirne's few
mistakes -- cf next entry.
Winning Ways. 1982. P. 407 says Wythoff's game is also called Chinese Nim or Tsyan-shizi.
No reference given. See comment under Domoryad above. This says many authors
have done this incorrectly.
D. St. P. Barnard. 50 Daily Telegraph Brain-Twisters. Javelin Books, Poole, Dorset, 1985.
Prob. 30: All buttoned up, pp. 49-50, 91 & 115. He suggests three pile game where one
can take any number from one pile or an equal number from any two or all three piles.
[See my note to Wythoff, above.]
Matthias Mala. Schnelle Spiele. Hugendubel, Munich, 1988. San Shan, p. 66. This
describes a nim-like game named San Shan and says it was played in ancient China.
Jagannath V. Badami. Musings on Arithmetical Numbers Plus Delightful Magic Squares.
Published by the author, Bangalore, India, nd [Preface dated 9 Sep 1999]. Section 4.16:
The game of Nim, pp. 124-125. This is a rather confused description of one pile games
(21, 5) and (41, 5), but he refers to solving them by (mentally) dividing the pile into
piles. This makes me think of combining the two games, i.e. playing Nim with several
SOURCES - page 24
piles but with a limit on the number one can take in a move.
4.A.5.GENERAL THEORY
Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as
Add. MS 37202, c1820. ??NX. Ff. 134-144 are: Essay 10 Part 5. See 4.B.1 for more
details. At the top of f. 134.r, he has added a note: "This is probably my earliest Note
on Games of Skill. I do not recollect the date. 3 March 1865". He then describes Tit
Tat To and makes some simple analysis, but he never uses a name for it.
Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205.
??NX. See 4.B.1 for more details. On f. 304, he starts on analysis of games.
Ff. 310-383 are almost entirely devoted to Tit-Tat-To, with some general discussions.
F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in
general. F. 324-333, Oct 1844, studies "General laws for all games of Skill between
two players" and draws flow charts showing the basic recursive analysis of a game tree
(ff. 325.v & 325.r). On f. 332, he counts the number of positions in Tit Tat To as
9! + 8! + ... + 1! = 409,113. F. 333 has an idea of the tree structure of a game.
John M. Dubbey. The Mathematical Work of Charles Babbage. CUP, 1978, pp. 96-97 &
125-130. See 4.B.1 for more details. He discusses the above Babbage material. On
p. 127, Dubbey has: "The basic problem is one that appears not to have been previously
considered in the history of mathematics." Dubbey, on p. 129, says: "This analysis ...
must count as the first recorded stochastic process in the history of mathematics."
However, it is really a deterministic two-person game.
E. Zermelo. Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. Proc.
5th ICM (1912), CUP, 1913, vol. II, 501-504. Gives general idea of first and second
person games.
Ahrens. A&N. 1918. P. 154, note. Says that each particular Dots and Boxes board, with
rational play, has a definite outcome.
W. Rivier. Archives des Sciences Physiques et Naturelles (Nov/Dec 1921). ??NYS -- cited
by Rivier (1935) who says that the later article is a new and simpler version of this one.
H. Steinhaus. Difinicje potrzebne do teorji gry i pościgu (Definitions for a theory of games
and pursuit). Myśl Akademicka (Lwów) 1:1 (Dec 1925) 13-14 (in Polish). Translated,
with an introduction by Kuhn and a letter from Steinhaus in: Naval Research Logistics
Quarterly 7 (1960) 105-108.
Dénès König. Über eine Schlussweise aus dem Endlichen ins Unendliche. Mitteilungen der
Universitä Szeged 3 (1927) 121-130. ??NYS -- cited by Rivier (1935). Kalmár cites it
to the same Acta as his article.
László Kalmár. Zur Theorie der abstracten Spiele. Acta Litt. Sci. Regia Univ. Hungaricae
Francisco-Josephine (Szeged) 4 (1927) 62-85. Says there is a gap in Zermelo which has
been mended by König. Lengthy approach, but clearly gets the idea of first and second
person games.
Max Euwe. Proc. Koninklijke Akadamie van Wetenschappen te Amsterdam 32:5 (1929).
??NYS -- cited by Rivier (1935).
Emanuel Lasker. Brettspiele der Völker. Rätsel- und mathematische Spiele. A. Scherl,
Berlin, 1931, pp. 170-203. Studies the one pile game (100, 5) and the sum of two
one-pile games: (100, 5) + (50, 3). Discusses Nimm, "an old Chinese game according
to Ahrens" and says the solver is unknown. Gives Lasker's Nim -- one can take any
amount from a pile or split it in two -- and several other variants. Notes that 2nd
person + 2nd person is 2nd person while 2nd person + 1st person is 1st person.
Gives the idea of equivalent positions. Studies three (and more) person games,
assuming the pay-offs are all different. Studies some probabilistic games. Jörg
Bewersdorff [email of 6 Jun 1999] observes that Lasker's analysis of his Nim got very
close to the idea of the Sprague-Grundy number. See: Jörg Bewersdorff; Glück, Logik
und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen; Vieweg, 1998,
Section 2.5 Lasker-Nim: Gewinn auf verborgenem Weg, pp. 118-124.
W. Rivier. Une theorie mathématique des jeux de combinaisions. Comptes-Rendus du
Premier Congrès International de Récréation Mathématique, Bruxelles, 1935. Sphinx,
Bruxelles, 1935, pp. 106-113. A revised and simplified version of his 1921 article. He
cites and briefly discusses Zermelo, König and Euwe. He seems to be classifying
games as first player or second player.
René de Possel. Sur la Théorie Mathématique des Jeux de Hasard et de Réflexion. Actualités
SOURCES - page 25
Scientifiques et Industrielles 436. Hermann, Paris, 1936. Gives the theory of Nim and
also the misère version. Shows that any combinatorial game is a win, loss or draw and
describes the nature of first and second person positions. He then goes on to consider
games with chance and/or bluffing, based on von Neumann's 1927 paper.
R. Sprague. Über mathematische Kampfspiele. Tôhoku Math. J. 41 (1935/36) 438-444.
P. M. Grundy. Mathematics and games. Eureka 2 (1939) 6-8. Reprinted, ibid. 27 (1964)
9-11. These two papers develop the Sprague-Grundy Number of a game.
D. W. Davies. A theory of chess and noughts and crosses. Penguin Science News 16 (Jun
1950) 40-64. Sketches general ideas of tree structure, Sprague-Grundy number, rational
play, etc.
H. Steinhaus. Games, an informal talk. AMM 72 (1965) 457-468. Discusses Zermelo and
says he wasn't aware of Zermelo in 1925. Gives Mycielski's formulation and proof via
de Morgan's laws. Goes into pursuit and infinite games and their relation to the Axiom
of Choice.
H. Steinhaus. (Proof that a game without ties has a strategy.) In: M. Kac; Hugo Steinhaus -a reminiscence and a tribute; AMM 81 (1974) 572-581. Repeats idea of his 1965 talk.
4.B. PARTICULAR GAMES
See 5.M for Sim and 5.R.5 for Fox and Geese, etc.
Most of the board games described here are classic and have been extensively described
and illustrated in the various standard books on board games, particularly the works of
Robert C. Bell, especially his Board and Table Games from Many Civilizations; OUP, vol. I,
1960, vol. II, 1969; combined and revised ed., Dover, 1979 and the older work of Edward G.
Falkener; Games Ancient and Oriental and How to Play Them; Longmans, Green, 1892;
Dover, 1961. The works by Culin (see 4.A.4, 4.B.5 and 4.B.9) are often useful. Several
general works on games are cited in 4.B.1 and 4.B.5 -- I have read Murray's History of Board
Games Other than Chess, but not yet entered the material. Note that many of these works are
more concerned with the game than with its history and have a tendency to exaggerate the
ages of games by assuming, e.g. that a 3 x 3 board must have been used for Tic-Tac-Toe. I
will not try to duplicate the descriptions by Bell, Falkener and others, but will try to outline
the earliest history, especially when it is at variance with common belief. The most detailed
mathematical analyses are generally in Winning Ways.
4.B.1. TIC-TAC-TOE = NOUGHTS AND CROSSES
Popular belief is that the game is ancient and universal -- e.g. see Brandreth, 1976.
However the game appears to have evolved from earlier three-in-a-row games, e.g. Nine
Holes or Three Men's Morris, in the early 19C. See also the historical material in 4.B.5. The
game is not mentioned in Strutt nor most other 19C books on games, not even in Kate
Greenaway's Book of Games (1889), nor in Halliwell's section on slate games (op. cit. in
7.L.1, 1849, pp. 103-104), but there may be an 1875 description in Strutt-Cox of 1903.
Babbage refers to it in his unpublished MSS of c1820 as a children's game, but without giving
it a name. In 1842, he calls it Tit Tat To and he uses slight variations on this name in his
extended studies of the game -- see below. The OED's earliest references are: 1849 for
Tip-tap-toe; 1855 for Tit-tat-toe; 1861 for Oughts and Crosses. However, the first two
entries may be referring to some other game -- e.g. the entries for Tick-tack-toe for 1884 &
1899 are clearly to the game that Gomme calls Tit-tat-toe. Von der Lasa cites a 1838-39
Swedish book for Tripp, Trapp, Trull. Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak,
Tol as the Dutch name. Using the works of Strutt, Gomme, Strutt-Cox, Fiske, Murray, the
OED and some personal communications, I have compiled a separate index of 121 variant
names which refer to 5 basic games, with a few variants and a few unknown games. The
Murray and Parker material is given first, as it deals generally with the ancient history. Then I
list several standard sources and then summarize their content. Other material follows that.
Fiske says that van der Linde and von der Lasa (see 5.F.1) mention early appearances of
Morris games, but rather briefly and I don't always have that material.
The usual # shape board will be so indicated. If one is setting down pieces, then the
board is often drawn as a 'crossed square', i.e. a square with its horizontal and vertical
midlines drawn, and one plays on the intersections. Fiske 127 says this form is common in
Germany, but unknown in England and the US. In addition, the diagonals are often drawn,
producing a 'doubly crossed square'. The squares are sometime drawn as circles giving a
SOURCES - page 26
'crossed circle' and a 'doubly crossed circle', though it is hard to identify the corners in a
crossed circle. The 3 x 3 array of dots sometimes occurs. The standard # pattern is
sometimes surrounded by a square producing a '3 x 3 chessboard'.
Fiske 129 says the English play with O and +, while the Swedes play with O and 1.
My experience is that English and Americans play with O and X. One English friend said
that where she grew up, it was called 'Exeter's Nose' as a deliberate corruption of 'Xs and Os'.
The first clear references to the standard game of Noughts and Crosses are Babbage
(1820) and the items discussed under Tic-tac-toe below. Further clear references are:
Cassell's, Berg, A wrangler ..., Dudeney, White and everything entered below after White.
Misère version: Gardner (1957); Scotts (1975);
Murray mentions Morris, which he generally calls Merels, many times. Besides the many
specific references mentioned below and in 4.B.5, he shows, on p. 614, under Nine
Holes and Three Men's Morris, a number of 3 x 3 diagrams.
Kurna, Egypt, (-14C) -- a double crossed square and a double crossed circle -- see
Parker below.
Ptolemaic Egypt (in the BM, no. 14315) -- a square with # drawn inside. See
below where I describe this, from a recent exhibition, as just a # board.
Ceylon -- a doubly crossed square -- see Parker below.
Rome and Pompeii -- doubly crossed circles.
Under Nine Holes, he says a piece can be moved to any vacant point; under Three
Men's Morris, he says a man can only be moved along a marked line to an adjacent
point, i.e. horizontally, vertically or along a main diagonal.
Under Nine Holes, he shows the # board for English Noughts and Crosses. He
specifically notes that the pieces do not move. His only other mention of this board is
for a Swedish game called Tripp, Trapp, Trull, but he does not state that the pieces do
not move. He gives no other examples of the # board nor of non-moving pieces.
He also mentions Five (or Six) Men's Morris, of which little is known. On p.
133, he mentions a 3 x 3 "board of nine points used for a game essentially identical
with the 'three men's merels', which has existed in China from at least the time of the
Liang dynasty (A.D. 502-557). The 'Swei shu' (first half of the 7th c.) gives the names
of twenty books on this game."
H. Parker. Ancient Ceylon. ??, London, 1909; Asian Educational Services, New Delhi,
1981. Nerenchi keliya, pp. 577-580 & 644. There is a crossed square with small holes
at the intersections at the Temple of Kurna, Upper Egypt, -14C. [Rohrbough, loc. cit. in
4.B.5, says this temple was started by Ramses I and completed by Seti in -1336/-1333,
citing J. Royal Asiatic Soc. (1783) 17.] On p. 644, he shows 34 mason's diagrams from
Kurna, which include #, # in a circle, crossed square with small holes at the
intersections, doubly crossed square, doubly crossed circle. He cites Bell, Arch. Survey
of Ceylon, Third Progress Report, p. 5 note, for for a doubly crossed square in Ceylon,
c1C, but Noughts and Crosses is not found in the interior of Ceylon. The doubly
crossed square was used in 18C Ireland. On pp. 643-665, he discusses appearances of
the crossed square and doubly crossed circle as designs or characters and claims they
have mystic significance. On p. 662, he lists many early appearances of the # pattern.
Murray 440, note 63, includes a reference to Soutendam; Keurboek van Delft; Delft, c1425, f.
78 (or p. 78?); who says games of subtlety are allowed, e.g. ... ticktacken. There is no
indication if this may be our game and the OED indicates that such names were used for
backgammon back to 1558. The OED doesn't cite: W. Shakespeare; Measure for
Measure, c1604. Act I, scene ii, line 180 (or 196): "foolishly lost at a game of
ticktack". Later it was more common as Tric-trac.
Murray 746 notes a Welsh game Gwyddbwyll mentioned in the Mabinogion (14C). The name
is cognate with the Irish Fidchell and may be a Three Men's Morris, but the game was
already forgotten by the 15C.
STANDARD SOURCES ON GAMES
Joseph Strutt. The Sports and Pastimes of the People of England. (With title starting:
Glig-Gamena Angel-Ðeod., or the Sports ...; J. White, London, 1791, 1801, 1810).
A new edition, with a copious index, by William Hone. Tegg, London, 1830, 1831,
1833, 1834, 1838, 1841, 1850, 1855, 1875, 1876, 1891. [The 1830 ed. has a preface,
omitted in 1833, stating that the 1810 ed. is the same as the 1801 ed. and that Hone has
SOURCES - page 27
only changed it by adding the Index and incorporating some footnotes into the text.]
[Hall, BCB 263-266 are: 1801, 1810, 1830, 1831. Toole Stott 647-656 are: 1791;
1801; 1810; 1828-1830 in 10 monthly parts with Index by Hone; 1830; 1830; 1833;
1838; 1841; 1876, an expanded ed, ed by Hone. Heyl 300-302 gives 1830; 1838;
1850. Toole Stott 653 says the sheets were remaindered to Hone, who omitted the first
8pp and issued it in 1833, 1834, 1838, 1841. I have seen an 1855 ed. C&B list 1801,
1810, 1830, 1903. BMC has 1801, 1810, 1830, 1833, 1834, 1838, 1841, 1875, 1876,
1898.]
Strutt-Cox. The Sports and Pastimes of the People of England. By Joseph Strutt.
1801. A new edition, much enlarged and corrected by J. Charles Cox. Methuen, 1903.
The Preface sketches Strutt's life and says this is based on the 'original' 1801 in quarto,
with separate plates which were often hand coloured, but not consistently, while the
1810 reissue had them all done in a terra-cotta shade. Hone reissued it in octavo in
1830 with the plates replaced by woodcuts in the text and this was reissued in 1837,
1841 and 1875. (From above we see that there were other reissues.) "Mr. Strutt has
been left for the most part to speak in his own characteristic fashion .... A few obvious
mistakes and rash conclusions have been corrected, ... certain unimportant omissions
have been made. ... Nearly a third of the book is new." Reprinted in 1969 and in the
1960s?
J. T. Micklethwaite. On the indoor games of school boys in the middle ages. Archaeological
Journal 49 (Dec 1892) 319-328. Describes various 3 x 3 boards and games on them,
including Nine Holes and "tick, tack, toe; or oughts and crosses, which I suppose still
survives wherever slate and pencil are used as implements of education", Three Men's
Morris and also Nine Men's Morris, Fox and Geese, etc.
Alice B. Gomme. The Traditional Games of England, Scotland, and Ireland. 2 vols., David
Nutt, London, 1894 & 1898. Reprinted in one vol., Thames & Hudson, London, 1984.
Willard Fiske. Chess in Iceland and in Icelandic Literature with Historical Notes on Other
Table-Games. The Florentine Typographical Society, Florence, 1905. Esp. pp. 97-156
of the Stray Notes. P. 122 lists a number of works on ancient games.
These and the OED have several entries on Noughts and Crosses and Tic-tac-toe and
many on related games, which are summarised below. Gomme often cites or quotes Strutt.
The OED often gives the same quotes as Gomme. Gomme's references are highly abbreviated
but full details of the sources can usually be found in the OED.
(Nine Men's) Morris, where Morris is spelled about 30 different ways, e.g. Marl,
Merelles, Mill, Miracles, Morals, and Nine Men's may be given as, e.g. Nine-peg, Nine Penny,
Nine Pin. Also known as Peg Morris and Shepherd's Mill. Gomme I 80 & 414-419 and Strutt
317-318 (c= Strutt-Cox 256-258 & plate opp. 246, which adds reference to Micklethwaite) are
the main entries. See 4.B.5 for material more specifically on this game.
Nine Holes, also known as Bubble-justice, Bumble-puppy, Crates, and possibly
Troll-madam, Troule-in-Madame. Gomme I 413-414 and Strutt 274-275 & 384
(c= Strutt-Cox 222-223 & 304) are the main entries. Twelve Holes is similar [Gomme II 321
gives a quote from 1611]. There seem to be cases where Nine Men's Morris was used in
referring to Nine Holes [Gomme I 414-419]. There are two forms of the game: one form has
holes in an upright board that one must roll a ball or marble through; the other form has holes
in the ground, usually in a 3 x 3 array, that one must roll balls into. Unfortunately, none of
the references implies that one has to get three in a row -- see Every Little Boys Book for a
version where this is certainly not the case. There are references going back to 1572 for
Crates (but mentioning eleven holes) [Gomme I 81 & II 309] and 1573 [OED] for Nine
Holes. Botermans et al.; The World of Games; op. cit. in 4.B.5; 1989; p. 213, shows a 17C
engraving by Ménian showing Le Jeu de Troumadame as having a board with holes in it, held
vertically on a table and one must roll marbles through the holes. They say it is nowadays
known as 'bridge'.
Three Men's Morris. This is less common, but occurs in several variant spellings
corresponding to the variants of Nine Men's Morris, including, e.g. Three-penny Morris,
Tremerel. The game is played on a 3 x 3 board and each player has three men. After
making three plays each, consisting of setting men on the cells, further play consists of
picking up one of your own men and placing it on a vacant cell, with the object of getting
three in a row. There are several versions of this game, depending on which cells one may
play to, but the descriptions given rarely make this clear. [Gomme I 414-419] quotes from F.
Douce; Illustrations of Shakespeare and of Ancient Manners; 1807, i.184. "In the French
SOURCES - page 28
merelles each party had three counters only, which were to be placed in a line to win the
game. It appears to have been the tremerel mentioned in an old fabliau. See Le Grand,
Fabliaux et Contes, ii.208. Dr. Hyde thinks the morris, or merrils, was known during the time
that the Normans continued in possession of England, and that the name was afterwards
corrupted into three men's morals, or nine men's morals." [Hyde. Hist. Nederluddi [sic], p.
202.] In practice, the board is often or usually drawn as a crossed square. If one can move
along all winning lines, then it would be natural to draw a doubly crossed square. See under
Alfonso MS (1283) in 4.B.5 for versions called marro, tres en raya and riga di tre. Again,
much of the material on this game is in 4.B.5.
Five-penny Morris. None of the references make it clear, but this seems to be (a form
of) Three Men's Morris. Gomme I 122 and the OED [under Morrell] quote: W. Hawkins;
Apollo Shroving (a play of 1627), act III, scene iv, pp. 48-49.
"..., Ovid hath honour'd my exercises. He describes in verse our boyes play.
Twise three stones, set in a crossed square where he wins the game
That can set his three along in a row,
And that is fippeny morrell I trow."
Most of the references (and myself) are perplexed by the reference to five, though the fact that
one has at most five moves in Tic-tac-toe might have something to do with it?? Since Three
Men's Morris is less well known, some writers have assumed Five-penny Morris was Nine
Men's Morris and others have called all such games by the same name. A few lines later,
Hawkins has: "I challenge him at all games from blowpoint upward to football, and so on to
mumchance, and ticketacke. ... rather than sit out, I will give Apollo three of the nine at
Ticketacke, ..."
Corsicrown [Gomme I 80] seems to be a version of Three Men's Morris, but using
seven of the nine cells, omitting two opposite side cells. Gomme quotes from J. Mactaggart;
The Scottish Gallovidian Encyclopedia; (1871 or possibly 1824?): "each has three men ....
there are seven points for these men to move about on, six on the edges of the square and one
at the centre."
Tic-tac-toe. The earliest clearly described versions are given in Babbage (with no name
given), c1820, and Gomme I 311, under Kit-cat-cannio, where she quotes from: Edward
Moor; Suffolk Words and Phrases; 1823 (This word does not occur in the OED). Gomme
also gives entries for Noughts and Crosses [I 420-421] and Tip-tap-toe [II 295-296] with
variants Tick-tack-toe and Tit-tat-toe. In 1842-1865, Babbage uses Tit Tat To and slight
variants. Under Tip-tap-toe, Gomme says the players make squares and crosses and that a tie
game is a score for Old Nick or Old Tom. (When I was young, we called it Cat's Game, and
this is an old Scottish term [James T. R. Ritchie; The Singing Street Scottish Children's
Games, Rhymes and Sayings; (O&B, 1964); Mercat Press, Edinburgh, 2000, p. 61].) She
quotes regional glossaries for Tip-tap-toe (1877), Tit-tat-toe (1866 & 1888), Tick-tack-toe
(1892). The OED entry for Oughts and Crosses seems to be this game and gives an 1861
quote. Von der Lasa cites a 1838-39 Swedish book for Tripp, Trapp, Trull. Van der Linde
(1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as the Dutch name.
Tit-tat-toe [Gomme II 296-298]. This is a game using a slate marked with a circle and
numbered sectors. The player closes his eyes and taps three times with a pencil and tries to
land on a good sector. Gomme gives the verse:
Tit, tat, toe, my first go,
Three jolly butcher boys all in a row
Stick one up, stick one down,
Stick one in the old man's ground.
But cf Games and Sports for Young Boys, 1859, below.
The OED entries under Tick-tack, Tip-tap and Tit give a number of variant spellings
and several quotations, which are often clearly to this game, but are sometimes unclear. Also
some forms seem to refer to backgammon.
In her 'Memoir on the study of children's games' [Gomme II 472-473], Gomme gives a
somewhat Victorian explanation of the origin of Old Nick as the winner of a tie game as
stemming from "the primitive custom of assigning a certain proportion of the crops or pieces
of land to the devil, or other earth spirit."
Franco Agostini & Nicola Alberto De Carlo. Intelligence Games. (As: Giochi della
Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. P. 81 says
examples of boards were discovered in the lowest level of Troy and in the Bronze Age
tombs in Co. Wicklow, Ireland. Their description is a bit vague but indicates that the
SOURCES - page 29
Italian version of Tic-tac-toe is actually Three Men's Morris.
Anonymous. Play the game. Guardian Education section (21 Sep 1993) 18-19. Shows a
stone board with the # incised on it 'from Bet Shamesh, Israel, 2000 BC'. This might
be the same as the first board below??
A small exhibition of board games organized by Irving Finkel at the British Museum, 1991,
displayed the following.
Stone slab with the usual # Tick-Tac-Toe board incised on it, but really a 4 x 3
board. With nine stone men. From Giza, >-850. BM items EA 14315 & 14309,
donated by W. M. Flinders Petrie. Now on display in Room 63, Case C.
Stone Nine Holes board from the Temple of Artemis, Ephesus, 2C-4C. Item BM GR
1873.5.5.150. This is a 3 x 3 array of depressions. Now on display in Room
69, Case 9.
Robbie Bell & Michael Cornelius. Board Games Round the World. CUP, 1988. P. 6 states
that the crossed square board has been found at Kurna (c-1400) and at the Ptolemaic
temple at Komombo (c-300). They state that Three Men's Morris is the game
mentioned by Ovid in Ars Amatoria. They say that it was known to the Chinese at the
time of Confucius (c-500) under the name of Yih, but is now known as Luk tsut k'i.
They also say the game is also known as Nine Holes -- which seems wrong to me.
The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year
1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as
the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete
reproduction in 194 Phototypic Plates. 2 vols., Karl W. Hirsemann, Leipzig, 1913.
(There was also an edition by Arnald Steiger, Geneva, 1941.) See 4.B.5 for more
details of this work. Vol. 2, f. 93v, p. CLXXXVI, shows a doubly crossed square board.
??NX -- need to study text.
Pieter Bruegel (the Elder). Children's Games. Painting dated 1560 at the Kunsthistorisches
Museum, Vienna. In the right background, children are playing a game involving
throwing balls into holes in the ground, but the holes appear to be in a straight line.
Anonymous. Games of the 16th Century. 1950. Op. cit. in 4.A.3. P. 134 describes nineholes, quoting an unknown poet of 1611: "To play at loggats, Nine-holes, or Tenpinnes". The author doesn't specify what positions the balls are to be rolled into. P. 152
describes Troll-my-dames or Troule-in-madame: "they may have in the end of a bench
eleven holes made, into which to troll pummets, or bowls of lead, ...."
William Wordsworth. The Prelude, Book 1. Completed 1805, published 1850. Lines
509-513.
At evening, when with pencil, and smooth slate
In square divisions parcelled out and all
With crosses and with cyphers scribbled o'er,
We schemed and puzzled, head opposed to head
In strife too humble to be named in verse.
It is not clear if this is referring to Noughts and Crosses.
Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as
Add. MS 37202, c1820. ??NX. F. 4r is part of the Table of Contents. It shows
Noughts and Crosses games played on the # board and on a 4 x 4 board adjacent to
entry 4: The Mill. Ff. 124-146 are: Essay 10 -- Of questions requiring the invention of
new modes of analysis. On f. 128.r, he refers to a game in which "the relative positions
of three of the marks is the object of inquiry." Though the reference is incomplete, a
Noughts and Crosses game is drawn on the facing page, f. 127.v. Ff. 134-144 are:
Essay 10 Part 5. At the top of f. 134.r, he has added a note: "This is probably my
earliest Note on Games of Skill. I do not recollect the date. 3 March 1865". The Essay
begins: "Amongst the simplest of those games requiring any degree of skill which
amuse our early years is one which is played at in the following manner." He then
describes the game in detail and makes some simple analysis, but he never uses a name
for it.
Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205.
??NX. On f. 304, he starts on analysis of games. Ff. 310-383 are almost entirely
devoted to Tit-Tat-To, with some general discussions. Most of this material comprises
a few sheets of working, carefully dated, sometimes amended and with the date of the
amendment. A number of sheets describe parts of the automaton that he was planning
to build which would play the game, but no such machine was built until 1949. The
sheets are not always in strict chronological order.
SOURCES - page 30
F. 310.r is the first discussion of the game, called Tit Tat To, dated 17 Sep 1842.
On F. 312.r, 20 Sep 1843, he says he has "Reduced the 3024 cases D to 199 which
include many Duplicates by Symmetry." F. 321.r, 10 Sep 1860, is the beginning of a
summary of his work on games of skill in general. He refers to Tit-tat-too. F. 322.r
continues and he says: "I have found no game of skill more simple that that which
children often play and which they call Tit-tat-to." F. 324-333, Oct 1844, studies
"General laws for all games of Skill between two players" and draws flow charts
showing the basic recursive analysis of a game tree (ff. 325.v & 325.r). On f. 332, he
counts the number of positions as 9! + 8! + ... + 1! = 409,113. F. 333 has an idea of
the tree structure of a game. On ff. 337-338, 8 Sep 1848, he has Tit-tat too. On ff.
347.r-347.v, he suggests Nine Men's Morris boards in triangular and pentagonal shapes
and does various counting on the different shapes. On ff. 348-349, 26 Oct 1859, he
uses Tit-Tat-To.
John M. Dubbey. The Mathematical Work of Charles Babbage. CUP, 1978, pp. 96-97 &
125-130. He discusses the above Babbage material. On p. 127, Dubbey has: "After a
surprisingly lengthy explanation of the rules, he attempts a mathematical formulation.
The basic problem is one that appears not to have been previously considered in the
history of mathematics." Babbage represents the game using roots of unity. Dubbey,
on p. 129, says: "This analysis ... must count as the first recorded stochastic process in
the history of mathematics." However, it is really a deterministic two-person game.
Games and Sports for Young Boys. Routledge, nd [1859 - BLC]. P. 70, under Rhymes and
Calls: "In the game of Tit-tat-toe, which is played by very young boys with slate and
pencil, this jingle is used:-Tit, tat, toe, my first go:
Three jolly butcher boys all in a row;
Stick one up, stick one down.
Stick one on the old man's crown."
Baron Tassilo von Heydebrand und von der Lasa. Ueber die griechischen und römischen
Spiele, welche einige ähnlichkeit mit dem Schach hatten. Deutsche Schachzeitung
(1863) 162-172, 198-199, 225-234, 257-264. ??NYS -- described on Fiske 121-122 &
137, who says van der Linde I 40-47 copies much of it. Von der Lasa asserts that the
Parva Tabella of Ovid is Kleine Mühle (Three Men's Morris). He says the game is
called Tripp, Trapp, Trull in the Swedish book Hand-Bibliothek för Sällkapsnöjen, of
1839, vol. II, p. 65 (or 57) -- ??NYS. Van der Linde says that the Dutch name is Tik,
Tak, Tol. Fiske notes that both of these refer to Noughts and Crosses, but it is unclear if
von der Lasa or van der Linde recognised the difference between Three Men's Morris
and Noughts and Crosses.
C. Babbage. Passages from the Life of a Philosopher. 1864. Chapter XXXIV -- section on
Games of Skill, pp. 465-471. (= pp. 152-156 in: Charles Babbage and His Calculating
Engines, Dover, 1961.) Partial analysis. He calls it tit-tat-to.
The Play Room: or, In-door Games for Boys and Girls. Dick & Fitzgerald(?), 1866.
[Reprinted as: How to Amuse an Evening Party. Dick & Fitzgerald, NY, 1869.] ??NX
-- the 1869 was seen at Shortz's. P. 22: Tit-tat-to. Uses O and +. "This is a game that
small boys enjoy, and some big ones who won't own it."
Anonymous. Every Little Boy's Book A Complete Cyclopædia of in and outdoor games with
and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and
fifty illustrations. Routledge, London, nd. HPL gives c1850, but the text is clearly
derived from Every Boy's Book, whose first edition was 1856. But the main part of the
text considered here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as
unnamed editor), but is in the 8th ed of 1868 (published for Christmas 1867), which
was the first seriously revised edition, with Edmund Routledge as editor. So this may
be c1868. This is the first published use of the term Noughts and Crosses found so far - the OED's 1861 quote is to Oughts and Crosses..
Pp. 46-47: Slate games: Noughts and crosses. "This is a capital game, and one
which every school-boy truly enjoys." Though the example shown is a draw, there is no
mention of the fact that the game should always be a tie.
Pp. 85-86: Nine-holes. This has nine holes in a row and each player has a hole.
The ball is rolled to them and the person in whose hole it lands must run and pick up the
ball and try to hit one of the others who are running away. So this has nothing to do
with our games or other forms of Nine Holes.
P. 106: Nine-holes or Bridge-board. This has nine holes in an upright board and
SOURCES - page 31
the object is get one's marbles through the holes. (This material is in the 1856 ed. of
Every Boy's Book.)
Correspondent to Notes and Queries (1875) ??NYS -- quoted by Strutt-Cox 257. Describes a
game called Three Mans' Marriage [sic] in Derbyshire which seems to be Noughts and
Crosses played on a crossed square board. Pieces are not described as moving, but in
the next description of a Nine Men's Morris, they are specifically described as moving.
However, the use of a crossed square board may indicate that diagonals were not
considered.
Cassell's. 1881. Slate Games: Noughts and Crosses, or Tit-Tat-To, p. 84, with cross reference
under Tit-Tat-To, p. 87. = Manson, 1911, pp. 202-203 & 208.
Albert Norman. Ungdomens Bok [Book for Youth] (in Swedish). 2nd ed., Stockholm, 1883.
Vol. I, p. 162++. ??NYS -- quoted and described in Fiske 134-136. Description of
Tripp, Trapp, Trull, with winning cry: "Tripp, trapp, trull, min qvarn är full." (Qvarn =
mill.)
Lucas. RM2, 1883. Pp. 73-99. Analysis of Three Men's Morris, on a board with the main
diagonals drawn, with moves of only one square along a winning line. He shows this is
a first person game. If the first player is not permitted to play in the centre, then it is a
tie game. No mention of Tic-Tac-Toe.
Albert Ellery Berg, ed. The Universal Self-Instructor. Thomas Kelly, NY, 1883. Tit-tat-to,
p. 379. Brief description.
Mark Twain. The Adventures of Huckleberry Finn. 1884. Chap. XXXIV, about half-way
through. "It's as simple as tit-tat-toe, three-in-a-row, ..., Huck Finn."
"A wrangler and late master at Harrow school." The science of naughts and crosses. Boy's
Own Paper 10: (No. 498) (28 Jul 1888) 702-703; (No. 499) (4 Aug 1888) 717;
(No. 500) (11 Aug 1888) 735; (No. 501) (18 Aug 1888) 743. Exhaustive analysis,
including odds of second player making a correct response to each opening. For first
move in: middle, side, corner, the odds of a correct response are: 1/2, 1/2, 1/8. He
implies that the analysis is not widely known.
"Tom Wilson". Illustred Spelbok (in Swedish). Nd [late 1880s??]. ??NYS -- described by
Fiske 136-137. This gives Tripp, Trapp, Trull as a Three Men's Morris game on the
crossed square, with moves "according to one way of playing, to whatever points they
please, but according to another, only to the nearest point along the lines on which the
pieces stand. This last method is always employed when the board has, in addition to
the right lines, or lines joining the middles of the exterior lines, also diagonals
connecting the angles". He then describes a drawn version using the # board and 0
and + (or 1 and 2 in the North) which seems to be genuinely Noughts and Crosses.
Fiske says the book seems to be based on an early edition of the Encyclopédie des Jeux
or a similar book, so it is uncertain how much the above represents the current Swedish
game. Fiske was unable to determine the author's real name, though he was still living
in Stockholm at the time.
Il Libro del Giuochi. Florence, 1894. ??NYS -- described in Fiske, pp. 109-110. Gives
doubly crossed square board and mentions a Three Men's Morris game.
T. de Moulidars. Grande Encyclopédie des Jeux. Montgredien or Librairie Illustree, Paris,
nd. ??NYS -- Fiske 115 (in 1905) says it appeared 'not very long ago' and that Gelli
seems to be based on it. Fiske quotes the clear description of Three Men's Morris as
Marelle Simple, using a doubly crossed square, saying that pieces move to adjoining
cells, following a line, and that the first player should win if he plays in the centre.
Fiske notes that Noughts and Crosses is not mentioned.
J. Gelli. Come Posso Divertirmi? Milan, 1900. ??NYS -- described in Fiske 107. Fiske
quotes the description of Three Men's Morris as Mulinello Semplice, essentially a
translation from Moulidars.
Dudeney. CP. 1907. Prob. 109: Noughts and crosses, pp. 156 & 248. (c= MP, prob. 202:
Noughts and crosses, pp. 89 & 175-176. = 536, prob. 471: Tic tac toe, pp. 185 &
390-392. Asserts the game is a tie, but gives only a sketchy analysis. MP gives a
reasonably exhaustive analysis. Looks at Ovid's game.
A. C. White. Tit-tat-toe. British Chess Magazine (Jul 1919) 217-220. Attempt at a complete
analysis, but has a mistake. See Gardner, SA (Mar 1957) = 1st Book, chap. 4.
D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940. Section V Games and
Playthings, pp. 148-165. On p. 160, he quotes Ovid and says it is Noughts and Crosses,
or in Ireland, Tip-top-castle.
The Home Book of Quizzes, Games and Jokes. Blue Ribbon Books, NY, 1941. This is
SOURCES - page 32
excerpted from several books -- this material is most likely taken from: Clement Wood
& Gloria Goddard; Complete Book of Games; same publisher, nd [late 1930s]. P. 150:
Tit-tat-toe, noughts and crosses. Brief description of the game on the # board. "To
win requires great ingenuity."
G. E. Felton & R. H. Macmillan. Noughts and Crosses. Eureka 11 (Jan 1949) 5-9.
Mentions Dudeney's work on the 3 x 3 board and asks for generalizations. Mentions
pegotty = go-bang. Then studies the 4 x 4 x 4 game -- see 4.B.1.a. Adds some
remarks on pegotty, citing Falkener, Lucas and Tarry.
Stanley Byard. Robots which play games. Penguin Science News 16 (Jun 1950) 65-77. On
p. 73, he says D. W. Davies, of the National Physical Laboratory, has built, and
exhibited to the Royal Society in May 1949, an electro-mechanical noughts and crosses
machine. A photo of the machine is plate 16. He also mentions Babbage's interest in
such a machine and an 1874 paper to the US National Academy by a Dr. Rogers -??NYS.
P. C. Parks. Building a noughts and crosses machine. Eureka 14 (Oct 1951) 15-17. Cites
Babbage, Rogers, Davies, Byard. Parks built a simple machine with wire and tin cans
in 1950 at a cost of about £6. Says G. Eastell of Thetford, Norfolk, built a machine
using sixty valves for the Festival of Britain.
Gardner. Ticktacktoe. SA (Mar 1957) c= 1st Book, chap. 4. Quotes Wordsworth, discusses
Three Men's Morris (citing Ovid) and its variants (including versions on 4 x 4 and
5 x 5 boards), the misère version (the person who makes three in a row loses), three
and n dimensional forms (giving L. Moser's result on the number of winning lines on a
kn board), go-moku, Babbage's proposed machine, A. C. White's article. Addendum
mentions the Opies' assertion that the name comes from the rhyme starting "Tit, tat,
toe, My first go,".
C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. A ticktacktoe
machine, pp. 384-385. Noel Elliott gives a brief description of a relay logic machine to
play the game.
Donald Michie. Trial and error. Penguin Science Survey 2 (1961) 129-145. ??NYS.
Describes his matchbox and bead learning machine, MENACE (Matchbox educable
noughts and crosses engine), for the game.
Gardner. A matchbox game-learning machine. SA (Mar 1962) c= Unexpected, chap. 8.
Describes Michie's MENACE. Says it used 300 matchboxes. Gardner adapts it to
Hexapawn, which is much simpler, requiring only 24 matchboxes. Discusses other
games playable by 'computers'. Addendum discusses results sent in by readers
including other games.
Barnard. 50 Observer Brain-Twisters. 1962. Prob. 34: Noughts and crosses, pp. 39-40, 64 &
93-94. Asserts there are 1884 final winning positions. He doesn't consider
equivalence by symmetry and he allows either player to start.
Donald Michie & R. A. Chambers. Boxes: an experiment in adaptive control. Machine
Intelligence 2 (1968) 136-152. Discusses MENACE, with photo of the pile of boxes.
Says there are 288 boxes, but doesn't explain exactly how he found them. Chambers
has implemented MENACE as a general game-learning computer program using
adaptive control techniques designed by Michie. Results for various games are given.
S. Sivasankaranarayana Pillai. A pastime common among South Indian school children. In:
P. K. Srinivasan, ed.; Ramanujan Memorial Volumes: 1: Ramanujan -- Letters and
Reminiscences; 2: Ramanujan -- An Inspiration; Muthialpet High School, Number
Friends Society, Old Boys' Committee, Madras, 1968. Vol. 2, pp. 81-85. [Taken from
Mathematics Student, but no date or details given -- ??] Shows ordinary tic-tac-toe is a
draw and considers trying to get t in a row on an n x n board. Shows that n = t  3 is
a draw and that if t  n + 1 - (n/6), then the game is a draw.
L. A. Graham. The Surprise Attack in Mathematical Problems. Dover, 1968. Tic-tac-toe for
gamblers, prob. 8, pp. 19-22. Proposed by F. E. Clark, solutions by Robert A.
Harrington & Robert E. Corby. Find the probability of the first player winning if the
game is played at random. Two detailed analyses shows that the probabilities for first
player, second player, tie are (737, 363, 160)/1260.
[Henry] Joseph & Lenore Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books
(Charter Communications), NY, 1975. P. 143: Ha-ho-ha. Misère version of noughts
and crosses proposed. No discussion.
Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Noughts and
Crosses, pp. 11-12. Asserts "It's been played all around the world for hundreds, if not
SOURCES - page 33
thousands, of years ...." I've included it as a typical example of popular belief about the
game. = Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989,
Tic-Tac-Toe, pp. 11-12.
Winning Ways. 1982. Pp. 667-680. Complete and careful analysis, including various
uncommon traps. Several equivalent games. Discusses extensions of board size and
dimension.
Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Tic-tac-toe
squared, pp. 88-89. Get 3 in a row on the 4 x 4 board. Also considers Tic-tac-toe-toe
-- get 4 in a row on 5 x 5 board.
George Markovsky. Numerical tic-tac-toe -- I. JRM 22:2 (1990) 114-123. Describes and
studies two versions where the moves are numbered 1, 2, .... One is due to Ron
Graham, the other to P. H. Nygaard and Markowsky sketches the histories.
Ira Rosenholtz. Solving some variations on a variant of tic-tac-toe using invariant subsets.
JRM 25:2 (1993) 128-135. The basic variant is to avoid making three in a row on a
4 x 4 board. By playing symmetrically, the second player avoids losing and 252 of the
256 centrally symmetric positions give a win for the second player. Extends analysis to
2n x 2n, 5 x 5, 4 x 4 x 4, etc.
Bernhard Wiezorke. Sliding caution. CFF 32 (Aug 1993) 24-25 & 33 (Feb 1994) 32. This
describes a sliding piece puzzle on the doubly crossed square board -- see under 5.A.
See: Yuri I. Averbakh; Board games and real events; 1995; in 5.R.5, for a possible connection.
4.B.1.a
IN HIGHER DIMENSIONS
C. Planck. Four-fold magics. Part 2 of chap. XIV, pp. 363-375, of W. S. Andrews, et al.;
Magic Squares and Cubes; 2nd ed., Open Court, 1917; Dover, 1960. On p. 370, he
notes that the number of m-dimensional directions through a cell of the n-dimensional
board is the m-th term of the binomial expansion of ½(1+2)n.
Maurice Wilkes says he played 3-D noughts and crosses at Cambridge in the late 1930s, but
the game was to get the most lines on a 3 x 3 x 3 board. I recall seeing a commercial
version, called Plato?, of this in 1970.
Cedric Smith says he played 3-D and 4-D versions at Cambridge in the early 1940s.
Arthur Stone (letter to me of 9 Aug 1985) says '3 and 4 dimensional forms of tic-tac-toe
produced by Brooks, Smith, Tutte and myself', but it's not quite clear if they invented
these. Tutte became expert on the 43 board and thought it was a first person game.
They only played the 54 game once - it took a long time.
Funkenbusch & Eagle, National Mathematics Mag. (1944) ??NYR.
G. E. Felton & R. H. Macmillan. Noughts and crosses. Eureka 11 (1949) 5-9. They say
they first met the 4 x 4 x 4 game at Cambridge in 1940 and they give some analysis of
it, with tactics and problems.
William Funkenbusch & Edwin Eagle. Hyper-spacial tit-tat-toe or tit-tat-toe in four
dimensions. National Mathematics Magazine 19:3 (Dec 1944) 119-122. ??NYR
A. L. Rubinoff, proposer; L. Moser, solver. Problem E773 -- Noughts and crosses. AMM 54
(1947) 281 & 55 (1948) 99. Number of winning lines on a kn board is
{(k+2)n - kn}/2. Putting k = 1 gives Planck's result.
L. Buxton. Four dimensions for the fourth form. MG 26 (1964) 38-39. 3 x 3 x 3 and
3 x 3 x 3 x 3 games are obviously first person, but he proposes playing for most lines
and with the centre blocked on the 3 x 3 x 3 x 3 board. Suggests 3n and 4 x 4 x 4
games.
Anon. Puzzle page: Noughts and crosses. MTg 33 (1965) 35. Says practice shows that the
4 x 4 x 4 game is a draw. [I only ever had one drawn game!] Conjectures nn is first
player and (n+1)n is a draw.
Roland Silver. The group of automorphisms of the game of 3-dimensional ticktacktoe. AMM
74 (1967) 247-254. Finds the group of permutations of cells that preserve winning lines
is generated by the rigid motions of the cube and certain 'eviscerations'. [It is believed
that this is true for the kn board, but I don't know of a simple proof.]
Ross Honsberger. Mathematical Morsels. MAA, 1978. Prob. 13: X's and O's, p. 26. Obtains
L. Moser's result.
Kathleen Ollerenshaw. Presidential Address: The magic of mathematics. Bull. Inst. Math.
Appl. 15:1 (Jan 1979) 2-12. P. 6 discusses my rediscovery of L. Moser's 1948 result.
Paul Taylor. Counting lines and planes in generalised noughts and crosses. MG 63 (No. 424)
(Jun 1979) 77-82. Determines the number pr(k) of r-sections of a kn board by means
SOURCES - page 34
of a recurrence pr(k) = [pr-1(k+2) - pr-1(k)]/2r which generalises L. Moser's 1948
result. He then gets an explicit sum for it. Studies some other relationships. This work
was done while he was a sixth form student.
Oren Patashnik. Qubic: 4 x 4 x 4 tic-tac-toe. MM 53 (1980) 202-216. Computer assisted
proof that 4 x 4 x 4 game is a first player win.
Winning Ways. 1982. Pp. 673-679, esp. 678-679. Discusses getting k in a row on a n x n
board. Discusses 43 game (Tic-Toc-Tac-Toe) and kn game.
Victor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A
Second Mensa Puzzle Book, 1985, Muller, London, as: The Mensa Puzzle Book,
Treasure Press, London, 1991.) Chapter 7: Conceptual conflict in multi-dimensional
space, pp. 80-94 (1991: 98-112) & answers on pp. 99, 100, 106 & 131 (1991: 115, 116,
122 & 147). He considers various higher dimensional noughts and crosses on the 33,
34 and 35 boards. He finds that there are 49 winning lines on the 33 and he finds
how to determine the number of d-facets on an n-cube as the coefficients in the
expansion of (2x + 1)n. He also considers games where one has to complete a 3 x 3
plane to win and gives a problem: OXO three hypercube planes, p. 91 (1991: 109) &
Answer 29, p. 106 (1991: 122) which asks for the number of planes in the hypercube
34. The answer says there are 123 of them, but in 1985 I found 154 and the general
formula for the number of d-sections of a kn board. When I wrote to Serebriakoff, he
responded that he could not follow the mathematics and that "I arrived at the figures ...
from a simple formula published in one of Art [sic] Gardner's books which checked out
as far as I could take it. Several other mathematicians have looked through it and not
disagreed." I wrote for a reference to Gardner but never had a response. I presented my
work to the British Mathematical Colloquium at Cambridge on 2 Apr 1985 and
discovered that the results were known -- I had found the explicit sum given by Taylor
above, but not the recurrence.
4.B.2. HEX
David Fielker sent some pages from a Danish book on games, but the TP is not present in his
copies, so we don't have details. This says that Hein introduced the game in a lecture to
students at the Institute for Theoretical Physics (now the Niels Bohr Institute) in
Copenhagen in 1942. After its appearance in Politiken, specially printed pads for
playing the game were sold, and a game board was marketed in the US as Hex in 1952.
Piet Hein. Article or column in Politiken (Copenhagen) (26 Dec 1942). ??NYR, but the
diagrams show a board of hexagons.
Gardner (1957) and others have related that the game was independently invented by John
Nash at Princeton in 1948-1949. Gardner had considerable correspondence after his
article which I have examined. The key point is that one of Niels Bohr's sons, who had
known the game in Copenhagen, was a visitor at the Institute for Advanced Study at the
time and showed it to friends. I concluded that it was likely that some idea of the game
had permeated to Nash who had forgotten this and later recalled and extensively
developed the idea, thinking it was new to him. I met Harold Kuhn in 1998, who was a
student with Nash at the time and he has no doubt that Nash invented the idea. In
particular, Nash started with the triangular lattice, i.e. the dual of Hein's board, for some
time before realising the convenience of the hexagonal lattice. Nash came to Princeton
as a graduate student in autumn 1948 and had invented the game by the spring of 1949.
Kuhn says he observed Nash developing the ideas and recognising the connections with
the Jordan Curve Theorem, etc. Kuhn also says that there was not much connection
between students at Princeton and at the Institute and relates that von Neumann saw the
game at Princeton and asked what it was, indicating that it was not well known at the
Institute. In view of this, it seems most likely that Nash's invention was independent,
but I know from my own experience that it can be difficult to remember the sources of
one's ideas -- a casual remark about a hexagonal game could have re-emerged weeks or
months later when Nash was studying games, as the idea of looking at hexagonal boards
in some form, from which the game would be re-invented. Sylvester was notorious for
publishing ideas which he had actually refereed or edited some years earlier, but had
completely forgotten the earlier sources. In situations like Hex, we will never know
exactly what happened -- even if we were present at the time, it is difficult to know
what is going on in the mind of the protagonist and the protagonist himself may not
know what subconscious connections his mind is making. Even if we could discover
SOURCES - page 35
that Nash had been told something about a hexagonal game, we cannot tell how his
mind dealt with this information and we cannot assume this was what inspired his work.
In other words, even a time machine will not settle such historical questions -- we need
something that displays the conscious and the unconscious workings of a person's mind.
Parker Brothers. Literature on Hex, 1952. ??NYS or NYR.
Claude E. Shannon. Computers and automata. Proc. Institute of Radio Engineers 41
(Oct 1953) 1234-1241. Describes his Hex machine on p. 1237.
M. Gardner. The game of Hex. SA (Jul 1957) = 1st Book, chap. 8. Description of Shannon's
8 by 7 'Hoax' machine, pp. 81-82, and its second person strategy, p. 79.
Anatole Beck, Michael N. Bleicher & Donald W. Crowe. Excursions into Mathematics.
Worth Publishers, NY, 1969. Chap. 5: Games (by Beck), Section 3: The game of Hex,
pp. 327-339 (with photo of Hein on p. 328). Says it has been attributed to Hein and
Nash. At Yale in 1952, they played on a 14 x 14 board. Shows it is a first player win,
invoking the Jordan Curve Theorem
David Gale. The game of Hex and the Brouwer fixed-point theorem. AMM 86:10 (Dec
1979) 818-827. Shows that the non-existence of ties (Hex Theorem) is equivalent to
the Brouwer Fixed-Point Theorem in two and in n dimensions. Says the use of the
Jordan Curve Theorem is unnecessary.
Winning Ways. 1982. Pp. 679-680 sketches the game and the strategy stealing argument
which is attributed to Nash.
C. E. Shannon. Photo of his Hoax machine sent to me in 1983.
Cameron Browne. Hex Strategy: Making the Right Connections. A. K. Peters, Natick,
Massachusetts, 2000.
4.B.3. DOTS AND BOXES
Lucas. Le jeu de l'École Polytechnique. RM2, 1883, pp. 90-91. He gives a brief description,
starting: "Depuis quelques années, les élèves de l'École Polytechnique ont imaginé un
nouveaux jeu de combinaison assez original." He clearly describes drawing the edges
of the game board and that the completer of a box gets to go again. He concludes: "Ce
jeu nous a paru assez curieux pour en donner ici la description; mais, jusqu'a présent,
nous ne connaissons pas encore d'observations ni de remarques assez importantes pour
en dire davantage."
Lucas. Nouveaux jeux scientifiques de M. Édouard Lucas. La Nature 17 (1889) 301-303.
Clearly describes a game version of La Pipopipette on p. 302, picture on p. 301, "... un
nouveau jeu ... dédié aux élèves de l'école Polytechnique." This is dots and boxes with
the outer edges already drawn in.
Lucas. L'Arithmétique Amusante. 1895. Note III: Les jeux scientifiques de Lucas,
pp. 203-209 -- includes his booklet: La Pipopipette, Nouveau jeu de combinaisons,
Dédié aux élèves de l'École Polytechnique, Par un Antique de la promotion de 1861,
(1889), on pp. 204-208. On p. 207, he says the game was devised by several of his
former pupils at the École Polytechnique. On p. 37, he remarks that "Pipo est la
désignation abrégée de Polytechnique, par les élèves de l'X, ...."
Robert Marquard & Georg Frieckert. German Patent 108,830 -- Gesellschaftsspiel. Patented:
15 Jun 1899. 1p + 1p diagrams. 8 x 8 array of boxes on a board with slots for inserting
edges. No indication that the player who completes a box gets to play again. They have
some squares with values but also allow all squares to have equal value.
C. Ganse. The dot game. Ladies' Home Journal (Jun 1903) 41. Describes the game and
states that one who makes a box gets to go again.
Loyd. The boxer's puzzle. Cyclopedia, 1914, pp. 104 & 352. = MPSL1, prob. 91, pp. 88-89
& 152-153. c= SLAHP: Oriental tit-tat-toe, pp. 28 & 92-93. Loyd doesn't start with the
boundaries drawn. He asserts it is 'from the East'.
Ahrens. A&N. 1918. Chap. XIV: Pipopipette, pp. 147-155, describes it in more detail than
Lucas does. He says the game appeared recently.
Blyth. Match-Stick Magic. 1921. Boxes, pp. 84-85. "The above game is familiar to most
boys and girls ...." No indication that the completer of a box gets to play again.
Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig
Voggenreiter, Potsdam, 1930. Pp. 84-85: Die Käsekiste. Describes a version for two or
more players. The first player must start at a corner and players must always connect to
previously drawn lines. A player who completes a box gets to play again.
Meyer. Big Fun Book. 1940. Boxes, p. 661. Brief description, somewhat vaguely stating
SOURCES - page 36
that a player who completes a box can play again.
The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. P. 151: Dots and
squares. Clearly says the completer gets to play again. "The game calls for great
ingenuity."
"Zodiastar". Fun with Matches and Match Boxes. (Cover says: Match Tricks From the
1880s to the 1940s.) Universal Publications, London, nd [late 1940s?]. The game of
boxes, pp. 48-49. Starts by laying out four matches in a square and players put down
matches which must touch the previous matches. Completing a box gives another play.
No indication that matches must be on lattice lines, but perhaps this is intended.
Readers' Research Department. RMM 2 (Apr 1961) 38-41, 3 (Jun 1961) 51-52, 4 (Aug 1961)
52-55. On pp. 40-41 of No. 2, it says that Martin Gardner suggests seeking the best
strategy. Editor notes there are two versions of the rules -- where the one who makes a
box gets an extra turn, and where he doesn't -- and that the game can be played on other
arrays. On p. 51 of No. 3, there is a symmetry analysis of the no-extra-turn game on a
board with an odd number of squares. On pp. 52-54 of No. 4, there is some analysis of
the extra-turn case on a board with an odd number of boxes.
Everett V. Jackson. Dots and cubes. JRM 6:4 (Fall 1973) 273-279. Studies 3-dimensional
game where a play is a square in the cubical lattice.
Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Worm, pp. 18-19.
This is a sort of 'anti-boxes' -- one draws segments on the lattice forming a path without
any cycles -- last player wins. = Pencil & Paper Puzzle Games; Watermill Press,
Mahwah, New Jersey, 1989, pp. 18-19.
Winning Ways. 1982. Chap. 16: Dots-and-Boxes, pp. 507-550
David B. Lewis. Eureka! Perigee (Putnam), NY, 1983. Pp. 44-45 suggests playing on the
triangular lattice.
Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987.
Eternal triangles, pp. 80-81. Gives the game on the triangular lattice.
Snakes, pp. 81-82. Same as Brandreth's Worm. I think 'snake' would be a better title as
only one path is drawn.
4.B.4. SPROUTS
M. Gardner. SA (Jul 1967) = Carnival, chap. 1. Describes Michael Stewart Paterson and
John Horton Conway's invention of the game on 21 Feb 1967 at tea time in the
Department common room at Cambridge. The idea of adding a spot was due to
Paterson and they agreed the credit for the game should be 60% Paterson to 40%
Conway.
Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Sprouts, p. 13. "...
actually born in Cambridge about ten years ago." c= Pencil & Paper Puzzle Games;
Watermill Press, Mahwah, New Jersey, 1989, p. 13: "... was invented about ten years
ago."
Winning Ways. 1982. Sprouts, pp. 564-570 & 573. Says the game was "introduced by M. S.
Paterson and J. H. Conway some time ago". Also describes Brussels Sprouts and Starsand-Stripes. An answer for Brussels Sprouts and some references are on p. 573.
Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Sprouts, pp. 95-97.
Karl-Heinz Koch. Pencil & Paper Games. (As: Spiele mit Papier und Bleistift, no details);
translated by Elisabeth E. Reinersmann. Sterling, NY, 1992. Sprouts, pp. 36-37, says it
was invented by J. H. Conway & M. S. Paterson on 21 Feb 1976 [sic -- misprint of
1967] during their five o'clock tea hour.
4.B.5. OVID'S GAME AND NINE MEN'S MORRIS
See also 4.B.1 for historical material.
The classic Nine Men's Morris board consists of three concentric squares with their
midpoints joined by four lines. The corners are sometimes also joined by another four
diagonal lines, but this seems to be used with twelve men per side and is sometimes called
Twelve Men's Morris -- see 1891 below. Fiske 108 says this is common in America but
infrequent in Europe, though on 127 he says both forms were known in England before 1600,
and both were carried to the US, though the Nine form is probably older.
Murray 615 discusses Nine Men's Morris. He cites Kurna, Egypt (-14C), medieval Spain
SOURCES - page 37
(Alquerque de Nueve), the Gokstad ship and the steps of the Acropolis of Athens. He
says the board sometimes has diagonals added and then is played with 9, 11 or 12
pieces.
Dudeney. AM. 1917. Introduction to Moving Counter Problems, pp. 58-59. This gives a
brief survey, mentioning a number of details that I have not seen elsewhere, e.g. its
occurrence in Poland and on the Amazon. Says the board was found on a Roman tile at
Silchester and on the steps of the Acropolis in Athens among other sites.
J. A. Cuddon. The Macmillan Dictionary of Sports and Games. Macmillan, London, 1980.
Pp. 563-564. Discusses the history. Says there is a c-1400 board cut in stone at Kurna,
Egypt and similar boards were made in years 9 to 21 at Mihintale, Ceylon. Says Ars
Amatoria may be describing Three Men's Morris and Tristia may be describing a kind
of Tic-tac-toe. Cites numerous medieval descriptions and variants.
Claudia Zaslavsky. Tic Tac Toe and Other Three-in-a-Row Games from Ancient Egypt to the
Modern Computer. Crowell, NY, 1982. This is really a book for children and there are
no references for the historical statements. I have found most of them elsewhere, and
the author has kindly send me a list of source books, but I have not yet tracked down the
following items -- ??.
There is an English court record of 1699 of punishment for playing Nine Holes in
church.
There is a Nine Men's Morris board on a stone on the temple of Seti I
(presumably this is at Kurna). There is a picture in the 13C Spanish 'Book of Games'
(presumably the Alfonso MS -- see below) of children playing Alquerque de Tres (c=
Three Men's Morris). A 14C inventory of the Duc de Berry lists tables for Mérelles (=?
Nine Men's Morris) (see Fiske 113-115 below) and a book by Petrarch shows two apes
playing the game.
H. Parker. Ancient Ceylon. Loc. cit. in 4.B.1. Nine Men's Morris board in the Temple of
Kurna, Egypt, -14C. [Rohrbough, below, says this temple was started by Ramses I and
completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.] Two
diagrams for Nine Men's Morris are cut into the great flight of steps at Mihintale,
Ceylon and these are dated c1C. He cites Bell; Arch. Survey of Ceylon, Third Progress
Report, p. 5 note, for another diagram of similar age.
Jack Botermans, Tony Burrett, Pieter van Delft & Carla van Spluntern. The World of Games.
(In Dutch, 1987); Facts on File, NY, 1989.
P. 35 describes Yih, a form of Three Men's Morris, played on a doubly crossed
square with a man moving "one step along any line". A note adds that only the French
have a rule forbidding the first player to play in the centre, which makes the game more
challenging and is recommended.
Pp. 103-107 is the beginning of a section: Games of alignment and configuration
and discusses various games, but rather vaguely and without references. They mention
Al-Qurna, Mihintale, Gokstad and some other early sites. They say Yih was described
by Confucius, was played c-500 and is "the game, that we now know as tic-tac-toe, or
three men's morris." They describe Noughts and Crosses in the usual way. They then
distinguish Tic-Tac-Toe, saying "In Britain it is generally known as three men's morris
...." and say it is the same as Yih, "which was known in ancient Egypt". They say "Ovid
mentions tic-tac-toe" in Ars Amatoria, that several Roman boards have survived and
that it was very popular in 14C England with several boards for this and Nine Men's
Morris cut into cloister seats. They then describe Three-in-a-Row, which allows pieces
to move one step in any direction, as a game played in Egypt. They then describe Five
or Six Men's Morris, Nine Men's Morris, Twelve Men's Morris and Nine Men's Morris
with Dice, with nice 13C & 15C illustration of Nine Men's Morris.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Pp. 6-8. They
discuss the crossed square board -- see 4.B.1 -- and describe Three Men's Morris with
moves only along the lines to an adjacent vacant point. They then describe Achi, from
Ghana, on the doubly crossed square with the same rules. They then describe Six Men's
Morris which was apparently popular in medieval Europe but became obsolete by
c1600.
Ovid. Ars Amatoria. -1. II, 203-208 & III, 353-366. Translated by J. H. Mozley; Loeb
Classical Library, 1929, pp. 80-81 & 142-145. Translated by B. P. Moore, 1935, used
in A. D. Melville; Ovid The Love Poems; OUP, 1990, pp. 113, 137, 229 & 241.
II, 203-208 are three couplets apparently referring to three games: two dice games
SOURCES - page 38
and Ludus Latrunculorum. Mozley's prose translation is:
"If she be gaming, and throwing with her hand the ivory dice, do you throw
amiss and move your throws amiss; or if is the large dice you are throwing, let no
forfeit follow if she lose; see that the ruinous dogs often fall to you; or if the
piece be marching under the semblance of a robbers' band, let your warrior fall
before his glassy foe."
'Dogs' is the worst throw in Roman dice games.
Moore's verse translation of 207-208 is:
"And when the raiding chessmen take the field, Your champion to his
crystal foe must yield."
Melville's note says the original has 'bandits' and says the game is Ludus
Latrunculorum.
III, 357-360 is probably a reference to the same game since 'robbers' occurs again,
though translated as brigands by Mozley, and again it immediately follows a reference
to throwing dice. Mozley's translation of 353-366 is:
"I am ashamed to advise in little things, that she should know the throws of
the dice, and thy powers, O flung counter. Now let her throw three dice, and now
reflect which side she may fitly join in her cunning, and which challenge, Let her
cautiously and not foolishly play the battle of the brigands, when one piece falls
before his double foe and the warrior caught without his mate fights on, and the
enemy retraces many a time the path he has begun. And let smooth balls be flung
into the open net, nor must any ball be moved save that which you will take out.
There is a sort of game confined by subtle method into as many lines as the
slippery year has months: a small board has three counters on either side,
whereon to join your pieces together is to conquer."
Moore's translation of 357-360 is:
"To guide with wary skill the chessmen's fight, When foemen twain
o'erpower the single knight, And caught without his queen the king must face
The foe and oft his eager steps retrace".
This is clearly not a morris game -- Mozley's note above and the next entry make it clear
it is Ludus Latrunculorum, which had a number of forms. Mozley's note on pp. 142143 refers to Tristia II, 478 and cites a number of other references for Ludus
Latrunculorum.
Moore's translation of 363-366 is:
"A game there is marked out in slender zones As many as the fleeting year
has moons; A smaller board with three a side is manned, And victory's his who
first aligns his band."
Mozley's notes and Melville's notes say the first two lines refer to the Roman game of
Ludus Duodecim Scriptorum -- the Twelve Line Game -- which is the ancestor of
Backgammon. Mozley says the game in the latter two lines is mentioned in Tristia, "but
we have no information about it." Melville says it is "a 'position' game, something like
Nine Men's Morris" and cites R. C. Bell's article on 'Board and tile games' in the
Encyclopaedia Britannica, 15th ed., Macropaedia ii.1152-1153, ??NYS.
Ovid. Tristia. c10. II, 471-484. Translated by A. L. Wheeler. Loeb Classical Library, 1945,
pp. 88-91. This mentions several games and the text parallels that of Ars Amatoria III.
"Others have written of the arts of playing at dice -- this was no light sin in
the eyes of our ancestors -- what is the value of the tali, with what throw one can
make the highest point, avoiding the ruinous dogs; how the tessera is counted,
and when the opponent is challenged, how it is fitting to throw, how to move
according to the throws; how the variegated soldier steals to the attack along the
straight path when the piece between two enemies is lost, and how he
understands warfare by pursuit and how to recall the man before him and to
retreat in safety not without escort; how a small board is provided with three men
on a side and victory lies in keeping one's men abreast; and the other games -- I
will not describe them all -- which are wont to waste that precious thing, our
time."
A note says some see a reference to Ludus Duodecim Scriptorum at the beginning of
this. The next note says the next text refers to Ludus Latrunculorum, a game on a
squared board with 30 men on a side, with at least two kinds of men. The note for the
last game says "This game seems to have resembled a game of draughts played with few
men." and refers to Ars Amatoria and the German Mühlespiel, which he describes as 'a
SOURCES - page 39
sort of draughts', but which is Nine Men's Morris.
R. G. Austin. Roman board games -- I & II. Greece and Rome 4 (No. 10) (Oct 1934) 24-34
& 4 (No. 11) (Feb 1935) 76-82. Claims the Ovid references are to Ludus
Latrunculorum (a kind of Draughts?), Ludus Duodecim Scriptorum (later Tabula, an
ancestor of Backgammon) and (Ars Amatoria.iii.365-366) a kind of Three Men's
Morris. In the last, he shows a doubly crossed 3 x 3 board, but it is not clear which
rule he adopts for the later movement of pieces, but he says: "the first player is always
able to force a win if he places his first man on the centre point, and this suggests that
the dice may have been used to determine priority of play, although there is no evidence
of this." He says no Roman name for this game has survived. He discusses various
known artifacts for all the game, citing several Roman 8 x 8 boards found in Britain.
He gives an informal bibliography with comments as to the value of the works.
D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940. Section V Games and
Playthings, pp. 148-165. On p. 160, he quotes Ovid, Ars Amatoria.iii.365-366 and says
it is Noughts and Crosses, or in Ireland, Tip-top-castle.
The British Museum has a Nine Men's Morris board from the Temple of Artemis, Ephesus,
2C-4C. Item BM GR 1872,8-3,44. This was in a small exhibition of board games in
1990. I didn't see it on display in late 1996.
Murray, p. 189. There was an Arabic game called Qirq, which Murray identifies with Morris.
"Fourteen was a game played with small stones on a wooden board which had three
rows of holes (al-Qâbûnî)." Abû-Hanîfa [the H should have a dot under it], c750, held
that Fourteen was illegal and Qirq was held illegal by writers soon afterward. On p.
194, Murray gives a 10C passage mentioning Qirq being played at Mecca.
Fiske 255 cites the Kitāb al Aghāni, c960, for a reference to qirkat, i.e. morris boards.
Paul B. Du Chaillu. The Viking Age. Two vols., John Murray, London, 1889. Vol. II, p.168,
fig. 992 -- Fragments of wood from Gokstad ship. Shows a partial board for Nine
Men's Morris found in the Gokstad ship burial. There is no description of this
illustration and there is only a vague indication that this is 10C, but other sources say it
is c900.
Gutorm Gjessing. The Viking Ship Finds. Revised ed., Universitets Oldsaksamling, Oslo,
1957. P. 8: "... there are two boards which were used for two kinds of games; on one
side figures appear for use in a game which is frequently played even now (known as
"Mølle")."
Thorlief Sjøvold. The Viking Ships in Oslo. Universitets Oldsaksamling, Oslo, 1979. P. 54:
"... a gaming board with one antler gaming piece, ...."
In medieval Europe, the game is called Ludus Marellorum or Merellorum or just Marelli or
Merelli or Merels, meaning the game of counters. Murray 399 says the connection with
Qirq is unclear. However, medieval Spain played various games called Alquerque,
which is obviously derived from Qirq. Alquerque de Nueve seems to be Nine Men's
Morris. However, in Italy and in medieval France, Marelle or Merels could mean
Alquerque (de Doze), a draughts-like game with 12 men on a side played on a 5 x 5
board (Murray 615). Also Marro, Marella can refer to Draughts which seems to
originate in Europe somewhat before 1400.
Stewart Culin. Korean Games, with Notes on the Corresponding Games of China and Japan.
University of Pennsylvania, Philadelphia, 1895. Reprinted as: Games of the Orient;
Tuttle, Rutland, Vermont, 1958. Reprinted under the original title, Dover and The
Brooklyn Museum, 1991. P. 102, section 80: Kon-tjil -- merrells. This is the usual
Nine Men's Morris. The Chinese name is Sám-k'i (Three Chess). "I am told by a
Chinese merchant that this game was invented by Chao Kw'ang-yin (917-975), founder
of the Sung dynasty." This is the only indication of an oriental source that I have seen.
Gerhard Leopold. Skulptierte Werkstücke in der Krypta der Wipertikirche zu Quedlinburg.
IN: Friedrich Möbius & Ernst Schubert, eds.; Skulptur des Mittelalters; Hermann
Böhlaus Nachfolger, Weimar, 1987, pp. 27-43; esp. pp. 37 & 43. Describes and gives
photos of several Nine-Men's-Morris boards carved on a pillar of the crypt of the
Wipertikirche, Quedlinburg, Sachsen-Anhalt, probably from the 10/11 C.
Richard de Fournivall. De Vetula. 13C. This describes various games, including Merels.
Indeed the French title is: Ci parle du gieu des Merelles .... ??NYS -- cited by Murray,
pp. 439, 507, 520, 628. Murray 620 cites several MSS and publications of the text.
"Bonus Socius" [Nicolas de Nicolaï?]. This is a collection of chess problems, compiled
c1275, which exists in many manuscript forms and languages. See 5.F.1 for more
details of these MSS. See Murray 618-642. On pp. 619-624 & 627, he mentions
SOURCES - page 40
several MSS which include 23, 24, 25 or 28 Merels problems. On p. 621, he cites
"Merelles a Neuf" from 14C. Fiske 104 & 110-111 discusses some MSS of this
collection.
The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year
1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as
the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete
reproduction in 194 Phototropic Plates. 2 vols., Karl W. Horseman, Leipzig, 1913.
(See in 4.A.1 for another ed.) This is a collection of chess problems produced for
Alfonso X, the Wise, King of Castile (Castilla). Vol. 2, ff. 92v-93r,
pp. CLXXXIV-CLXXXV, shows Nine Men's Morris boards. ??NX -- need to study
text. See: Murray 568-573; van der Linde I 137 & 279 ??NYS & Quellenstudien 73
& 277-278, ??NYS (both cited by Fiske 98); van der Lasa 116, ??NYS (cited by
Fiske 99).
Fiske 98-99 says that the MS also mentions Alquerque, Cercar de Liebre and
Alquerque de Neuve (with 12 men against one). Fiske 253-255 gives a more detailed
study of the MS based on a transcript. He also quotes a communication citing al
Querque or al Kirk in Kazirmirski's Arabic dictionary and in the Kitāb al Aghāni, c960.
José Brunet y Bellet. El Ajedrez. Barcelona, 1890. ??NYS -- described by Fiske
98. This has a chapter on the Alfonso MS and refers to Alquerque de Doce, saying that
it is known as Tres en Raya in Castilian and Marro in Catalan (Fiske 102 says this word
is no longer used in Spanish). Brunet notes that there are five miniatures pertaining to
alquerque. Fiske says that all this information leaves us uncertain as to what the games
were. Fiske says Brunet's chapter has an appendix dealing with Carrera's 1617
discussion of 'line games' and describing Riga di Tre as the same as Marro or Tres en
Raya as a form of Three Men's Morris
Murray gives many brief references to the game, which I will note here simply by his page
number and the date of the item.
438-439 (12C); 446 (14C);
449 (c1400 -- 'un marrelier', i.e. a Merels board);
431 (c1430); 447 (1491); 446 (1538).
Anon. Romance of Alexander. 1338. (Bodleian Library, Mss Bodl. 264). ??NYS. Nice
illustration clearly showing Nine Men's Morris board. I. Disraeli (Amenities of
Literature, vol. I, p. 86) also cites British Museum, Bib. Reg. 15, E.6 as a prose MS
version with illustrations. Prof. D. J. A. Ross tells me there is nothing in the text
corresponding to the illustrations and that the Bodleian text was edited by M. R. James,
c1920, ??NYS. Illustration reproduced in: A. C. Horth; 101 Games to Make and Play;
Batsford, London, (1943; 2nd ed., 1944); 3rd ed., 1946; plate VI facing p. 44, in
B&W. Also in: Pia Hsiao et al.; Games You Make and Play; Macdonald and Jane's,
London, 1975, p. 7, in colour.
Fiske 113-115 gives a number of quotations from medieval French sources as far back as mid
14C, including an inventory of the Duc de Berry in 1416 listing two boards. Fiske notes
that the game has given rise to several French phrases. He quotes a 1412 source calling
it Ludus Sanct Mederici or Jeu Saint Marry and also mentions references in city statutes
of 1404 and 1414.
MS, Montpellier, Faculty of Medicine, H279 (Fonts de Boulier, E.93). 14C. This is a version
of the Bonus Socius collection. Described in Murray 623-624, denoted M, and in van
der Linde I 301, denoted K. Lucas, RM2, 1883, pp. 98-99 mentions it and RM4, 1894,
Quatrième Récréation: Le jeu des mérelles au XIIIe siècle, pp. 67-85 discusses it
extensively. This includes 28 Merels problems which are given and analysed by Lucas.
Lucas dates the MS to the 13C.
Household accounts of Edward IV, c1470. ??NYS -- see Murray 617. Record of purchase of
"two foxis and 46 hounds" to form two sets of "marelles".
Civis Bononiae [Citizen of Bologna]. This is a collection of chess problems compiled c1475,
which exists in several MSS. See Murray 643-703. It has 48 or 53 merels problems.
On p. 644, 'merelleorum' is quoted.
A Hundred Sons. Chinese scroll of Ming period (1368-1644). 18C copy in BM. ??NYS -extensively reproduced and described in: Marguerite Fawdry; Chinese Childhood;
Pollock's Toy Theatres, London, 1977. On p. 12 of Fawdry is a scene, apparently from
the scroll, in which some children appear to be playing on a Twelve Men's Morris
board.
Elaborate boards from Germany (c1530) and Venice (16C) survive in the National Museum,
SOURCES - page 41
Munich and in South Kensington (Murray 757-758). Murray shows the first in B&W
facing p. 757.
William Shakespeare. A Midsummer Night's Dream. c1610. Act II, scene I, lines 98-100:
"The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton green
For lack of tread are indistinguishable." Fiske 126 opines that the latter two lines may
indicate that the board was made in the turf, though he admits that they may refer just to
dancers' tracks, but to me it clearly refers to turf mazes.
J. C. Bulenger. De Ludis Privatis ac Domesticus Veterum. Lyons, 1627. ??NYS Fiske 115
& 119 quote his description of and philological note on Madrellas (Three Men's
Morris).
Paul Fleming (1609-1640). In one of his lyrics, he has Mühlen. ??NYS -- quoted by Fiske
132, who says this is the first German mention of Morris.
Fiske 133 gives the earliest Russian reference to Morris as 1675.
Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford,
1694. Historia Triodii, pp. 202-214, is on morris games. (Described in Fiske 118-124,
who says there is further material in the Elenchus at the end of the volume -- ??NYS)
Hyde asserts that the game was well known to the Romans, though he cannot find a
Roman name for it! He cites and discusses Bulenger, but disagrees with his philology.
Gives lots of names for the game, ranging as far as Russian and Armenian. He gives
both the Nine and Twelve Men's Morris boards on p. 210, but he has not found the
Twelve board in Eastern works. On p. 211, he gives the doubly crossed square board
with a title in Chinese characters, pronounced 'Che-lo', meaning 'six places', and having
three white and three black men already placed along two sides. He says the Irish name
is Cashlan Gherra (Short Castle) and that the name Copped Crown is common in
Cumberland and Westmorland. He then describes playing the Twelve Man and Nine
Man games, and then he considers the game on the doubly crossed square board. He
seems to say there are different rules as to how one can move. ??need to study the Latin
in detail. This is said to throw light on the Ovid passages. Hyde believes the game was
well known to the Romans and hence must be much older. Fiske remarks that this is
history by guesswork.
Murray 383 describes Russian chess. He says Amelung identifies the Russian game "saki
with Hölzchenspiel (?merels)". Saki is mentioned on this page as being played at the
Tsar's court, c1675.
Archiv der Spiele. 3 volumes, Berlin, 1819-1821. Vol. 2 (1820) 21-27. ??NYS Described
and quoted by Fiske 129-132. This only describes the crossed square and the Nine
Men's Morris boards. It says that the Three Men's Morris on the crossed square board is
a tie, i.e. continues without end, but it is not clear how the pieces are allowed to move.
Fiske says this gives the most complete explanation he knows of the rules for Nine
Men's Morris.
Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205.
??NX. For more details, see 4.B.1. On ff. 347.r-347.v, 8 Sep 1848, he suggests Nine
Men's Morris boards in triangular and pentagonal shapes and does various counting on
the different shapes.
The Family Friend (1856) 57. Puzzle 17. -- Two and a Bushel. Shows the standard # board.
"This very simple and amusing games, -- which we do not remember to have seen
described in any book of games, -- is played, like draughts, by two persons with
counters. Each player must have three, ... and the game is won when one of the players
succeeds in placing his three men in a row; ...." There is no specification of how the
men move. The word 'bushel' occurs in some old descriptions of Three Men's Morris
and Nine Men's Morris as the name of the central area.
The Sociable. 1858. Merelles: or, nine men's morris, pp. 279-280. Brief description, notable
for the use of Merelles in an English book.
Von der Lasa. Ueber die griechischen und römischen Spiele, welche einige ähnlichkeit mit
dem Schach hatten. Deutsche Schachzeitung (1863) 162-172, 198-199, 225-234,
257-264. ??NYS -- described on Fiske 121-122 & 137, who says van der Linde I 40-47
copies much of it. He asserts that the Parva Tabella of Ovid is Kleine Mühle (Three
Men's Morris). Von der Lasa says the game is called Tripp, Trapp, Trull in the Swedish
book Hand-Bibliothek för Sällkapsnöjen, of 1839, vol. II, p. 65 (or 57??). Van der
Linde says that the Dutch name is Tik, Tak, Tol. Fiske notes that both of these refer to
Noughts and Crosses, but it is unclear if von der Lasa or van der Linde recognised the
SOURCES - page 42
difference between Three Men's Morris and Noughts and Crosses.
Albert Norman. Ungdomens Bok [Book for Youth] (in Swedish). 2nd ed., Stockholm, 1883.
Vol. I, p. 162++. ??NYS -- quoted and described in Fiske 134-136. Plays Nine Men's
Morris on a Twelve Men's Morris board.
Webster's Dictionary. 1891. ??NYS -- Fiske 118 quotes a definition (not clear which) which
includes "twelve men's morris". Fiske says: "Here we have almost the only, and
certainly the first mention of the game by its most common New England name,
"twelve men's morris," and also the only hint we have found in print that the more
complicated of the morris boards -- with the diagonal lines ... -- is used with twelve
men, instead of nine, on each side." Fiske 127 says the name only appears in American
dictionaries.
Dudeney. CP. 1907. Prob. 110: Ovid's game, pp. 156-157 & 248. Says the game "is
distinctly mentioned in the works of Ovid." He gives Three Men's Morris, with moves
to adjacent cells horizontally or diagonally, and says it is a first player win.
Blyth. Match-Stick Magic. 1921. Black versus white, pp. 79-80. 4 x 4 board with four men
each. But the men must be initially placed WBWB in the first row and BWBW in
the last row. They can move one square "in any direction" and the object is to get four
in a row of your colour.
Games and Tricks -- to make the Party "Go". Supplement to "Pearson's Weekly", Nov. 7th,
no year indicated [1930s??]. A matchstick game, p. 11. On a 4 x 4 board, place eight
men, WBWB on the top row and BWBW on the bottom row. Players alternately
move one of their men by one square in any direction -- the object is to make four in a
line.
Lynn Rohrbough, ed. Ancient Games. Handy Series, Kit N, Cooperative Recreation Service,
Delaware, Ohio, (1938), 1939.
Morris was Player [sic] 3,300 Years Ago, p. 27. Says the temple of Kurna was started
by Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc.
(1783) 17.
Three Men's Morris, p. 27. After placing their three men, players 'then move trying to
get three men in a row.' Contributor says he played it in Cardiff more than 50
years ago.
Winning Ways. 1982. Pp. 672-673. Says Ovid's Game is conjectured to be Three Men's
Morris. The current version allows moves by one square orthogonally and is a first
person win if the first person plays in the centre. If the first player cannot play in the
centre, it is a draw. They use Three Men's Morris for the case with one step moves
along winning lines, i.e. orthogonally or along main diagonals. An American Indian
game, Hopscotch, permits one step moves orthogonally or diagonally (along any
diagonal). A French game, Les Pendus, allows any move to a vacant cell. All of these
are draws, even allowing the first player to play in the centre. They briefly describe Six
and Nine Men's Morris.
Ralph Gasser & J. Nievergelt. Es ist entscheiden: Das Muehle-Spiel ist unentscheiden.
Informatik Spektrum 17 (1994) 314-317. ??NYS -- cited by Jörg Bewersdorff [email of
6 Jun 1999].
L. V. Allis. Beating the World Champion -- The state of the art in computer game playing.
IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian
Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995;
pp. 155-175. On p. 163, he states that Ralph Gasser showed that Nine Men's Morris is a
draw in Oct 1993, but the only reference is to a letter from Gasser.
Ralph Gasser. Solving Nine Men's Morris. IN: Games of No Chance; ed. by Richard
Nowakowski; CUP, 1996, pp. 101-113. ??NYS -- cited by Bewersdorff [loc. cit.] and
described in William Hartston; What mathematicians get up to; The Independent Long
Weekend (29 Mar 1997) 2. Demonstrates that Nine Men's Morris is a draw. Gasser's
abstract: "We describe the combination of two search methods used to solve Nine Men's
Morris. An improved analysis algorithm computes endgame databases comprising
about 1010 states. An 18-ply alpha-beta search the used these databases to prove that the
value of the initial position is a draw. Nine Men's Morris is the first non-trivial game to
be solved that does not seem to benefit from knowledge-based methods." I'm not sure
about the last statement -- 4 x 4 x 4 noughts and crosses (see 4.B.1.a) and Connect-4
were solved in 1980 and 1988, though the first was a computer aided proof and the
original brute force solution of Connect-4 by James Allen in Sep 1988 was improved to
a knowledge-based approach by L. V. Allis by Aug 1989. The five-in-a-row version of
SOURCES - page 43
Connect-4 was shown to be a first person win in 1993. Bewersdorff [email of 11 Jun
1999] clarifies this by observing that draw here means a game that continues forever -one cannot come to a stalemate where neither side can move.
4.B.6. PHUTBALL
Winning Ways. 1982. Philosopher's football, pp. 688-691. In 1985, Guy said this was the
only published description of the game.
4.B.7. BRIDG-IT
This is best viewed as played on a n x n array of squares. The n(n+1) vertical edges
belong to one player, say red, while the n(n+1) horizontal edges belong to black. Players
alternate marking a square with a line of their colour between edges of their colour. A square
cannot be marked twice. The object is to complete a path across the board. In practice, the
edges are replaced by coloured dots which are joined by lines. As with Hex, there can be no
ties and there must be a first person strategy.
M. Gardner. SA (Oct 1958) c= 2nd Book, Chap. 7. Introduces David Gales's game, later
called Bridg-it. Addendum in the book notes that it is identical to Shannon's 'Bird Cage'
game of 1951 and that it was marketed as Bridg-it in 1960.
M. Gardner. SA (Jul 1961) c= New MD, Chap. 18. Describes Oliver Gross's simple strategy
for the first player to win. Addendum in the book gives references to other solutions
and mentions.
M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Article says Bridg-it was still on the
market.
Winning Ways. 1982. Pp. 680-682. Covers Bridg-it and Shannon Switching Game.
In Oct 2000, I bought a second-hand copy of a 5 x 5 version called Connections, attributed to
Tom McNamara, but with no date.
4.B.8. CHOMP
Fred Schuh. Spel van delers (Game of divisors). Nieuw Tijdschrift vor Wiskunde 39
(1951-52) 299-304. ??NYS -- cited by Gardner, below.
M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Gives David Gale's description of his game
and results on it. Addendum in Knotted points out that it is equivalent to Schuh's game
and gives other references.
David Gale. A curious Nim-type game. AMM 81 (1974) 876-879. Describes the game and
the basic results. Wonders if the winning move is unique. Considers three dimensional
and infinite forms. A note added in proof refers to Gardner's article, says two
programmers have consequently found that the 8 x 10 game has two winning first
moves and mentions Schuh's game.
Winning Ways. 1982. Pp. 598-600. Brief description with extensive table of good moves.
Cites an earlier paper of Gale and Stewart which does not deal with this game.
4.B.9. SNAKES AND LADDERS
I have included this because it has an interesting history and because I found a nice way
to express it as a kind of Markov process or random walk, and this gives an expression for the
average time the game lasts. I then found that the paper by Daykin et al. gives most of these
ideas.
The game has two or three rules for finishing.
A.
One finishes by going exactly to the last square, or beyond it.
B.
One finishes by going exactly to the last square. If one throws too much, then
one stands still.
C.
One finishes by going exactly to the last square. If one throws too much, one
must count back from the last square. E.g., if there are 100 squares and one is at 98 and one
throws 6, then one counts: 99, 100, 99, 98, 97, 96 and winds up on 96. (I learned this from a
neighbour's child but have only seen it in one place -- in the first Culin item below.)
Games of this generic form are often called promotion games. If one considers the
game with no snakes or ladders, then it is a straightforward race game, and these date back to
SOURCES - page 44
Egyptian and Babylonian times, if not earlier.
In fact, the same theory applies to random walks of various sorts, e.g. random walks of
pieces on a chessboard, where the ending is arrival exactly at the desired square.
In the British Museum, Room 52, Case 24 has a Babylonian ceramic board (WA 1991-7.20,I)
for a kind of snakes and ladders from c-1000. The label says this game was popular
during the second and first millennia BC.
Sheng-kuan t'u [The game of promotion]. 7C. Chinese game. This is described in: Nagao
Tatsuzo; Shina Minzoku-shi [Manners and Customs of the Chinese]; Tokyo, 19401942, perhaps vol. 2, p. 707, ??NYS This is cited in: Marguerite Fawdry; Chinese
Childhood; Pollock's Toy Theatres, London, 1977, p. 183, where the game is described
as "played on a board or plan representing an official career from the lowest to the
highest grade, according to the imperial examination system. It is a kind of Snakes and
Ladders, played with four dice; the object of each player being to secure promotion over
the others."
Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford,
1694. De ludo promotionis mandarinorum, pp. 70-101 -- ??NX. This is a long
description of Shing quon tu, a game on a board of 98 spaces, each of which has a
specific description which Hyde gives. There is a folding plate showing the Chinese
board, but the copy in the Graves collection is too fragile to photocopy. I did not see
any date given for the game.
Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S.
National Museum for 1893, pp. 489-537. Pp. 502-507 describes several versions of the
Japanese Sugoroku (Double Sixes) which is a generic name for games using dice to
determine moves, including backgammon and simple race games, as well as Snakes and
Ladders games. One version has ending in the form C. Then says Shing Kún T’ò
(The Game of the Promotion of Officials) is described by Hyde as The Game of the
Promotion of the Mandarins and gives an extended description of it. There is a legend
that the game was invented when the Emperor Kienlung (1736-1796) heard a candidate
playing dice and the candidate was summoned to explain. He made up a story about the
game, saying that it was a way for him and his friends to learn the different ranks of the
civil service. He was sent off to bring back the game and then made up a board
overnight. However Hyde had described the game a century before this date. It seems
that this is not really a Snakes and Ladders game as the moves are determined by the
throw of the dice and the position -- there are no interconnections between cells. But
Culin notes that the game is complicated by being played for money or counters which
permit bribery and rewards, etc.
Culin. Chess and Playing Cards. Op. cit. in 4.A.4. 1898.
Pp. 820-822 & plates 24 & 25 between 821 & 822. Says Shing Kún T’ò (The
Game of the Promotion of Officials) is described by Hyde as The Game of the
Promotion of the Mandarins and refers to the above for an extended description.
Describes the Korean version: Tjyong-Kyeng-To (The Game of Dignitaries) and
several others from Korea and Tibet, with 108, 144, 169 and 64 squares.
Pp. 840-842 & plate 28, opp. p. 841 describes Chong ün Ch’au (Game of the
Chief of the Literati) as 'in many respects analogous' to Shing Kún T’ò and the
Japanese game Sugoroku (Double Sixes) -- in several versions. Then mentions modern
western versions -- Jeu de L'Oie, Giuoco dell'Oca, Juego de la Oca, Snake Game.
Pp. 843-848 is a table listing 122 versions of the game in the University of Pennsylvania
Museum of Archaeology and Paleontology. These are in 11 languages, varying from 22
to 409 squares.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Snakes and
Ladders and the Chinese Promotion Game, pp. 65-67. They describe the Hindu version
of Snakes and Ladders, called Moksha-patamu. Then they discuss Shing Kun t'o
(Promotion of the Mandarins), which was played in the Ming (1368-1616) with four or
more players racing on a board with 98 spaces and throwing 6 dice to see how many
equal faces appeared. They describe numerous modern variants.
Deepak Shimkhada. A preliminary study of the game of Karma in India, Nepal, and Tibet.
Artibus Asiae 44 (1983) 4. ??NYS - cited in Belloli et al, p. 68.
Andrew Topsfield. The Indian game of snakes and ladders. Artibus Asiae 46:3 (1985)
203-214 + 14 figures. Basically a catalogue of extant Indian boards. He says the game
SOURCES - page 45
is called Gyān caupad [the d should have an underdot] or Gyān chaupar in Hindi.
He states that Moksha-patamu sounds like it is Telugu and that this name appeared in
Grunfield's Games of the World (1975) with no reference to a source and that Bell has
repeated this. Game boards were drawn or painted on paper or cloth and hence were
perishable. The oldest extant version is believed to be an 84 square board of 1735, in
the Museum of Indology, Jaipur. There were Hindu, Jain, Muslim and Tibetan versions
representing a kind of Pilgrim's Progress, finally arriving at God or Heaven or Nirvana.
The number of squares varies from 72 to 360.
He gives many references and further details. An Indian version of the game was
described by F. E. Pargiter; An Indian game: Heaven or Hell; J. Royal Asiatic Soc.
(1916) 539-542, ??NYS. He cites the version by Ayres (and Love's reproduction of it -see below) as the first English version. He cites several other late 19C versions.
F. H. Ayres. [Snakes and ladders game.] No. 200682 Regd. Example in the Bethnal Green
Museum, Misc. 8 - 1974. Reproduced in: Brian Love; Play The Game; Michael
Joseph, London, 1978; Snakes & Ladders 1, pp. 22-23. This is the earliest known
English version of the game, with 100 cells in a spiral and 5 snakes and 5 ladders.
N. W. Bazely & P. J. Davis. Accuracy of Monte Carlo methods in computing finite Markov
chains. J. of Res. of the Nat. Bureau of Standards -- Mathematics and Mathematical
Physics 64B:4 (Oct-Dec 1960) 211-215. ??NYS -- cited by Davis & Chinn and
Bewersdorff. Bewersdorff [email of 6 Jun 1999] brought these items to my attention
and says it is an analysis based on absorbing Markov chains.
D. E. Daykin, J. E. Jeacocke & D. G. Neal. Markov chains and snakes and ladders. MG 51
(No. 378) (Dec 1967) 313-317. Shows that the game can be modelled as a Markov
process and works out the expected length of play for one player (47.98 moves) or two
players (27.44 moves) on a particular board with finishing rule A.
Philip J. Davis & William G. Chinn. 3.1416 and All That. S&S, 1969, ??NYS; 2nd ed,
Birkhäuser, 1985, chap. 23 (by Davis): "Mr. Milton, Mr. Bradley -- meet Andrey
Andreyevich Markov", pp. 164-171. Simply describes how to set up the Markov chain
transition matrix for a game with 100 cells and ending B. Doesn't give any results.
Lewis Carroll. Board game for one. In: Lewis Carroll's Bedside Book; ed. by Gyles
Brandreth (under the pseud. Edgar Cuthwellis); Methuen, 1979, pp. 19-21. ??look for
source; not in Carroll-Wakeling, Carroll-Wakeling II or Carroll-Gardner. Board of 27
cells with pictures in the odd cells. If you land on any odd cell, except the last one, you
have to return to square 1. "Sleep is almost certain to have overwhelmed the player
before he reaches the final square." Ending A is probably intended. (The average
duration of this game should be computable.)
S. C. Althoen, L. King & K. Schilling. How long is a game of snakes and ladders? MG 77
(No. 478) (Mar 1993) 71-76. Similar analysis to Daykin, Jeacocke & Neal, using
finishing rule B, getting 39.2 moves. They also use a simulation to find the number of
moves is about 39.1.
David Singmaster. Letter [on Snakes and ladders]. MG 79 (No. 485) (Jul 1995) 396-397. In
response to Althoen et al. Discusses history, other ending rules and wonders how the
length depends on the number of snakes and ladders.
Irving L. Finkel. Notes on two Tibetan dice games. IN: Alexander J. de Voogt, ed.; New
Approaches to Board Games Research: Asian Origins and Future Perspectives;
International Institute for Asian Studies, Leiden, 1995; pp. 24-47. Part II: The Tibetan
'Game of Liberation', pp. 34-47, discusses promotion games with many references to the
literature and describes a particular game.
Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse
und Grenzen. Vieweg, 1998. Das Leiterspiel, pp. 67-68 & Das Leiterspiel als
Markow-Kette. Discusses setting up the Markov chain, citing Bazley & Davis, with the
same board as in Davis & Chinn, then states that the average duration is 39.224
moves.
Jay Belloli, ed. The Universe A Convergence of Art, Music, and Science. [Catalogue for a
group of exhibitions and concerts in Pasadena and San Marino, Sep 2000 - Jun 2001.]
Armory Center for the Arts, Pasadena, 2001. P. 68 has a discussion of the Jain versions
of the game, called 'gyanbazi', with a colour plate of a 19C example with a 9 x 9 board
with three extra cells.
SOURCES - page 46
4.B.10.
MU TORERE
This is a Maori game which can be found in several books on board games. I have
included it because it has been completely analysed. There are eight (or 2n) points around a
central area. Each player has four (or n) markers, originally placed on consecutive points.
One can move from a point to an adjacent point or to the centre, or one can move from the
centre to a point, provided the position moved to is empty. The first player who cannot move
is the loser. To prevent the game becoming trivial, it is necessary to require that the first two
(or one) moves of each player involve his end pieces, though other restrictions are sometimes
given.
Marcia Ascher. Mu Torere: An analysis of a Maori game. MM 60 (1987) 90-100. Analyses
the game with 2n points. For n = 1, there are 6 inequivalent positions (where
equivalence is by rotation or reflection of the board) and play is trivially cyclic. For
n = 2, there are 12 inequivalent positions, but there are no winning positions. For
n = 3, there are 30 inequivalent positions, some of which are wins, but the game is a
tie. Obtains the number of positions for general n. For the traditional version with
n = 4, there are 92 inequivalent positions, some of which are wins, but the game is a
tie, though this is not at all obvious to an inexperienced player. In 1856, it was reported
that no foreigner could win against a Maori. For n = 5, there are 272 inequivalent
positions, but the game is a easy win for the first player -- the constraints on first moves
need to be revised. Ascher gives references to the ethnographic literature for
descriptions of the game.
Marcia Ascher. Ethnomathematics. Brooks/Cole Publishing, Pacific Grove, California, 1991.
Sections 4.4-4.7, pp. 95-109 & Notes 4-7, pp. 118-119. Amplified version of her MM
article.
4.B.11.
MASTERMIND, ETC.
There were a number of earlier guessing games of the Mastermind type before the
popular version devised by Marco Meirovitz in 1973 -- see: Reddi. One of these was the
English Bulls and Cows, but I haven't seen anything written on this and it doesn't appear in
Bell, Falkener or Gomme. Since 1975 there have been several books on the game and a
number of papers on optimal strategies. I include a few of the latter.
NOTATION. If there are h holes and c choices at each hole, then I abbreviate this as
ch.
A. K. Austin. How do You play 'Master Mind'. MTg 71 (Jun 1975) 46-47. How to state the
rules correctly.
S. S. Reddi. A game of permutations. JRM 8:1 (1975) 8-11. Mastermind type guessing of a
permutation of 1,2,3,4 can win in 5 guesses.
Donald E. Knuth. The computer as Master Mind. JRM 9:1 (1976-77) 1-6. 64 can be won in
5 guesses.
Robert W. Irving. Towards an optimum Mastermind strategy, JRM 11:2 (1978-79) 81-87.
Knuth's algorithm takes an average of 5804/1296 = 4.478 guesses. The author
presents a better strategy that takes an average of 5662/1296 = 4.369 guesses, but
requires six guesses in one case. A simple adaptation eliminates this, but increases the
average number of guesses to 5664/1296 = 4.370. An intelligent setter will choose a
pattern with a single repetition, for which the average number of guesses is 3151/720 =
4.376.
A. K. Austin. Strategies for Mastermind. G&P 71 (Winter 1978) 14-16. Presents Knuth's
results and some other work.
Merrill M. Flood. Mastermind strategy. JRM 18:3 (1985-86) 194-202. Cites five earlier
papers on strategy, including Knuth and Irving. He considers it as a two-person game
and considers the setter's strategy. He has several further papers in JRM developing his
ideas.
Antonio M. Lopez, Jr. A PROLOG Mastermind program. JRM 23:2 (1991) 81-93. Cites
Knuth, Irving, Flood and two other papers on strategy.
Kenji Koyama and Tony W. Lai. An optimal Mastermind strategy. JRM 25:4 (1994)
251-256. Using exhaustive search, they find the strategy that minimizes the expected
number of guesses, getting expected number 5625/1296 = 4.340. However, the worst
SOURCES - page 47
case in this problem requires 6 guesses. By a slight adjustment, they find the optimal
strategy with worst case requiring 5 guesses and its expected number of guesses is
5626/1296 = 4.341. 10 references to previous work, not including all of the above.
Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse
und Grenzen. Vieweg, 1998. Section 2.15 Mastermind: Auf Nummer sicher, pp. 227234 & Section 3.13 Mastermind: Farbcodes und Minimax, pp. 316-319. Surveys the
work on finding optimal strategies. Then studies Mastermind as a two-person game.
Finds the minimax strategy for the 32 game and describes Flood's approach.
4.B.12.
RITHMOMACHIA = THE PHILOSOPHERS' GAME
I have generally not tried to include board games in any comprehensive manner, but I
have recently seen some general material on this which will be useful to anyone interested in
the game. The game is one of the older and more mathematical of board games, dating from
c1000, but generally abandoned about the end of the 16C along with the Neo-Pythagorean
number theory of Boethius on which the game was based.
Arno Borst. Das mittelalterliche Zahlenkampfspiel. Sitzungsberichten der Heidelberger
Akademie der Wissenschaften, Philosophisch-historische Klasse 5 (1986) Supplemente.
Available separately: Carl Winter, Heidelberg, 1986. Edits the surviving manuscripts
on the game. ??NYS -- cited by Stigter & Folkerts.
Detlef Illmer, Nora Gädeke, Elisabeth Henge, Helen Pfeiffer & Monika Spicker-Beck.
Rhythmomachia. Hugendubel, Munich, 1987.
Jurgen Stigter. Emanuel Lasker: A Bibliography AND Rithmomachia, the Philosophers'
Game: a reference list. Corrected, 1988 with annotations to 1989, 1 + 15 + 16pp
preprint available from the author, Molslaan 168, NL-2611 CZ Delft, Netherlands.
Bibliography of the game.
Jurgen Stigter. The history and rules of Rithmomachia, the Philosophers' Game. 14pp
preprint available from the author, as above.
Menso Folkerts. 'Rithmimachia'. In: Die deutsche Litteratur des Mittelalters:
Verfasserlexikon; 2nd ed., De Gruyter, Berlin, 1990; vol. 8, pp. 86-94. Sketches history
and describes the 10 oldest texts.
Menso Folkerts. Die Rithmachia des Werinher von Tegernsee. In: Vestigia Mathematica, ed.
by M. Folkerts & J. P. Hogendijk, Rodopi, Amsterdam, 1993, pp. 107-142. Discusses
Werinher's work (12C), preserved in one MS of c1200, and gives an edition of it.
4.B.13.
MANCALA GAMES
This is a very broad field and I will only mention a few early items. Four row mancala
games are played in south and east Africa. Three row games are played in Ethiopia and
adjacent parts of Somaliland. Two row games are played everywhere else in Africa, the
Middle East and south and south-east Asia. See the standard books by R. C. Bell and
Falkener for many examples. Many general books mention the game, but I only know a few
specific books on the game -- these are listed first below.
One article says that game boards have been found in the pyramids of Khamit (-1580)
and there are numerous old boards carved in rocks in several parts of Africa.
An anonymous article, by a member of the Oware Society in London, [Wanted: skill,
speed, strategy; West Africa (16-22 Sep 1996) 1486-1487] lists the following names for
variants of the game: Aditoe (Volta region of Ghana), Awaoley (Côte d'Ivoire), Ayo (Nigeria),
Chongkak (Johore), Choro (Sudan), Congclak (Indonesia), Dakon (Philippines),
Guitihi (Kenya), Kiarabu (Zanzibar), Madji (Benin), Mancala (Egypt), Mankaleh (Syria),
Mbau (Angola), Mongola (Congo), Naranji (Sri Lanka), Qai (Haiti), Ware (Burkina Faso),
Wari (Timbuktu), Warri (Antigua),
Stewart Culin. Mancala, The National Game of Africa. IN: US National Museum Annual
Report 1894, Washington, 1896, pp. 595-607.
Chief A. O. Odeleye. Ayo A Popular Yoruba Game. University Press Ltd., Ibadan, Nigeria,
1979. No history.
Laurence Russ. Mancala Games. Reference Publications, Algonac, Michigan, 1984.
Photocopy from Russ, 1995.
Kofi Tall. Oware The Abapa Version. Kofi Tall Enterprise, Kumasi, Ghana, 1991.
SOURCES - page 48
Salimata Doumbia & J. C. Pil. Les Jeux de Cauris. Institut de Recherches Mathématiques,
08 BP 2030, Abidjan 08, Côte d'Ivoire, 1992.
Pascal Reysset & François Pingaud. L'Awélé. Le jeu des semailles africaines. 2nd ed.,
Chiron, Paris, 1995 (bought in Dec 1994). Not much history.
François Pingaud. L'awélé jeu de strategie africain. Bornemann, 1996.
Alexander J. de Voogt. Mancala Board Games. British Museum Press, 1997. ??NYR.
Larry (= Laurence) Russ. The Complete Mancala Games Book How to Play the World's
Oldest Board Games. Foreword by Alex de Voogt. Marlowe & Co., NY, 2000. His
1984 book is described as an earlier edition of this.
William Flinders Petrie. Objects of Daily Use. (1929); Aris & Phillips, London??, 1974.
P. 55 & plate XLVII. ??NYS -- described with plate reproduced in Bell, below. Shows
and describes a 3 x 14 board from Memphis, ancient Egypt, but with no date given, but
Bell indicates that the context implies it is probably earlier than -1500. Petrie calls it
'The game of forty-two and pool' because of the 42 holes and a large hole on the side,
apparently for storing pieces either during play or between games.
R. C. Bell. Games to Play. Michael Joseph (Penguin), 1988. Chap. 4, pp. 54-61, Mancala
games. On pp. 54-55, he shows the ancient Egyptian board from Petrie and his own
photo of a 3 x 6 board cut into the roof of a temple at Deir-el-Medina, probably about
-87.
Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford,
1694. De Ludo Mancala, pp. 226-232. Have X of part of this.
R. H. Macmillan. Wari. Eureka 13 (Oct 1950) 12. 2 x 6 board with each cup having four to
start. Says it is played on the Gold Coast.
Vernon A. Eagle. On some newly described mancala games from Yunnan province, China,
and the definition of a genus in the family of mancala games. IN: Alexander J. de
Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future
Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 48-62.
Discusses the game in general, with many references. Attempts a classification in
general. Describes six forms found in Yunnan.
Ulrich Schädler. Mancala in Roman Asia Minor? Board Games Studies International
Journal for the Study of Board Games 1 (1998) 10-25. Notes that mancala could have
been played on a flat board of two parallel rows of squares, i.e. something like a 2 x n
chessboard, but that archaeologists have tended to view such patterns as boards for race
games, etc. Describes 52 examples from Asia Minor. Some general discussion of
Greek and Roman games.
John Romein & Henri E. Bal (Vrije Universiteit, Amsterdam). New computer cluster solves
3500-year old game. Posted on www.alphagalileo.org on 29 Aug 2002. They show
that Awari is a tie game. They determined all 889,063,398,406 possible positions and
stored them in a 778 GByte database. They then used a 144 processor cluster to analyse
the graph, which 'only' took 51 hours.
4.B.14.
DOMINOES, ETC.
R. C. Bell. Games to Play. 1988. Op. cit. in 4.B.13. P. 136 gives some history. The
Académie Français adopted the word for both the pieces and the game in 1790 and it
was generally thought that they were an 18C invention. However, a domino was found
on the Mary Rose, which sank in 1545, and a record of Henry VIII (reigned 1509-1547)
losing £450 at dominoes has been found.
Bell, p. 131, describes the modern variant Tri-Ominos which are triangular pieces with values
at the corners. They were marketed c1970 and marked © Pressman Toy Corporation,
NY.
Hexadoms are hexagonal pieces with numbers on the edges -- opposite edges have the same
numbers. These were also marketed in the early 1970s -- I have a set made by Louis
Marx, Swansea, but there is no date on it.
4.B.15.
SVOYI KOSIRI
Anonymous [R. S. & J. M. B[rew ?]]. Svoyi kosiri is an easy game. Eureka 16 (Oct 1953)
8-12. This is an intriguing game of pure strategy commonly played in Russia and
SOURCES - page 49
introduced to Cambridge by Besicovitch. It translates roughly as 'One's own trumps'.
There are two players and the hands are exposed, with one's spades and clubs being the
same as the other's hearts and diamonds. At Cambridge, the cards below 6 are
removed, leaving 36 cards in the deck. The article doesn't explain how trumps are
chosen, but if one has spades as trumps, then the other has hearts as trumps! Players
alternate playing to a central discard pile. A player can take the pile and start a new pile
with any card, or he can 'cover' the top card and then play any card on that. 'Covering' is
done by playing a higher card of the same suit or one of the player's own trumps -- if
this cannot be done, e.g. if the ace of the player's own trumps has been played, the
player has to take the pile. The object is to get rid of all one's cards.
SOURCES - page 50
5.
COMBINATORIAL RECREATIONS
7.AZ is actually combinatorial rather than arithmetical and I may shift it.
5.A. THE 15 PUZZLE, ETC.
Pictorial versions: The Premier (1880), Lemon (1890), Stein (1898), King (1927).
Double-sided versions: The Premier (1880), Brown (1891).
Relation to Magic Squares: Loyd (1896), Cremer (1880), Tissandier (1880 & 1880?),
Cassell's (1881), Hutchison (1891).
Making a magic square with the Fifteen Puzzle: Dudeney (1898), Anon & Dudeney (1899),
Loyd (1914), Dudeney (1917), Gordon (1988). See also: Ollerenshaw & Bondi in
7.N.
GENERAL
Peter Hajek. 1995 report of his 1992 visit to the Museum of Money, Montevideo, Uruguay,
with later pictures by Jaime Poniachik. In this Museum is a metal chest made in
England in 1870 for the National State Bank of Uruguay. The front has a 7 x 7 array
of metal squares with bolt heads. These have to be slid in a 12 move sequence to reveal
the three keyholes for opening the chest. This opens up a whole new possible
background for the 15 Puzzle -- can anyone provide details of other such sliding
devices?
S&B, pp. 126-129, shows several versions of the puzzle.
L. Edward Hordern. Sliding Piece Puzzles. OUP, 1986. Chap. 2: History of the sliding block
puzzle, pp. 18-30. This is the most extensive survey of the history. He concludes that
Loyd did not invent the general puzzle where the 15 pieces are placed at random, which
became popular in 1879(?). Loyd may have invented the 14-15 version or he may have
offered the $1000 prize for it, but there is no evidence of when (1881??) or where.
However, see the entries for Loyd's Tit-Bits article and Dudeney's 1904 article which
seem to add weight to Loyd's claims. Most of the puzzles considered here are described
by Hordern and have code numbers beginning with a letter, e.g. E23, which I will give.
I contributed a note about computer techniques of solving such puzzles and
hoping that programmers would attack them as computer power increased.
In 1993-1995, I produced four Sliding Block Puzzle Circulars, totalling 24 pages (since
reformatted to 21), largely devoted to reporting on computer solutions of puzzles in
Hordern. Since then, a large number of solution programs have appeared and many
more puzzles have appeared. The best place to look is on Nick Baxter's Sliding Block
Home Page: http://www.johnrausch.com/slidingblockpuzzles/index.html .
EARLY ALPHABETIC VERSIONS
Embossing Co. Puzzle labelled "No. 2 Patent Embossed puzzle of Fifteen and Magic Sixteen.
Manufactured by the Embossing Co. Patented Oct 24 1865". Illustrated in S&B, p.
127. Examples are in the collections of Slocum and Hordern. Hordern, p. 25, says that
searching has not turned up such a patent.
Edward F. [but drawing gives E.] Gilbert. US Patent 91,737 -- Alphabetical Instruction
Puzzle. Patented 22 Jun 1869. 1p + 1p diagrams. Described by Hordern, p. 26. This is
not really a puzzle -- it has the sliding block concept, but along several tracks and with
many blank spaces. I recall a similar toy from c1950.
Ernest U. Kinsey. US Patent 207,124 -- Puzzle-Blocks. Applied: 22 Nov 1877; patented:
20 Aug 1878. 2pp + 1p diagrams. Described by Hordern, p. 27. 6 x 6 square sliding
block puzzle with one vacant space and tongue & grooving to prevent falling out. Has
letters to spell words. He suggests use of triangular and diamond-shaped pieces. This
seems to be the most likely origin of the Fifteen Puzzle craze.
Montgomery Ward & Co. Catalogue. 1889. Reproduced in: Joseph J. Schroeder, Jr.; The
Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield,
Illinois, 1977?, p. 34. Spelling Boards. Like Gilbert's idea, but a more compact layout.
SOURCES - page 51
LOYD
Loyd prize puzzle: One hundred pounds. Tit-Bits (14 Oct 1893) 25 & (18 Nov 1893) 111.
Loyd is described as "author of "Fifteen Puzzle," ...."
Loyd. Tit-Bits 31 (24 Oct 1896) 57. Loyd asserts he developed the 15 puzzle from a 4 x 4
magic square. "[The fifteen block puzzle] had such a phenomenal run some twenty
years ago. ... There was one of the periodical revivals of the ancient Hindu "magic
square" problem, and it occurred to me to utilize a set of movable blocks, numbered
consecutively from 1 to 16, the conditions being to remove one of them and slide the
others around until a magic square was formed. The "Fifteen Block Puzzle" was at
once developed and became a craze.
I give it as originally promulgated in 1872 ..." and he shows it with the 15 and 14
interchanged. "The puzzle was never patented" so someone used round blocks instead
of square ones. He says he would solve such puzzles by turning over the 6 and the 9.
"Sphinx" [= Dudeney] says he well remembers the sensation and hopes "Mr. Loyd is
duly penitent."
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the "Fifteen Puzzle" that in 1872 and
1873 was sold by millions, .... When this puzzle was brought out by its inventor, Mr.
Sam Loyd, ... he thought so little of it that he did not even take any steps to protect his
idea, and never derived a penny profit from it.... We have recently tried all over the
metropolis to obtain a single example of the puzzle, without success." Dudeney says
the puzzle came with 16 pieces and you removed the 16. He also says he recently could
not find a single example in London.
Loyd. The 14-15 puzzle in puzzleland. Cyclopedia, 1914, pp. 235 & 371 (= MPSL1,
prob. 21, pp. 19-20 & 128). He says he introduced it 'in the early seventies'. One
problem asks to move from the wrong position to a magic square with sum = 30 (i.e.
the blank is counted as 0). This is c= SLAHP, pp. 17-18 & 89.
G. G. Bain. Op. cit. in 1, 1907. Story of Loyd being unable to patent it.
Anonymous & Sam Loyd. Loyd's puzzles, op. cit. in 1, 1896. Loyd "owns up to the great sin
of having invented the "15 block puzzle"", but doesn't refer to the patent story or the
date.
W. P. Eaton. Loc. cit. in 1, 1911. Loyd refers to it as the 'Fifteen block' puzzle, but doesn't
say he couldn't patent it.
Loyd Jr. SLAHP. 1928. Pp. 1-3 & 87. "It was in the early 80's, ... that the world-disturbing
"14-15 Puzzle" flashed across the horizon, and the Loyds were among its earliest
victims." He gives many of the stories in the Cyclopedia and two of the same problems.
He doesn't mention the patent story.
THE 15 PUZZLE
W. W. Johnson. Notes on the 15-Puzzle -- I. Amer. J. Math. 2 (1879) 397-399.
W. E. Story. Notes on the 15-Puzzle -- II. Ibid., 399-404.
J. J. Sylvester. Editorial comment. Ibid., 404.
(This issue may have been delayed to early 1880?? Johnson & Story are not terribly
readable, but Sylvester is interesting, asserting that this is the first time that the parity of
a permutation has become a popular concept.)
Anonymous. Untitled editorial. New York Times (23 Feb 1880) 4. "... just now the chief
amusement of the New York mind, ... a mental epidemic .... In a month from now, the
whole population of North America will be at it, and when the 15 puzzle crosses the
seas, it is sure to become an English mania."
Anonymous. EUREKA! The Popular but Perplexing Problem Solved at Last. "THIRTEEN - FOURTEEN -- FIFTEEN" New York Herald (28 Feb 1880) 8. ""Fifteen" is a puzzle
of seeming simplicity, but is constructed with diabolical cunning. At first sight the
victim feels little or no interest; but if he stops for a single moment to try it, or to look at
any one else who is trying it, the mania strikes him. ... As to the last two numbers, it
depends entirely upon the way in which the blocks happen to fall in the first place ....
Two or three enterprising gamblers took up the puzzle and for a time made an excellent
living.... The subject was brought up in the Academy of Sciences by the veteran
scientist Dr. P. H. Vander Weyde", who showed it could not be solved. The Herald
SOURCES - page 52
reporter discovered that the problem is solvable if one turns the board 90o, i.e. runs the
numbers down instead of across, and Vander Weyde was impressed. The article
implies the puzzle had already been widely known for some time.
Mary T. Foote. US Patent 227,159 -- Game apparatus. Filed: 4 Mar 1880; patented: 4 May
1880. 1p + 1p diagrams. The patent is for a box with sliding numbered blocks for
teaching the multiplication tables. Lines 57-63: "I am aware that it is not novel to
produce a game apparatus in which blocks are to be mixed and then replaced by a series
of moves; also, that it is not novel to number such blocks, as in the "game of 15," so
called, where the fifteen numbers are first mixed and then moved into place."
Persifor Frazer Jr. Three methods and forty-eight solutions of the Fifteen Problem. Proc.
Amer. Philos. Soc. 18 (1878-1880) 505-510. Meeting of 5 Mar 1880. Rather cryptic
presentation of some possible patterns. Asserts his 26 Feb article in the Bulletin
(??NYS -- ??where -- Philadelphia??) was the first "solution for the 13, 15, 14 case".
J. A. Wales. 15 - 14 - 13 -- The Great Presidential Puzzle. Puck 7 (No. 158) (17 Mar 1880)
back cover.
Anonymous. Editorial: "Fifteen". New York Times (22 Mar 1880) 4. "No pestilence has
ever visited this or any other country which has spread with the awful celerity of what is
popularly called the "Fifteen Puzzle." It is only a few months ago that it made its
appearance in Boston, and it has now spread over the entire country." Asserts that an
unregenerate Southern sympathiser has introduced it into the White House and thereby
disrupted a meeting of President Hayes' cabinet.
Sch. [H. Schubert]. The Boss Puzzle. Hamburgischer Correspondent (= Staats- und Gelehrte
Zeitung des Hamburgischen unpartheyeischen Correspondent) No. 82 (6 Apr 1880) 11,
with response on 87 (11 Apr 1880) 12 (Sprechsaal). Gives a fairly careful description
of odd and even permutations and shows the puzzle is solvable if and only if it is in an
even permutation. The response is signed X and says that when the problem is
insoluble, just turn the box by 90o to see another side of the problem!
Gebr. Spiro, Hofliefer (Court supplier), Jungfernsteig 3(?--hard to read), Hamburg.
Hamburgischer Correspondent (= Staats- und Gelehrte Zeitung des Hamburgischen
unpartheyeischen Correspondent) No. 88 (13 Apr 1880) 7. Advertises Boss Puzzles:
"Kaiser-Spiel 50Pf. Bismarck-Spiel 50 Pf. Spiel der 15 u. 16, 50 Pf. Spiel der 16
separat, 15 Pf. System und Lösung, 20 Pf."
G. W. Warren. Letter: Clew to the Fifteen Puzzle. The Nation 30 (No. 774) (29 Apr 1880)
326.
Anon. Shavings. The London Figaro (1 May 1880) 12. "The "15 Puzzle," which has for
some months past been making a sensation in New York equal to that aroused by
"H. M. S. Pinafore" last year, has at length reached this country, and bids fair to become
the rage here also." (Complete item!)
George Augustus Sala. Echoes of the Week. Illustrated London News 76 (No. 2138) (22
May 1880) 491.
Mary T. Foote. US Patent 227,159 -- Game Apparatus. Applied: 4 Mar 1880; patented:
4 May 1880. 1p + 1p diagrams. Described in Hordern, p. 27. 3 x 12 puzzles based on
multiplication tables. Refers to the "game of 15" and Kinsey.
Arthur Black. ?? Brighton Herald (22 May 1880). ??NYS -- mentioned by Black in a letter
to Knowledge 1 (2 Dec 1881) 100.
Anonymous. Our latest gift to England. From the London Figaro. New York Times (11 Jun
1880) 2(?). ??page
The Premier. First (?) double-sided version, with pictures of Gladstone and Beaconsfield,
apparently produced for the 1880 UK election. Described in Hordern, pp. 32-33 &
plate I.
Ahrens. MUS II 227. 1918. Story of Reichstag being distracted in 1880.
P. G. Tait. Note on the Theory of the "15 Puzzle". Proc. Roy. Soc. Edin. 10 (1880) 664-665.
Brief but valid analysis. Mentions Johnson & Story. First mention of the possibility of
a 3D version.
T. P. Kirkman. Question 6489 and Note on the solution of the 15-puzzle in question 6489.
Mathematical Questions with their Solutions from the Educational Times 34 (1880)
113-114 & 35 (1881) 29-30. The question considers the n x n problem. The note is
rather cryptic. (No use??)
Messrs. Cremer (210 Regent St. and 27 New Bond St., London). Brilliant Melancholia.
Albrecht Durer's Game of the Thirty Four and "Boss" Game of the Fifteen. 1880.
Small booklet, 16pp + covers, apparently instructions to fit in a box with pieces
SOURCES - page 53
numbered 1 to 16 to be used for making magic squares as well as for the 15 puzzle.
Explains that only half the positions of the 15 puzzle are obtainable and describes them
by examples. (Photo in The Hordern Collection of Hoffmann Puzzles, p. 74, and in
Hordern, op. cit. above, plate IV.) Possibly written by "Cavendish" (Henry Jones).
H. Schubert. Theoretische Entscheidung über das Boss-Puzzle Spiel. 2nd ed., Hamburg,
1880. ??NYS (MUS, II, p. 227)
Gaston Tissandier. Les carrés magiques -- à propos du "Taquin," jeu mathématique. La
Nature 8 (No. 371) (10 Jul 1880) 81-82. Simple description of the puzzle called
'Taquin' which came from America and has had a very great success for several weeks.
Says it had 16 squares and was usable as a sliding piece puzzle or a magic square
puzzle. Cites Frénicle's 880 magic squares of order 4.
Anon. & C. Henry. Gaz. Anecdotique Littéraire, Artistique et Bibliographique. (Pub. by
G. d'Heylli, Paris) Year 5, t. II, 1880, pp. 58-59 & 87-92. ??NYS
Piarron de Mondésir. Le dernier mot du taquin. La Nature 8 (No. 382) (25 Sep 1880)
284-285. Simple description of parity decision for the 15 puzzle. Says 'la Presse
illustrée' offered 500 francs for achieving the standard pattern from a random pattern,
but it was impossible, or rather it was possible in only half the cases.
Jasper W. Snowdon. The "Fifteen" Puzzle. Leisure Hour 29 (1880) 493-495.
Gwen White. Antique Toys. Batsford, London, 1971; reprinted by Chancellor Press,
London, nd [1982?]. On p. 118, she says: "The French game of Taquin was played in
1880, in which 15 pieces had to be moved into 16 compartments in as few moves as
possible; the word 'taquin' means 'a teaser'." She gives no references.
Tissandier. Récréations Scientifiques. 1880?
2nd ed., 1881 -- unlabelled section, pp. 143-153. As: Le taquin et les carrés
magiques; seen in 1883 ed., ??NX; 1888: pp. 208-215. Adapted from the 1880 La
Nature articles of Tissandier and de Mondésir. 1881 says it came from America -'récemment une nouvelle apparition', but this is dropped in 1888 -- otherwise the two
versions are the same.
Translated in Popular Scientific Recreations, nd [c1890], pp. 731-735. Text says
"Mathematical games, ..., have recently obtained a new addition .... ... from America,
...." The references to contemporary reactions are deleted and the translation is
confused. E.g. the newspaper is now just "a French paper" and the English says the
problem is impossible in nine cases out of ten!
Lucas. Récréations scientifiques sur l'arithmétique et sur la géométrie de situation. Sixième
récréation: Sur le jeu du taquin ou du casse-tête américain. Revue scientifique de
France et de l'étranger (3) 27 (1881) 783-788. c= Le jeu du taquin, RM1, 1882, pp.
189-211. Revue says that Sylvester told him that it was invented 18 months ago by an
American deaf-mute. RM1 says "vers la fin de 1878". Cf Schubert, 1895.
Cassell's. 1881. Pp. 96-97: American puzzles "15" and "34". = Manson, pp. 246-248. Says
"articles ... have appeared in many periodicals, but no one has ... publish[ed] a solution."
Then sketches the parity concept and its application.
Richard A. Proctor. The fifteen puzzle. Gentlemen's Magazine 250 (No. 1801) (1881) 30-45.
"Boss". Letter: The fifteen puzzle. Knowledge 1 (11 Nov 1881) 37-38, item 13. This
magazine was edited by Proctor. The letter starts: "I am told that in a magazine article
which appeared some time ago, you have attempted to show that there are positions in
the Fifteen Puzzle from which the won position can never be obtained." I suspect the
letter was produced by Proctor. The response is signed Ed. and begins: "I thought the
Fifteen Puzzle was dead, and hoped I had had some share in killing the time-absorbing
monster." Notes that many people get to the position starting blank, 1, 2, 3 and view
this as a win. Sketches parity argument and suggests "Boss" work on the 3 x 3 or
3 x 2 or even the 2 x 2 version.
Editorial comment. The fifteen puzzle. Knowledge 1 (25 Nov 1881) 79. "I supposed every
one knew the Fifteen Puzzle." Proceeds to explain, obviously in response to readers
who didn't know it.
Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (2 Dec 1881) 100, item 80. Sketches
a proof which he says he published in the Brighton Herald of 22 May 1880.
"Yawnups". Letter: The fifteen puzzle. Knowledge 1 (30 Dec 1881) 185. Solution from the
15-14 position obtained by turning the box. Editorial comment says the solution uses
102 moves and the editor gets an easy solution in 57 moves. Adds that a 60 move
solution has been received.
Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (13 Jan 1882) 230. Finds a solution
SOURCES - page 54
from the 15-14 position in 39 moves by turning the box and asserts no shorter solution
is possible. Says he also gave this in the Brighton Herald in May 1880. An addition
says J. Watson has provided a similar solution, which takes 38 moves??
A. B. Letter: The fifteen puzzle. Knowledge 2 (20 Oct 1882) 345, item 598. Finds a boxturning solution in 39 moves.
C. J. Malmsten. Göteborg Handl 1882, p. 75. ??NYS -- cited by Ahrens in his Encyklopadie
article, op. cit. in 3.B, 1904.
Anonymous. Enquire Within upon Everything. Houlston and Sons, London. This was a
popular book with editions almost every year -- I don't know when the following
material was added. Section 2591: Boss; or the Fifteen Puzzle, p. 363. Place the pieces
'indifferently' in the box. Half the positions are unsolvable. Cites Cavendish for the
solution by turning the box 90o but notes this only works with round pieces. Goes on to
The thirty-four puzzle, citing Dürer. I found this material in the 66th ed., 862nd
thousand, of 1883, but I didn't find the material in the 86th ed of 1892.
Letters received and short answers. Knowledge 4 (16 Nov 1883) 310. 'Impossible'.
P. G. Tait. Listing's Topologie. Philosophical Mag. (5) 17 (No. 103) (Jan 1884) 30-46 &
plate opp. p. 80. Section 11, p. 39. Simple but cryptic solution.
Letters received and short answers. Letter from W. S. B. asks how to solve the problem when
the last row has 13, 14, 15 [sic!]; Answer by Ed. points out the misprint and says the
easiest solution is to remove the 15 and put it after the 14, or to invert the 6 and 9.
Knowledge 6 (No. 159) (14 Nov 1884) 412 & 6 (No. 160) (21 Nov 1884) 429.
Don Lemon. Everybody's Pocket Cyclopedia .... Saxon & Co., London, (1888), revised 8th
ed., 1890. P. 137: The fifteen puzzle. Brief description, with pieces placed randomly in
the box -- "to get the last three into order is often a puzzle indeed".
John D. Champlin & Arthur E. Bostwick. The Young Folk's Cyclopedia of Games and
Sports. 1890. ??NYS Cited in Rohrbough; Brain Resters and Testers; c1935; Fifteen
Puzzle, p. 20. Describes idea of parity of number of exchanges. [Another reference
provided more details of Champlin & Bostwick.]
Lemon. 1890. A trick puzzle, no. 202, pp. 31 & 105 (= Sphinx, no. 422, pp. 60 & 112). 15
puzzle with lines on the pieces to arrange as "a representation of a president with only
one eye". The solution is a spelling of the word 'president'. Attributed to Golden Days - ??. After The Premier puzzle of c1880, this is the second suggestion of using a picture
and the first publication of the idea that I have seen.
G. A. Hutchison, ed. Indoor Games and Recreations. The Boy's Own Bookshelf. (1888);
New ed., Religious Tract Society, London, 1891. (See M. Adams; Indoor Games for a
much revised version, but which doesn't contain this material.) Chap. 19: The
American Puzzles., pp. 240-241. "These puzzles, known as the 'Thirty-four Game' and
the 'Fifteen Game,' on their introduction amongst us some years ago ...." "The '15'
puzzle would appear to have been, on its coming to England a few years ago, strictly a
new introduction ...." He sketches the parity concept. [NOTE. I have seen a reference
to the editor as Hutchinson, but the book definitely omits the first n.]
Daniel V. Brown. US Patent 471,941 -- Puzzle. Applied: 23 Apr 1891; patented: 29 Mar
1892, 2pp + 1p diagrams. Double-sided 16 block puzzle to spell George Washington
on one side and Benjamin Harrison on the other. No sliding involved.
Berkeley & Rowland. Card Tricks and Puzzles. 1892. American fifteen puzzle, pp. 105-107.
"The Fifteen Puzzle was introduced by a shrewd American some ten years ago, ...."
Refers to Tait's 1880 paper. Says half the positions are impossible, but solves them by
turning the box 90o or by inverting the 6 and the 9.
Hoffmann. 1893. Chap IV, no. 69: The "Fifteen" or "Boss" puzzles, pp. 161-162 & 217-218
= Hoffmann-Hordern, pp. 142-144, with photo of five early examples, two or three of
which also are thirty-four puzzles. (Hordern Collection, p. 74, has a photo of a version
by Cremer, cf above.) "This, like a good many of the best puzzles, hails from America,
where, some years ago, it had an extraordinary vogue, which a little later spread to this
country, the British public growing nearly as excited over the mystic "Fifteen" as they
did at a later date over the less innocent "Missing Word" competitions." He
distinguishes between the ordinary Fifteen where one puts the pieces in at random, and
the Boss or Master puzzle which has the 14 and 15 reversed. "Notwithstanding the
enormous amount of energy that has been expended over the "Fifteen" Puzzle, no
absolute rule for its solution has yet been discovered and it appears to be now generally
agreed by mathematicians that out of the vast number of haphazard positions ... about
half admit [of solution]. To test whether ... the following rule has been suggested." He
SOURCES - page 55
then says to count the parity of the number of transpositions.
Hoffmann. 1893. Chap. IV, no. 70: The peg-away puzzle, pp. 163 & 218
= Hoffmann-Hordern, p. 145. This is a 3 x 3 version of the Fifteen puzzle, made by
Perry & Co. Start with a random pattern and get to standard form. "The possibility of
success in solving this puzzle appears to be governed by precisely the same rule as the
"Fifteen" Puzzle." Hoffmann-Hordern has no photo of this -- do any examples exist??
H. Schubert. Zwölf Geduldspiele. Dümmler, Berlin, 1895. [Taken from his columns in
Naturwissenschaftlichen Wochenschrift, 1891-1894.] Chap. VII: Boss-Puzzle oder
Fünfzehner-Spiel, pp. 75-94?? Pp. 75-77 sketches the history, saying it was called "Jeu
du Taquin" (Neck-Spiel) in France and was popular in 1879-1880 in Germany. Cites
Johnson & Story and his own 1880 booklet. Gives the story of a deaf and dumb
American inventing it in Dec 1878, saying "Sylvester communicated this at the annual
meeting of the Association Française pour l'Avancement des Sciences at Reims".
Cf Lucas, 1881. [There is a second edition, Teubner??, Leipzig, 1899, ??NYS.
However this material is almost identical to the beginning of Chap. 15 in Schubert's
Mathematische Mussestunden, 3rd ed., Göschen, Leipzig, 1909, vol. 2. The later
version omits only some of the Hamburg details of 1879-1880. Hence the 2nd ed. of
Zwölf Geduldspiele is probably very close to these versions.]
Dudeney. Problem 49: The Victoria Cross puzzle. Tit-Bits 32 (4 & 25 Sep 1897) 421 &
475. = AM, 1917, prob. 218, pp. 60 & 194. B7. 3 x 3 board with letters Victoria
going clockwise around the edges, leaving the middle empty, and starting with V in a
corner. Slide to get Victoria starting at an edge cell, in the fewest moves. Does it in 18
moves, by interchanging the i's and says there are 6 such solutions.
Dudeney. Problem 65: The Spanish dungeon. Tit-Bits 33 (1 Jan & 5 Feb 1898) 257 & 355.
= AM, 1917, prob. 403, pp. 122-123 & 244. B14. Convert 15 Puzzle, with pieces in
correct order, into a magic square. Does it in 37 moves.
Conrad F. Stein. US Design 29,649 -- Design for a Game-Board. Applied: 29 Sep 1898;
patented: 8 Nov 1898 as No. 692,242. 1p + 1p diagrams. This appears to be a 3 x 4
puzzle with a picture of a city with a Spanish flag on a tower. Apparently the object is
to move an American flag to the tower.
Anon. & Dudeney. A chat with the Puzzle King. The Captain 2 (Dec? 1899) 314-320; 2:6
(Mar 1900) 598-599 & 3:1 (Apr 1900) 89. The eight fat boys. 3 x 3 square with
pieces: 1 2 3; 4 X 5; 6 7 8 to be shifted into a magic square. Two solutions in 19
moves. Cf Dudeney, 1917.
Addison Coe. US Patent 785,665 -- Puzzle or Game Apparatus. Applied: 17 Nov 1904;
patented: 1 Mar 1905. 4pp + 3pp diagrams. Gives a 3 x 5 flat version and a
3-dimensional version -- cf 5.A.2.
Dudeney. AM. 1917.
Prob. 401: Eight jolly gaol birds, pp. 122 & 243. E23. Same as 'The eight fat boys' (see
Anon. & Dudeney, 1899) with the additional condition that one person refuses to
move, which occurs in one of the two previous solutions.
Prob. 403: The Spanish dungeon, pp. 122-123 & 244. = Tit-Bits prob. 65 (1898). B14.
Prob. 404: The Siberian dungeons, pp. 123 & 244. B16. 2 x 8 array with prisoners
1, 2, ..., 8 in top row and 9, 10, ..., 16 in bottom row. Two extra rows of 4
above the right hand end (i.e. above 5, 6, 7, 8) are empty. Slide the prisoners
into a magic square. Gives a solution in 14 moves, due to G. Wotherspoon,
which they feel is minimal. This allows long moves -- e.g. the first move moves
8 up two and left 3.
"H. E. Licks" [pseud. of Mansfield Merriman]. Recreations in Mathematics. Van Nostrand,
NY, 1917. Art. 28, pp. 20-21. 'About the year 1880 ... invented in 1878 by a deaf and
dumb man....'
[From sometime in the 1980s, I suspected the author's name was a pseudonym.
On pp. 132-138, he discusses the Diaphote Hoax, from a Pennsylvania daily newspaper
of 10 Feb 1880, which features the following people: H. E. Licks, M. E. Kannick, A. D.
A. Biatic, L. M. Niscate. The diaphote was essentially a television. He says this report
was picked up by the New York Times and the New York World. An email from Col.
George L. Sicherman on 5 Jun 2000 agrees that the name is false and suggested that the
author was "the eminent statistician Mansfield Merriman" who wrote the article on The
Cattle Problem of Archimedes in Popular Science Monthly (Nov 1905), which is
abridged on pp. 33-39 of the book, but omitting the author's name. Sichermann added
that Merriman was one of the authors of Pillsbury's List. William Hartston says this
SOURCES - page 56
was an extraordinary list of some 30 words which Pillsbury, who did memory feats, was
able to commit to memory quite rapidly. Sicherman continued to investigate Merriman
and got Prof. Andri Lange interested and Lange corresponded with a James A.
McLennan, author of a history of the physics department at Lehigh University where
Merriman had been. McLennan found Merriman's obituary from the American Society
of Civil Engineers which states that Merriman used H. E. Licks as a pseudonym.
[Email from Sicherman on 25 Feb 2002.]]
Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane,
The Bodley Head, 1939, p. 300. "But this puzzle stuff, as I say, is as old as human
thought. As soon as mankind began to have brains they must have loved to exercise
them for exercise' sake. The 'jig-saw' puzzles come from China where they had them
four thousand years ago. So did the famous 'sixteen puzzle' (fifteen movable squares
and one empty space) over which we racked our brains in the middle eighties."
G. Kowalewski. Boss-Puzzle und verwandte Spiele. K. F. Kohler Verlag, Leipzig, 1921
(reprinted 1939). Gives solution of general polygonal versions, i.e. on a graph with a
Hamilton circuit and one or more diagonals.
Hummerston. Fun, Mirth & Mystery. 1924.
1
2
9
Push, pp. 22 & 25. This is played on the board
shown at the left with its orthogonal lines, like
12
13
3, 10, 11, 4, and its diagonal lines, like
5 14 15 6
1, 9, 11, 13, 6. 10, 15 and 11, 14 are not
16
connected, so this is an octagram. Take 16
7 8
numbered counters and place them at random on
the board and remove counter 16. Move the pieces
to their correct locations. He asserts that 'unlike the original ["Sixteen" Puzzle],
no position can be set up in "Push" that cannot be solved'.
The six bulls puzzle, Puzzle no. 34, pp. 90 & 177. This uses the 2 x 3 + 1
0
board shown at the right, where the 0 is the blank space. Exchange
1 2 3
3 and 6 and 4 and 5. He does it in 20 moves. [This is Hordern's
4 5 6
B3, first known from 1977 under the name Bull Pen, but is a variant of
Hordern's B2, first known from 1973.]
Q. E. D. -- The sergeant's problem, Puzzle no. 40, pp. 106 & 178. Take a 2 x 3 board, with
the centre of one long side blank. Interchange the men along one short side. He does
this in 17 moves, but the blank is not in its initial position nor are the other men. [This
is Hordern's B1, first known from Loyd's Cyclopedia, 1914.]
King. Best 100. 1927. No. 26, p. 15. = Foulsham's, no. 9, pp. 7 & 10. "An entertaining
variation ... is to draw, and colour, if you like, a small picture; then cut it into sixteen
squares and discard the lower right hand square."
G. Kowalewski. Alte und neue mathematische Spiele. Teubner, Leipzig, 1930, pp. 61-81.
Gives solution of general polygonal versions.
Dudeney. PCP. 1932. The Angelica puzzle, prob. 253, pp. 76 & 167. = 435, prob. 378, pp.
136 & 340. B8. 3 x 3 problem -- convert: A C I L E G N A X to A N G E L I
C A X. Requires interchanging the As. Solution in 36 moves. In the answer in 435,
Gardner notes that it can be done in 30 moves.
H. V. Mallison. Note 1454: An array of squares. MG 24 (No. 259) (May 1940) 119-121.
Discusses 15 Puzzle and says any legal position can be achieved in at most about 150
moves. But if one fixes cells 6, 7, 11, then a simple problem requires about 900
moves.
McKay. At Home Tonight. 1940. Prob. 44: Changing the square, pp. 73 & 88. In the usual
formation, colour the pieces alternately blue and red, as on a chessboard, with the blank
at the lower right position 16 being a missing red, so there are 7 reds. Move so the
colours are still alternating but the blank is at the lower left, i.e. position 13. Takes 15
moves.
Sherley Ellis Stotts. US Patent 3,208,753 -- Shiftable Block Puzzle Game. Filed: 7 Oct 1963;
patented: 28 Sep 1965. 4pp + 2pp diagrams. Described in Hordern, pp. 152-153,
F10-12. Rectangular pieces of different sizes. One can also turn a piece.
Gardner. SA (Feb 1964) = 6th Book, chap. 7. Surveys sliding-block puzzles with non-square
pieces and notes there is no theory for them. Describes a number of early versions and
the minimum number of moves for solution, generally done by hand and then confirmed
3
10
11
4
SOURCES - page 57
by computer. Pennant Puzzle, C19; L'Âne Rouge, C27d; Line Up the Quinties, C4;
Ma's Puzzle, D1; a form of Stotts' Baby Tiger Puzzle, F10.
Gardner. SA (Mar & Jun 1965) c= 6th Book, chap. 20. Prob. 9: The eight-block puzzle. B5.
3 x 3 problem -- convert: 8 7 6 5 4 3 2 1 X to 1 2 3 4 5 6 7 8 X. Compares it
with Dudeney's Angelica puzzle (1932, B8) but says it can be done if fewer than 36
moves. Many readers found solutions in 30 moves; two even found all 10 minimal
solutions by hand! Says Schofield (see next entry) has been working on this and gives
the results below, but this did not quite resolve Gardner's problem. William F.
Dempster, at Lawrence Radiation Laboratory, programmed a IBM 7094 to find all
solutions, getting 10 solutions in 30 moves; 112 in 32 moves and 512 in 34
moves. Notes it is unknown if any problem with the blank in a side or corner requires
more than 30 moves. (The description of Schofield's work seems a bit incorrect in the
SA solution, and is changed in the book.)
Peter D. A. Schofield. Complete solution of the 'Eight-Puzzle'. Machine Intelligence 1
(1967) 125-133. This is the 3 x 3 version of the 15 Puzzle, with the blank space in the
centre. Works with the corner twists which take the blank around a 2 x 2 corner in
four moves. Shows that the 5-puzzle, which is the 3 x 2 version, has every position
reachable in at most 20 moves, from which he shows that an upper bound for the 8puzzle is 48 moves. Since the blank is in the middle, the 8!/2 = 20160 possible
positions fall into 2572 equivalence classes. He also considers having inverse
permutations being equivalent, which reduces to 1439 classes, but this was too
awkward to implement. An ATLAS program found that the maximum number of
moves required was 30 and 60 positions of 12 classes required this maximum
number, but no example is given -- but see previous entry.
A. L. Davies. Rotating the fifteen puzzle. MG 54 (No. 389) (Oct 1970) 237-240. Studies
versions where the numbers are printed diagonally so one can make a 90o turn of the
puzzle. Then any pattern can be brought to one of two 'natural' patterns. He then asks
when this is true for an m x n board and obtains a complicated solution. For an n x n
board, n must be divisible by 4.
R. M. Wilson. Graph puzzles, homotopy and the alternating group. J. Combinatorial Thy.,
Ser. B, 16 (1974) 86-96. Shows that a sliding block puzzle, on any graph of n + 1
points which is non-separable and not a cycle, has at least An as its group -- except for
one case on 7 points.
Alan G. & Dagmar R. Henney. Systematic solutions of the famous 15-14 puzzles. Pi Mu
Epsilon J. 6 (1976) 197-201. They develop a test-value which significantly prunes the
search tree. Kraitchik gave a problem which took him 114 moves -- the authors show
the best solution has 58 moves!
David Levy. Computer Gamesmanship. Century Publishing, London, 1983. [Most of the
material appeared in Personal Computer World, 1980-1981.] Pp. 16-29 discusses
8-puzzle and uses the Henney's test-value as an evaluation function. Cites Schofield.
Nigel Landon & Charles Snape. A Way with Maths. CUP, 1984. Cube moving, pp. 23 & 46.
Consider a 9-puzzle in the usual arrangement: 1 2 3, 4 5 6, 7 8 x. Move the 1 to the
blank position in the minimal number of moves, ignoring what happens to the other
pieces. Generalise. Their answer only says 13 is minimal for the 3 x 3 board.
My student Tom Henley asked me the m x n problem in 1993 and gave a
conjectural minimum, which I have corrected to: if m = n, then it can be done in
8m - 11 moves; but if n < m, then it can be done in 6m + 2n - 13 moves, using a
straightforward method. However, I don't see how to show this is minimal, though it
seems pretty clear that it must be. I call this a one-piece problem. See also Ransom,
1993.
Len Gordon. Sliding the 15-1 [sic, but 15-14 must have been meant] puzzle to magic squares.
G&PJ 4 (Mar 1988) 56. Reports on computer search to find minimal moves from
either ordinary or 15-14 forms to a magic square. However, he starts with the blank
before the 1, i.e. as a 0 rather than a 16.
Leonard J. Gordon. The 16-15 puzzle or trapezeloyd. G&PJ 10 (1989) 164. Introduces his
puzzle which has a trapezoidal shape with a triangular wedge in the 2nd and 3rd row so
the last row can hold 5 pieces, while the other rows hold four pieces. Reversing the last
two pieces can be done in 85 moves, but this may not be minimal.
George T. Gilbert & Loren C. Larson. A sliding block problem. CMJ 23:4 (Sep 1992)
315-319. Essentially the same results as obtained by R. Wilson (1974). Guy points this
out in 24:4 (Sep 1993) 355-356.
SOURCES - page 58
P. H. R. [Peter H. Ransom]. Adam's move. Mathematical Pie 128 (Spring 1993) 1017 &
Notes, p. 3. Considers the one piece problem of Langdon & Snape, 1984. Solution
says the minimal solution on a n x n board is 8n - 11, but doesn't give the answer for
the m x n board.
Bernhard Wiezorke. Sliding caution. CFF 32 (Aug 1993) 24-25 & 33 (Feb 1994) 32. In
1986, the German games company ASS (Altenburg Stralsunder Spielkarten AG)
produced a game called Vorsicht (= Caution). Basically this is a 3 x 3 board
considered as a doubly crossed square. It has pieces marked with + or x. The +
pieces can only move orthogonally; the x pieces can only move diagonally. The
pieces are coloured and eight are placed on the board to be played as a sliding piece
puzzle from given starts to given ends. The diagonal moves are awkward to make and
Wiezorke suggests the board be spread out enough for diagonal moves to be made. A
note at the end says he has received two similar games made by Y. A. D. Games in
Israel.
Bala Ravikumar. The Missing Link and the Top-Spin. Report TR94-228, Department of
Computer Science and Statistics, University of Rhode Island, Jan 1994. The Missing
Link is a cylindrical form of the Fifteen Puzzle, with four layers and four pieces in each
layer. The middle two layers are rigidly joined, but that makes little difference in
solving the puzzle. After outlining the relevant group theory and solving the Fifteen
Puzzle, he shows the state space of the Missing Link is S15.
Richard E. Korf & Ariel Felner. Disjoint pattern database heuristics. Artificial Intelligence
134 (2002) 9-22. Discusses heuristic methods of solving the Fifteen Puzzle, Rubik's
Cube, etc. The authors applied their method to 1000 random positions of the Fifteen
Puzzle. The optimal solution length averaged 52.522 and the average time required was
27 msec. They also did 50 random positions of the Twenty-Four Puzzle and found an
average optimal solution length of 100.78, with average time being two days on a
440MHz machine.
5.A.1.NON-SQUARE PIECES
S&B, pp. 130-133, show many versions.
See Kinsey, 1878, above, for mention of triangular and diamond-shaped pieces.
Henry Walton. US Patent 516,035 -- Puzzle. Applied: 14 Mar 1893; patented: 6 Mar 1894.
1p + 1p diagrams. Described in Hordern, pp. 27 & 68-69, C1. 4 x 4 area with five
1 x 2 & two 2 x 1 pieces.
Lorman P. Shriver. US Patent 526,544 -- Puzzle. Applied: 28 Jun 1894; patented: 25 Sep
1894, 2pp + 1p diagrams. Described in Hordern, p. 27. 4 x 5 area with two 2 x 1 &
15 1 x 1 pieces. Because there is only one vacant space, the rectangles can only move
lengthwise and so this is a dull puzzle.
Frank E. Moss. US Patent 668,386 -- Puzzle. Applied: 8 Jun 1900; patented: 19 Feb 1901.
2pp + 1p diagrams. Described in Hordern, pp. 27-28 & 75, C14. 4 x 4 area with six
1 x 1, two 1 x 2 & two 2 x 1 pieces, allowing sideways movement of the rectangles.
William H. E. Wehner. US Patent 771,514 -- Game Apparatus. Applied: 15 Feb 1904;
patented: 4 Oct 1904. 2pp + 1p diagrams. First to use L-shaped pieces. Described in
Hordern, pp. 28 & 107, D5.
Lewis W. Hardy. US Patent 1,017,752 -- Puzzle. Applied: 14 Dec 1907; patented: 20 Feb
1912. 3pp + 1p diagrams. Described in Hordern, pp. 29 & 89-90, C43-45. 4 x 5 area
with one 2 x 2, two 1 x 2, three 2 x 1 & four 1 x 1 pieces.
L. W. Hardy. Pennant Puzzle. Copyright 1909. Made by OK Novelty Co., Chicago. No
known patent. Described in Gardner, SA (Feb 1964) = 6th Book, chap. 7 and in
Hordern, pp. 28-29 & 78-79, C19. 4 x 5 area with one 2 x 2, two 1 x 2, four 2 x 1,
two 1 x 1 pieces.
Nob Yoshigahara designed Rush Hour in the late 1970s and it was produced in Japan as
Tokyo Parking Lot. Binary Arts introduced it to the US in 1996 and it became very
popular.
Winning Ways. 1982. Pp. 769-777: A trio of sliding block puzzles. This covers Dad's
Puzzler (c19, with piece 8 moved two places to the right), The Donkey (C27d, with all
the central pieces moved down one position) and The Century (C42), showing how one
can examine partial problems which allow one to consider many positions the same and
much reduce the number of positions to be studied. This allows the graph to be written
on a large sheet and solutions to be readily found.
SOURCES - page 59
Andrew N. Walker. Checkmate and other sliding-block puzzles. Mathematics Preprint
Series, University of Nottingham, no. 95-32, 1995, 8pp + covers. Describes a version
by W. G. H. [Wil] Strijbos made by Pussycat. 4 x 4 with an extra position below the
left column. Pieces are alternately black and white and have a black king, a white king
and a white rook on them and the object is to produce checkmate, but all positions must
be legal in chess, except that the black and white markings do not have to be correct in
the intermediate positions. However, one soon finds that one tile is fixed in place and
two other tiles are joined together. He discusses general computer solving techniques
and finds there are five optimal solutions in 68 moves. He then discusses other
problems, citing Winning Ways, Hordern and my Sliding Block Puzzle Circulars. He
gives the UNIX shell scripts that he used.
Ivars Peterson. Simple puzzles can give computers an unexpectedly strenuous workout.
Science News 162:7 (17 Aug 2002) 6pp PO from their website, http:''sciencenews.org .
Reports on recent work by Gary W. Flake & Eric B. Baum that Nob Yoshigahara's
Rush Hour puzzle is PSPACE complete, but is not polynomial time. Robert A. Hearn
and Erik D. Demaine have verified and extended this, showing other sliding block
puzzles are PSPACE complete, including the case where all pieces are dominoes and
can slide sideways as well as front and back.
5.A.2.THREE DIMENSIONAL VERSIONS
See Hordern, pp. 27, 156-160 & plates IX & X.
P. G. Tait. Note on the Theory of the "15 Puzzle". Proc. Roy. Soc. Edin. 10 (1880) 664-665.
"... conceivable, but scarcely realisable ..."
Charles I. Rice. US Patent 416,344 -- Puzzle. Applied: 9 Sep 1889; patented: 3 Dec 1889.
2pp + 1p diagrams. Described in Hordern, pp. 27 & 157-158, G2. 2 x 2 x 2 version
with peepholes in the faces.
Ball. MRE, 1st ed., 1892, p. 78. Mentions possibility.
Hoffmann. 1893. Chap. X, No. 1: The John Bull political puzzle, pp. 331 & 357-358
= Hoffmann-Hordern, pp. 215-216. A 3 x 3 board in the form of a cylinder, with an
extra cell attached to one bottom cell. Pieces can move back and forth around each
level, but the connections from one level to the next are all parallel to one of the
diagonals -- though this isn't really a complication compared to having vertical
connections. The pieces have two markings: three colours and three letters. When they
are randomly placed on the board, you have to move them so they form a pair of
orthogonal 3 x 3 Latin squares. Fortunately there are such arrangements which differ
by an odd permutation, so the puzzle can be solved from any random starting point.
Two examples done. Says the game is produced by Jaques & Son.
Addison Coe. US Patent 785,665 -- Puzzle or Game Apparatus. Applied: 17 Nov 1904;
patented: 1 Mar 1905. 4pp + 3pp diagrams. Mentioned in Hordern, pp. 158-159, G3.
Gives a 3 x 5 flat version and a 3 x 3 x 3 cubical version with 3 x 3 arrays of holes
in the six faces (in order to push the pieces) and a 3 x 5 cylindrical version.
Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY,
1909. The nine disks, pp. 29-34 & 60. Same as Hoffmann except pieces have colour
and shape.
Guy thinks Hein patented Bloxbox, but I have not found any US patent of it -- ??CHECK.
Gardner. SA (Feb 1973). First mention of Hein's Bloxbox.
Daniel Kosarek. US Patent 3,845,959 -- Three-Dimensional Block Puzzle. Filed: 14 Nov
1973; patented: 5 Nov 1974. 3pp + 1p diagrams (+ 1p abstract). Mentioned in
Hordern, pp. 158-159, G3. 3 x 3 x 3 box with 3 x 3 array of portholes on each face.
Mentions 4 x 4 x 4 and larger versions.
Gabriel Nagorny. US Patent 4,428,581 -- Tri-dimensional Puzzle. Filed: 16 Jun 1981;
patented: 31 Jan 1984. Cover page + 3pp + 3pp diagrams. Three dimensional sliding
cube puzzles with central pieces joined together. A 3 x 3 x 3 version was made in
Hungary and marketed as a Varikon Box. Inventor's address is in France and he cites
earlier French applications of 19 Jun 1980 and 19 Nov 1980. He also describes a
3 x 4 x 4 version with the central areas of each face joined to a 1 x 2 x 2 block in the
middle.
5.A.3.ROLLING PIECE PUZZLES
SOURCES - page 60
Here one has a set of solid pieces in a tray and one tilts or rolls a piece into the blank
space.
Thomas Henry Ward.
UK Patent 2,870 -- Apparatus for Playing Puzzle or Educational Games. Provisional: 8 Jun
1883; Complete as: An Improved Apparatus to be Employed in Playing Puzzle or
Educational Games, 6 Dec 1883. 3pp + 1p diagrams.
US Patent 287,352 -- Game Apparatus. Applied: 13 Sep 1883; patented: 23 Oct 1883. 1p +
1p diagrams. Hexagonal board of 19 triangles with 18 tetrahedra to tilt.
George Mitchell & George Springfield. UK Patent 6867 -- A novel puzzle, and
improvements in the construction of apparatus therefor. Applied: 16 Mar 1897;
accepted: 5 Jun 1897. 2pp + 1p diagrams. Rolling cubes puzzle, where the cube faces
are hollowed and fit onto domes in the tray. Basic form has four cubes in a row with
two extra spaces above the middle cubes, but other forms are shown.
Sven Bergling invented the rolling ball labyrinth puzzle/game and they began to be produced
in 1946. [Kenneth Wells; Wooden Puzzles and Games; David & Charles, Newton
Abbot, 1983, p. 114.]
Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by
T. H. O'Beirne as: Recreations in Mathematics, Blackie, London, 1963. Problem 3:
Schwere Kiste, pp. 3-4 & 22-23 (= Heavy boxes, pp. 4-5 & 25-26). Three problems
with 5 boxes some of which are so heavy that one has to tilt or roll them.
Gardner. SA (Dec 1963). = Sixth Book, chap. 8. Gives Sprague's first problem.
Gardner. SA (Nov 1965). c= Carnival, chap. 9. Prob. 1: The red-faced cube. Two problems
of John Harris involving one cube with one red face rolling on a chessboard. Gardner
says that the field is new and that only Harris has made any investigations of the
problem. The book chapter cites Harris's 1974 article, below, and a 1971 board game
called Relate with each player having four coloured cubes on a 4 x 4 board.
Charles W. Trigg. Tetrahedron rolled onto a plane. JRM 3:2 (Apr 1970) 82-87. A
tetrahedron rolled on the plane forms the triangular lattice with each cell corresponding
to a face of the tetrahedron. He also considers rolling on a mirror image tetrahedron
and rolling octahedra.
John Harris. Single vacancy rolling cube problems. JRM 7:3 (1974) 220-224. This seems to
be the first appearance of the problem with one vacant space. He considers cubes
rolling on a chessboard. Any even permutation of the pieces with the blank left in place
is easily obtained. From the simple observation that each roll is an odd permutation of
the pieces and an odd rotation of the faces of a cube, he shows that the parity of the
rotation of a cube is the same as the parity of the number of spaces it has moved. He
shows that any such rotation can be achieved on a 2 x 3 board. Rotating one cube 120o
about a diagonal takes 32 moves. If the blank is allowed to move, the the parity of the
permutation of the pieces is the parity of the number of spaces the blank moves, but
each cube still has to have the parity of its rotation the same as the parity of the number
of spaces it has moved. If the identical pieces are treated as indistinguishable, the parity
of the permutation is only shown by the location of the blank space. He suggests the
use of ridges on the board so that the cube will roll automatically -- this was later used
in commercial versions. He gives a number of problems with different colourings of
the cubes.
Gardner. SA (Mar 1975). = Time Travel, chap. 9. Prob. 8: Rolling cubes. This is the first of
Harris's problems. Computer analysis has found that it can be done in fewer moves
than Harris had. Gardner also reports on the last of Harris's problems, which has also
been resolved by computer.
A 3 x 3 array with 8 coloured cubes was available from Taiwan in the early 1980s. It was
called Color Cube Mental Game -- I called it 'Rolling Cubes'. The cubes had thick
faces, producing grooved edges which fit into ridges in the bottom of the plastic frame,
causing automatic rolling quite nicely. I wonder if this was inspired by Harris's article.
John Ewing & Czes Kośniowski. Puzzle it Out -- Cubes, Groups and Puzzles. CUP, 1982.
The 8 Cubes Puzzle, pp. 58-59. Analysis of the Rolling Cubes puzzle. The authors
show how to rotate a single cube about a diagonal in 36 moves.
5.A.4.PANEX PUZZLE
SOURCES - page 61
Invented by Toshio Akanuma (??SP). Manufactured by Tricks Co., Japan, in 1983.
Described in Hordern, pp. 144-145 & 220, E35, and in S&B, p. 135. This looks like a Tower
of Hanoi (cf 7.M.2) with two differently coloured piles of 10 pieces on the outside two tracks
of three tracks of height 12 joined like a letter E. This is made as a sliding block puzzle, but
with blockages -- a piece cannot slide down a track further than its original position.
Mark Manasse, Danny Sleator & Victor K. Wei. Some Results on the Panex Puzzle.
Preprint sent by Jerry Slocum, 23pp, nd [1983, but S&B gives 1985]. For piles of size
n, the minimum number of moves, T(n), to move one pile to the centre track is
determined by means of a 2nd order, non-homogeneous recurrence which has different
forms for odd and even n. Compensating for this leads to a 2nd order
non-homogeneous recurrence, giving T(10) = 4875 and T(n) ~ C(1 + 2)n. This
solution doesn't ever move the other pile. The minimum number of moves, X(n), to
exchange the piles is bounded above and below and determined exactly for n  6 by
computer search. X(5) = 343, compared to bounds of 320 and 343. X(6) = 881,
compared to the bounds of 796 and 881. For n = 7, the bounds are 1944 and 2189,
For n = 10, the bounds are 27,564 and 31,537. The larger bounds are considered as
probably correct.
Christoph Hausammann. US Patent 5,261,668 -- Logic Game. Filed: 6 Aug 1992; patented:
16 Nov 1993. 1p abstract + 2pp text + 3pp diagrams. Essentially identical to Panex.
Vladimir Dubrovsky. Nesting Puzzles -- Part I: Moving oriental towers. Quantum 6:3
(Jan/Feb 1996) 53-59 & 49-51. Says Panex was produced by the Japanese Magic
Company in the early 1980s. Discusses it and cites S&B for the bounds given above.
Sketches a number of standard configurations and problems, leading to "Problem 9.
Write out a complete solution to the Panex puzzle." He says his method is about 1700
moves longer than the upper bound given above.
Nick Baxter. Recent results for the Panex Puzzle. 4pp handout at G4G5, 2002. Describes the
puzzle and its history. David Bagley wrote a program to implement the Manasse,
Sleator & Wei methods. On 7 Feb 2002, this confirmed the conjecture that
X(7) = 2189. On 26 Mar 2002, it obtained X(8) = 5359, compared to bounds of 4716
and 5359. It is estimated that the cases n = 9 and 10 will take 10 and 1200 years! If
Moore's Law on the increase of computing power continues for another 20 years, the
latter answer may be available by then. He gives a simplified version of the algorithm
for the upper bound, which gets 31,544 for n = 10. He has a Panex page:
www.baxterweb.com/puzzles/panex/ and will be publishing an edited and annotated
version of the Manasse, Sleator & Wei paper on it.
5.B. CROSSING PROBLEMS
See MUS I 1-13, Tropfke 658 and also 5.N.
Wolf, goat and cabbages: Alcuin, Abbot Albert, Columbia Algorism, Munich 14684,
Folkerts, Chuquet, Pacioli, Tartaglia, van Etten, Merry Riddles, Ozanam, Dilworth,
Wingate/Dodson, Jackson, Endless Amusement II, Boy's Own Book, Nuts to Crack,
Taylor; The Riddler, Child, Fireside Amusements, Magician's Own Book,
Book of 500 Puzzles, Boy's Own Conjuring Book, Secret Out (UK), Mittenzwey,
Carroll 1873, Kamp, Carroll 1878, Berg, Lemon, Hoffmann, Brandreth Puzzle
Book, Carroll 1899, King, Voggenreiter, Stein, Stong, Zaslavsky, Ascher,
Weismantel (a film),
Verse version: Taylor,
Version with only one pair of incompatibles: Voggenreiter
Extension to four items: Gori, Phillips, M. Adams, Gibbs, Ascher
Adults and children: Alcuin, Kamp, Hoffmann, Parker?, Voggenreiter, Gibbs
Three jealous husbands: Alcuin, Abbot Albert, Columbia Algorism, Munich 14684,
Folkerts, Rara, Chuquet, Pacioli, Cardan, Tartaglia, H&S - Trenchant, Gori,
Bachet, van Etten, Wingate/Kersey, Ozanam, Minguét, Dilworth, Les Amusemens,
Wingate/Dodson, Jackson, Endless Amusement II, Nuts to Crack,
Young Man's Book, Family Friend, Magician's Own Book, The Sociable,
Book of 500 Puzzles, Boy's Own Conjuring Book, Vinot, Secret Out (UK), Lemon,
Hoffmann, Fourrey, H. D. Northrop, Mr. X, Loyd, Williams, Clark, Goodstein,
O'Beirne, Doubleday, Allen,
SOURCES - page 62
Verse mnemonic: Abbot Albert, Munich 14684,
Verse solution: Ozanam, Vinot,
Four or more jealous husbands: Pacioli, Filicaia, Tartaglia, Bachet, Delannoy, Ball,
Carroll-Collingwood, Dudeney, O'Beirne
Jealous husbands, with island in river: De Fontenay, Dudeney, Ball, Loyd, Dudeney,
Pressman & Singmaster
Missionaries and cannibals: Jackson, Mittenzwey, Cassell's, Lemon, Pocock, Hoffmann,
Brandreth Puzzle Book, H. D. Northrop, Schubert, Arbiter, H&S, Abraham,
Bile Beans, Goodstein, Beyer, O'Beirne, Pressman & Singmaster.
With only one cannibal who can row: Brandreth Puzzle Book, Abraham, Beyer.
Bigger boats: Pacioli, Filicaia?, Bachet(-Labosne), Delannoy, Ball, Dudeney, Abraham?,
Goodstein, Kaplan, O'Beirne,
Alcuin. 9C.
Prob. 17: Propositio de tribus fratribus singulas habentibus sorores. 3 couples, rather
earthily expressed.
Prob. 18: Propositio de lupo et capra et fasciculo cauli. Wolf, goat, cabbages.
Prob. 19: Propositio de viro et muliere ponderantibus plaustrum. Man, wife and two
small children.
Prob. 20: Propositio de ericiis. Rewording of Prob. 19.
Ahrens. MUS II 315-318, cites many sources, mostly from folklore and riddle collections,
with one from the 12C and several from the 14C. ??NYS.
Abbot Albert. c1240.
Prob. 5, p. 333. Wolf, goat & cabbages.
Prob. 6, p. 334. 3 couples, with verse mnemonic.
Columbia Algorism. c1350.
No. 122, pp. 130-131 & 191: wolf, goat, bundle of greens. See also Cowley 402 &
plate opposite. P. 191 and the Cowley plate are reproductions of the text with a
crude but delightful illustration. P. 130 gives a small sketch of the illustration. I
have a colour slide from the MS.
No. 124, p. 132: 3 couples. See also Cowley 403 & plate opposite. The plate shows
another crude but delightful illustration. I have a colour slide from the MS.
Munich 14684. 14C.
Prob. XXVI, pp. 82-83: 3 couples, with verse mnemonic.
Prob. XXVII, p. 83: wolf, goat, cabbage.
Folkerts. Aufgabensammlungen. 13-15C. 11 sources with wolf, goat, cabbage. 12 sources
with three jealous couples.
Rara, 459-465, cites two Florentine MSS of c1460 which include 'the jealous husbands'.
??NYS.
Chuquet. 1484.
Prob. 163: wolf, goat & cabbages. FHM 233 says that a 12C MS claims that every boy
of five knows this problem.
Prob. 164: 3 couples. FHM 233.
Pacioli. De Viribus. c1500.
Ff. 103v - 105v. LXI. C(apitolo). de .3. mariti et .3. mogli gelosi (About 3 husbands
and 3 wives). = Peirani 146-148. 3 couples. Says that 4 or 5 couples requires
a 3 person boat.
F. IIIv. = Peirani 6. The Index lists the above as Problem 66 and lists a Problem 65:
Del modo a salvare la capra el capriolo dal lupo al passar de un fiume ch' non
siano devorati (How to save the goat and the kid from the wolf in crossing a river
so they are not eaten).
Piero di Nicolao d'Antonio da Filicaia. Libro dicto giuochi mathematici. Early 16C -??NYS, mentioned in Franci, op. cit. in 3.A. Franci, p. 23, says Pacioli and Filicaia
deal with the case of four or five couples and that Pacioli considers bigger boats, but I'm
not clear if Filicaia also does so.
Cardan. Practica Arithmetice. 1539. Chap. 66, section 73, f. FF.v.v (p. 157). (The 73 is not
printed in the Opera Omnia). Three jealous husbands.
Tartaglia. General Trattato, 1556, art. 141-143, p. 257r- 257v.
Art. 141: wolf, goat and cabbages.
Art. 142: three couples.
Art. 143: four couples -- erroneously -- see Bachet.
SOURCES - page 63
H&S 51 says 3 couples occurs in Trenchant (1566), ??NYS.
Gori. Libro di arimetricha. 1571.
Ff. 71r-71v (p. 77). 3 couples.
F. 80v (p. 77). Dog, wolf, sheep, horse to cross river in boat which holds 2, but each
cannot abide his neighbours in the given list, so each cannot be alone with such a
neighbour.
Bachet. Problemes. 1612. Addl. prob. IV: Trois maris jaloux ..., 1612: 140-142;
1624: 212-215; 1884: 148-153. Three couples; four couples -- notes that Tartaglia is
wrong by showing that one can never get five persons on the far side. Labosne gives a
solution with a 3 person boat and does n couples with an n-1 person boat.
van Etten. 1624.
Prob. 14: Des trois maistres & trois valets, p. 14. 3 men and 3 valets. (The men hate
the other valets and will beat them if given a chance.) (Not in English editions.)
Prob. 15: Du loup, de la chevre & du chou, pp. 14-15. Wolf, goat & cabbages. (Not in
English editions.)
Book of Merry Riddles. 1629 72 Riddle, pp. 43-44. "Over a water I must passe, and I must
carry a lamb, a woolfe, and a bottle of hay if I carry any more than one at once my boat
will sink." Tony Augarde; The Oxford Guide to Word Games; OUP, 1984; p. 6 says
wolf, goat, cabbage appears in the 1629 ed.
Wingate/Kersey. 1678?. Prob. 6., p. 543. Three jealous couples. Cf 1760 ed.
Ozanam. 1725.
Prob. 2, 1725: 3-4. Prob. 18, 1778: 171; 1803: 171; 1814: 150. Prob. 17, 1840: 77.
Wolf, goat and cabbage.
Prob. 3, 1725: 4-5. Prob. 19, 1778: 171-172; 1803: 171-172; 1814: 150-151.
Prob. 18, 1840: 77. Jealous husbands. Latin verse solution. He also discusses
three masters and valets: "none of the the masters can endure the valets of the
other two; so that if any one of them were left with any of the other two valets, in
the absence of his master, he would infallibly cane him."
Minguet. 1733. Pp. 158-159 (1755: 114-115; 1822: 175-176; 1864: 151). Three jealous
couples.
Dilworth. Schoolmaster's Assistant. 1743. Part IV: Questions: A short Collection of pleasant
and diverting Questions, p. 168.
Problem 6: Fox, goose and peck of corn. = D. Adams; Scholar's Arithmetic; 1801,
p. 200, no. 8.
Problem 7: Three jealous husbands. (Dilworth cites Wingate for this -- but this is in
Kersey's additions -- cf Wingate/Kersey, 1678? above.) = D. Adams; Scholar's
Arithmetic; 1801, p. 200, no. 9.
Les Amusemens. 1749. Prob. 14, p. 136: Les Maris jaloux. Solution is incorrect and has
been corrected by hand in my copy.
Edmund Wingate (1596-1656). A Plain and Familiar Method for Attaining the Knowledge
and Practice of Common Arithmetic. .... 19th ed., previous ed. by John Kersey (16161677) and George Shell(e)y, now by James Dodson. C. Hitch and L. Hawes, et al.,
1760.
Art. 749. Prob. VI. P. 379. Three jealous husbands. As in 1678? ed.
Art. 750. Prob. VII. P. 379. Fox, goose and corn.
Jackson. Rational Amusement. 1821. Arithmetical Puzzles.
No. 7, pp. 2 & 52. Fox, goose and corn. One solution.
No. 13, pp. 4 & 54. Three jealous husbands.
No. 21, pp. 5 & 56. Three masters and servants, where the servants will murder the
masters if they outnumber them -- i.e. missionaries and cannibals. First
appearance of this type.
Endless Amusement II. 1826?
Prob. 17, pp. 198-199. Wolf, goat and cabbage.
Prob. 25, pp. 201-202. Three jealous husbands.
Boy's Own Book. The wolf, the goat and the cabbages. 1828: 418-419; 1828-2: 423;
1829 (US): 214; 1855: 570; 1868: 670.
Nuts to Crack III (1834).
No. 209. Fox, goose and peck of corn.
No. 214. Three jealous husbands.
The Riddler. 1835. The wolf, the goat and the cabbages, pp. 5-6. Identical to Boy's Own
Book.
SOURCES - page 64
Young Man's Book. 1839. Pp. 39-40. Three jealous Husbands ..., identical to
Wingate/Kersey.
Child. Girl's Own Book. 1842: Enigma 49, pp. 237-238; 1876: Enigma 40, p. 200. Fox,
goose and corn. Says it takes four trips instead of three -- but the solution has 7
crossings.
Walter Taylor. The Indian Juvenile Arithmetic, or Mental Calculator; to which is added an
appendix, containing arithmetical recreations and amusements for leisure hours .... For
the author at the American Press, Bombay, 1849. [Quaritch catalogue 1224, Jun 1996,
says their copy has a note in French that Ramanujan learned arithmetic from this and
that it is not in BMC nor NUC. Graves 14.c.35.] P. 211, No. 8. Wolf, goat and
cabbage in verse! No solution.
Upon a river's brink I stand, it is both deep and wide;
With a wolf, a goat, and cabbage, to take to the other side.
Tho' only one each time can find, room in my little boat;
I must not leave the goat and wolf, not the cabbage and the goat.
Lest one should eat the other up, -- now how can it be done -How can I take them safe across without the loss of one?
Fireside Amusements. 1850: No. 24, pp. 111 & 181; 1890: No. 24, p. 100. Fox, goose and
basket of corn.
Family Friend 3 (1850) 344 & 351. Enigmas, charades, etc. -- No. 17: The three jealous
husbands.
Magician's Own Book. 1857.
The three jealous husbands, p. 251.
The fox, goose, and corn, p. 253.
The Sociable. 1858. Prob. 33: The three gentlemen and their servants, pp. 296 & 314-315.
"None of the gentlemen shall be left in company with any of the servants, except when
his own servant is present" -- so this is like the Jealous Husbands. = Book of 500
Puzzles, 1859, prob. 33, pp. 14 & 32-33. = Illustrated Boy's Own Treasury, 1860,
prob. 11, pp. 427-428 & 431.
Book of 500 Puzzles. 1859.
Prob. 33: The three gentlemen and their servants, pp. 14 & 32-33. As in The Sociable.
The three jealous husbands, p. 65.
The fox, goose and corn, p. 67.
Both identical to Magician's Own Book.
Boy's Own Conjuring Book. 1860.
The three jealous husbands, pp. 222-223.
The fox, goose, and corn, pp. 225.
Both identical to Magician's Own Book.
Vinot. 1860. Art. XXXVII: Les trois maris jaloux, pp. 56-57. Three jealous husbands, with
verse solution taken from Ozanam.
The Secret Out (UK). c1860. A comical dilemma, p. 27. Wolf, goat and cabbage. Varies it
as fox, goose and corn and then as gentlemen and servants, which is jealous husbands,
rather than the same problem.
Lewis Carroll. Letter of 15 Mar 1873 to Helen Feilden. Pp. 212-215 (Collins: 154-155).
Fox, goose and bag of corn. "I rashly proposed to her to try the puzzle (I daresay you
know it) of "the fox, and goose, and bag of corn."" Cf Carroll-Collingwood, pp. 212215 (Collins: 154-155); Carroll-Wakeling, prob. 28, pp. 36-37 and Carroll-Gardner,
p. 51. Cf Carroll, 1878. Wakeling writes that this does not appear elsewhere in Carroll.
Bachet-Labosne. 1874. For details, see Bachet, 1612.
Jens Kamp. Danske Folkeminder, Aeventyr, Folksagen, Gaader, Rim og Folketro, Samlede
fra Folkemende. R. Neilsen, Odense, 1877. Marcia Ascher has kindly sent me a
photocopy of the relevant material with a translation by Viggo Andressen.
No. 18, pp. 326-327: Fox, lamb and cabbage.
No. 19, p. 327: Husband, wife and two half-size sons.
Lewis Carroll. Letter of 22 Jan 1878 to Jessie Sinclair. Fox, goose and bag of corn. Cf
Carroll-Collingwood, pp. 205-207 (Collins: 150); Carroll-Wakeling, prob. 26: The fox,
the goose and the bag of corn, pp. 34 & 72. Cf Carroll. 1872.
Mittenzwey. 1880. Prob. 227-228, pp. 42 & 92; 1895?: 254-255, pp. 46 & 94;
1917: 254-255, pp. 42 & 90. Bear, goat and cabbage, mentioning second solution;
three kings and three servants, where the servants will rob the kings if they outnumber
them, i.e. like missionaries and cannibals.
SOURCES - page 65
Cassell's. 1881. P. 105: The dishonest servants. The servants are rogues who will murder
masters if they outnumber them, so this is equivalent to the missionaries and cannibals
version.
Lucas. RM1. 1882. Pp. 1-18 is a general discussion of the problem.
De Fontenay. Unknown source and date -- 1882?? Described in RM1, 1882, pp. 15-18
(check 1st ed.??). n > 3 couples, 2 person boat, island in river, can be done in 8n - 8
passages. Lucas says this was suggested at the Congrès de l'Association française pour
l'avancement des sciences at Montpellier in 1879, ??NYS. (De Fontenay is unclear -sometimes he permits bank to bank crossings, other times he only permits bank to
island crossings. His argument really gives 8n - 6 if bank to bank crossings are
prohibited. See Pressman & Singmaster, below, for clarification.)
Albert Ellery Berg, ed. Op. cit. in 4.B.1. 1883. P. 377: Fox, goose & peck of corn.
Lemon. 1890.
Gentlemen and their servants, no. 101, pp. 17-18 & 101. This is the same as
missionaries and cannibals.
The three jealous husbands, no. 151, pp. 24 & 103 (= Sphinx, no. 478, pp. 66 & 114.)
The solution mentions Alcuin.
Crossing the river, no. 450, pp. 59 & 114. English travellers and native servants
= missionaries and cannibals.
Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 136,
no. 14. Fox, goose and corn. No solution.
Herbert Llewelyn Pocock. UK Patent 15,358 -- Improvements in Toy Puzzles. Applied:
29 Sep 1890; complete specification: 29 Jun 1891; accepted: 22 Aug 1891. 2pp + 1p
diagrams. Three whites and three blacks and the blacks must never outnumber the
whites, i.e. same as missionaries and cannibals. He describes the puzzle as "well
known".
Delannoy. Described in RM1, 1891, Note 1: Sur le jeu des traversées, pp. 221-222. ??check
1882 ed. Shows n couples can cross in an x person boat in N trips, for
n, x, N = 2, 2, 5; 3, 2, 11; 4, 3, 9; 5, 3, 11; n > 5, 4, 2n - 3. (He has 2n - 1 by
mistake. Simple modification shows we also have 5, 4, 7; 6, 5, 9; 7, 6, 5; 8, 7, 7;
n > 8, n - 1, 5.)
Ball. MRE, 1st ed., 1892, pp. 45-47, says Lucas posed the problem of minimizing x for a
given n and quotes the Delannoy solution (with erroneous 2n - 1) and also gives De
Fontenay's version and solution. (He spells it De Fonteney as does his French
translator, though Ahrens gives De Fontenay and the famous abbey in Burgundy is
Fontenay -- ??)
The Ballybunnion and Listowel Railway in County Kerry, Ireland, was a late 19C railway
using the Lartigue monorail system. This had a single rail, about three feet off the
ground, with a carriage hanging over both sides of the rail. The principle job of the
conductor/guard to make sure the passengers and goods were equally distributed on
both sides. Kerry legend asserts that a piano had to be sent on this railway and there
were not enough passengers or goods to balance it. So a cow was sent on the other side.
At the far end, the piano was unloaded and replaced with two large calves and the
carriage sent back. The cow was then unloaded and one calf moved to the other side, so
the carriage could be sent back to the far end and everyone was happy.
Hoffmann. 1893. Chap. IV, pp. 157-158 & 211-213 = Hoffmann-Hordern, pp. 136-138, with
photos.
No. 56: The three travellers. Masters and servants, equivalent to missionaries and
cannibals. Solution says Jaques & Son make a puzzle version with six figures,
three white and three black. Photos in Hoffmann-Hordern, pp. 136 & 137 -- the
latter shows Caught in the Rain, 1880-1905, where Preacher, Deacon, Janitor and
their wives have to get somewhere using one umbrella.
No. 57: The wolf, the goat, and the cabbages. Photo on p. 136 of La Chevre et le Chou.
with box, by Watilliaux, 1874-1895. Hordern Collection, p. 72, and S&B, p.
134, show the same puzzle.)
No. 58: The three jealous husbands.
No. 59: The captain and his company. This is Alcuin's prop. 19 with many adults.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895].
P. 7: The wolf, the goat and the cabbages. Identical to Hoffmann No. 57, with nice
colour picture. No solution.
P. 9: The missionaries' and cannibals' puzzle. Usual form, with nice colour picture, but
SOURCES - page 66
only one cannibal can row. No solution. This seems to be the first to use the
context of missionaries and cannibals and the first to restrict the number of
rowers.
Lucas. L'Arithmétique Amusante. 1895. Les vilains maris jaloux, pp. 125-144 & Note II,
pp. 198-202.
Prob. XXXVI: La traversée des trois ménages, pp. 125-130. 3 couples. Gives Bachet's
1624 reasoning for the essentially unique solution -- but attributes it to 1613.
Prob. XXXVII: La traversée des quatre ménages, pp. 130-132. 4 couples in a 3
person boat done in 9 crossings.
L'erreur de Tartaglia, pp. 133-134. Discusses Tartaglia's error and Bachet's notice of it
and gives an easy proof that 4 couples cannot be done with a 2 person boat.
Prob. XXXVIII: La station dans une île, pp. 135-140. 4 couples, 2 person boat, with
an island. Gives De Fontenay's solution in 24 crossings.
Prob. XXXIX: La traversée des cinq ménages, pp. 141-143. 5 couples, 3 person boat
in 11 crossings.
Énoncé général du problème des traversées, pp. 143-144. n couples, x person boat,
can be done in N crossings as given by Delannoy above. He corrects 2n - 1 to
2n - 3 here.
Note II: Sur les traversées, pp. 198-202. Gives Tarry's version with an island and with
n men having harems of size m, where the women are obviously unable to row.
He gives solutions in various cases. For the ordinary case, i.e. m = 1, he finds a
solution for 4 couples in 21 moves, using the basic ferrying technique that
Pressman and Singmaster found to be optimal, but the beginning and end take
longer because the women cannot row. He says this gives a solution for n
couples in 4n + 5 crossings. He then considers the case of n - 1 couples and a
ménage with m wives and finds a solution in 8n + 2m + 7 crossings. I now see
that this solution has the same defects as those in Pressman & Singmaster, qv.
Ball. MRE, 3rd ed., 1896, pp. 61-64, repeats 1st ed., but adds that Tarry has suggested the
problem for harems -- see above.
Dudeney. Problem 68: Two rural puzzles. Tit-Bits 33 (5 Feb & 5 Mar 1898) 355 & 432.
Three men with sacks of treasure and a boat that will hold just two men or a man and a
sack, with additional restrictions on who can be trusted with how much. Solution in 13
crossings.
Carroll-Collingwood. 1899. P. 317 (Collins: 231 or 232 (missing in my copy)) Cf CarrollWakeling II, prob. 10: Crossing the river, pp. 17 & 66. Four couples -- only posed, no
solution. Wakeling gives a solution, but this is incorrect. After one wife is taken
across, he has another couple coming across and from Bachet onward, this is considered
improper as the man could get out of the boat and attack the first, undefended, wife.
E. Fourrey. Op. cit. in 4.A.1, 1899. Section 211: Les trois maîtres et les trois valets. Says a
master cannot leave his valet with the other masters for fear that they will intimidate
him into revealing the master's secrets. Hence this is the same as the jealous couples.
H. D. Northrop. Popular Pastimes. 1901.
No. 5: The three gentlemen and their servants, pp. 67 & 72. = The Sociable.
No. 12: The dishonest servants, pp. 68 & 73. "... the servants on either side of the river
should not outnumber the masters", so this is the same as missionaries and
cannibals.
Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:2 (Jun 1903) 140-141. A matrimonial
difficulty. Three couples. No answer given.
Dudeney. Problem 523. Weekly Dispatch (15 & 29 Nov 1903), both p. 10, (= AM, prob.
375, pp. 113 & 236-237). 5 couples in a 3 person boat.
Johannes Bolte. Der Mann mit der Ziege, dem Wolf und dem Kohle. Zeitschrift des Vereins
für Volkskunde 13 (1903) 95-96 & 311. The first part is unaware of Alcuin and Albert.
He gives a 12C Latin solution: It capra, fertur olus, redit hec, lupus it, capra transit
[from Wattenbach; Neuen Archiv für ältere deutsche Geschichtskunde 2 (1877) 402,
from Vorauer MS 111, ??NYS] and a 14C solution: O natat, L sequitur, redit O, C
navigat ultra, / Nauta recurrit ad O, bisque natavit ovis (= ovis, lupus, ovis, caulis, ovis)
[from Mone; Anzeiger für Kunde der deutschen Vorzeit 45 (No. 105) (1838), from
Reims MS 743, ??NYS]. Cites Kamp and several other versions, some using a fox, a
sheep, or a lamb. The addendum cites and quotes Alcuin and Albert as well as
relatively recent French and Italian versions.
H. Parker. Ancient Ceylon. Op. cit. in 4.B.1. 1909. Crossing the river, p. 623.
SOURCES - page 67
A King, a Queen, a washerman and a washerwoman have to cross a river in a boat that
holds two. However the King and Queen cannot be left on a bank with the low
caste persons, though they can be rowed by the washerperson of the same sex.
Solution in 7 crossings.
Ferry-man must transport three leopards and three goats in a boat which holds himself
and two others. If leopards ever outnumber goats, then the goats get eaten. So
this is like missionaries and cannibals, but with a ferry-man. Solution in 9
crossings.
H. Schubert. Mathematische Mussestunde. Vol. 2, 3rd ed., Göschen, Leipzig, 1909.
Pp. 160-162: Der drei Herren und der drei Sklaven. (Same as missionaries and
cannibals.)
Arbiter Co. (Philadelphia). 1910. Capital and Labor Puzzle. Shown in S&B, p. 134.
Equivalent to missionaries and cannibals.
Ball. MRE, 5th ed., 1911, pp. 71-73, repeats 3rd ed., but omits the details of De Fonteney's
solution in 8(n-1) crossings.
Loyd. Cyclopedia, 1914.
Summer tourists, pp. 207 & 366. 3 couples, 2 person boat, with additional
complications -- the women cannot row and there have been some arguments.
Solution in 17 crossings.
The four elopements, pp. 266 & 375. 4 couples, 2 person boat, with an island and the
stronger constraint that no man is to get into the boat alone if there is a girl alone
on either the island or the other shore. "The [problem] presents so many
complications that the best or shortest answer seems to have been overlooked by
mathematicians and writers on the subject." "Contrary to published answers, ...
the feat can be performed in 17 trips, instead of 24."
Ball. MRE, 6th ed., 1914, pp. 71-73, repeats 5th ed., but adds that 6n - 7 trips suffices for n
couples with an island, though he gives no reference.
Williams. Home Entertainments. 1914. Alcuin's riddle, pp. 125-126. "This will be
recognized as perhaps the most ancient British riddle in existence, though there are
several others conceived on the same lines." Three jealous couples.
Clark. Mental Nuts. 1916, no. 67. The men and their wives. "... no man shall be left alone
with another's wife."
Dudeney. AM. 1917. Prob. 376: The four elopements, pp. 113 & 237. 4 couples, 2 person
boat, with island, can be done in 17 trips and that this cannot be improved. This is the
same solution as given by Loyd. (See Pressman and Singmaster, below.)
Ball. MRE, 8th ed., 1919, pp. 71-73 repeats 6th ed. and adds a citation to Dudeney's AM
prob. 376 for the solution in 6n - 7 trips for n couples.
Hummerston. Fun, Mirth & Mystery. 1924. Crossing the river puzzles, Puzzle no, 52, pp.
128 & 180. 'Puzzles of this type ... interested people who lived more than a thousand
years ago'.
No. 1: The eight travellers. Six men and two boys who weigh half as much.
No. 2: White and black. = Missionaries and cannibals.
No. 3: The fox, the goose, and the corn.
No. 4: the jealous husbands.
H&S, 1927, p. 51 says missionaries and cannibals is 'a modern variant'.
King. Best 100. 1927. No. 10, pp. 10 & 40. Dog, goose and corn.
Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig
Voggenreiter, Potsdam, 1930.
P. 106: Der Wolf, die Ziege und der Kohlkopf. Usual wolf, goat, cabbage.
Pp. 106-107: Die 100 Pfund-Familie. Parents weigh 100 pounds; the two children
weigh 100 pounds together.
P. 107: Der Landjäger and die Strolche [The policeman and the vagabonds]. Two of
the vagabonds hate each other so much that they cannot be left together. As far
as I recall, this formulation is novel and I was surprised to realise that it is
essentially equivalent to the wolf, goat and cabbage version.
Phillips. Week-End. 1932. Time tests of intelligence, no. 41, pp. 22 & 194. Rowing
explorer with 4 natives: A, B, C, D, who cannot abide their neighbours in this list. A
can row. They get across in seven trips.
Abraham. 1933. Prob. 54 -- The missionaries at the ferry, pp. 18 & 54 (14 & 115). 3
missionaries and 3 cannibals. Doesn't specify boat size, but says 'only one cannibal
can row'. 1933 solution says 'eight double journeys', 1964 says 'seven crossings'. This
SOURCES - page 68
seems to assume the boat holds 3. (For a 2 man boat, it takes 11 crossings with one
missionary and two cannibals who can row or 13 crossings with one missionary and
one cannibal who can row.)
The Bile Beans Puzzle Book. 1933. No. 34: Missionaries & cannibals. Three of each but
only one of each can row. Done in 13 crossings.
Phillips. Brush. 1936. Prob. L.2: Crossing the Limpopo, pp. 39-40 & 98. Same as in
Week-End, 1932.
M. Adams. Puzzle Book. 1939. Prob. C.63: Going to the dance, pp. 139 & 178. Same as
Week-End, 1932, phrased as travelling to a dance on a motorcycle which carries one
passenger.
R. L. Goodstein. Note 1778: Ferry puzzle. MG 28 (No. 282) (1944) 202-204. Gives a
graphical way of representing such problems and considers m soldiers and m
cannibals with an n person boat, 3 jealous husbands and how many rowers are
required.
David Stein. Party and Indoor Games. P. M. Productions, London, nd [c1950?]. P. 98,
prob. 5: Man with cat, parrot and bag of seeds.
C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960.
A puzzle-solving machine, pp. 377-384. Describes how Paul Bezold made a logic
machine from relays to solve the fox, goose, corn problem.
How to design a "Pircuit" or Puzzle circuit, pp. 388-394. On pp. 391-394, Harry
Rudloe describes relay circuits for solving the three jealous couples problem,
which he attributes to Tartaglia, and the missionaries and cannibals problem.
Nathan Altshiller Court. Mathematics in Fun and in Earnest. Mentor (New American
Library), NY, 1961. [John Fauvel sent some pages from a different printing which has
much different page numbers than my copy.] "River crossing" problems, pp. 168-171.
Discusses various forms of the problem and adds a problem with two parents weighing
160, two children weighing 80 and a dog weighing 12, with a boat holding 160.
E. A. Beyer, proposer; editorial solution. River-crossing dilemma. RMM 4 (Aug 1961) 46
& 5 (Oct 1961) 59. Explorers and natives (= missionaries and cannibals), with all the
explorers and one native who can row. Solves in 13 crossings, but doesn't note that
only one rowing explorer is needed. (See note at Abraham, 1933, above.)
Philip Kaplan. Posers. (Harper & Row, 1963); Macfadden Books, 1964. Prob. 36, pp. 41 &
91. 5 men and a 3 person boat on one side, 5 women on the other side. One man
and one woman can row. Men are not allowed to outnumber women on either side nor
in the boat. Exchange the men and the women in 7 crossings.
T. H. O'Beirne. Puzzles and Paradoxes, 1965, op. cit. in 4.A.4, chap. 1, One more river to
cross, pp. 1-19. Shows 2n - 1 couples (or 2n - 1 each of missionaries and cannibals
?) can cross in a n person boat in 11 trips. 2n - 2 can cross in 9 trips. He also
considers variants on Gori's second version.
Doubleday - 2. 1971. Family outing, pp. 49-50. Three couples, but one man has quarrelled
with the other men and his wife has quarrelled with the other women, so this man and
wife cannot go in the boat nor be left on a bank with others of their sex. Further men
cannot be outnumbered by women on either bank. Gives a solution in 9 crossings, but
I find the conditions unworkable -- e.g. the initial position is prohibited!
Claudia Zaslavsky. Africa Counts. Prindle, Weber & Schmidt, Boston, 1973. Pp. 109-110
says that leopard, goat and pile of cassava leaves is popular with the Kpelle children of
Liberia. However, Ascher's Ethnomathematics (see below), p. 120, notes that this is
based on an ambiguous description and that an earlier report of a Kpelle version has the
form described below.
Ball. MRE, 12th ed., 1974, p. 119, corrects Delannoy's 2n - 1 to 2n - 3 and corrects De
Fontenay's 8n - 8 to 8n - 6, but still gives the solution for n = 4 with 24 crossings.
W. Gibbs. Pebble Puzzles -- A Source Book of Simple Puzzles and Problems. Curriculum
Development Unit, Solomon Islands, 1982. ??NYS, o/o??. Excerpted in: Norman K.
Lowe, ed.; Games and Toys in the Teaching of Science and Technology; Science and
Technology Education, Document Series No. 29, UNESCO, Paris, 1988, pp. 54-57. On
pp. 56-57 is a series of river crossing problems. E.g. get people of weights 1, 2, 3
across with a boat that holds a weight of at most 3. Also people numbered 1, 2, 3, 4, 5
such that no two consecutive people can be in the boat or left together.
In about 1986, James Dalgety designed interactive puzzles for Techniquest in Cardiff. Their
version has a Welshman with a dragon, a sheep and a leek!
Ian Pressman & David Singmaster. Solutions of two river crossing problems: The jealous
SOURCES - page 69
husbands and the missionaries and the cannibals. Extended Preprint, April 1988, 14pp.
MG 73 (No. 464) (Jun 1989) 73-81. (The preprint contains historical and other detail
omitted from the article as well as some further information.) Observes that De
Fontenay seems to be excluding bank to bank crossings and that Lucas' presentation is
cryptic. Shows that De Fontenay's method should be 8n - 6 crossings for n > 3 and
that this is minimal. If bank to bank crossings are permitted, as by Loyd and Dudeney,
a computer search revealed a solution with 16 crossings for n = 4, using an ingenious
move that Dudeney could well have ignored. For n > 4, there is a simple solution in
4n + 1 crossings, and these numbers are minimal. [When this was written, I had
forgotten that Loyd had done the problem for 4 couples in 17 moves, which changes
the history somewhat. However, I now see that Loyd was copying from Dudeney's
Weekly Dispatch problem 270 of 23 Apr 1899 & 11 Jun 1899. Loyd states what
appears to be a stronger constraint but all the methods in our article do obey the stronger
constraint. However, one could make the constraints stronger -- e.g. our solutions have
a husband taking the boat from bank to bank while his wife and another wife are on the
island -- the solution of Loyd & Dudeney avoids this and may be minimal in this case -??.]
For the missionaries and cannibals problem, the 16 crossing solution reduces to
15 and gives a general solution in 4n - 1 crossings, which is shown to be minimal. If
bank to bank crossings are not permitted, then De Fontenay's amended 8n - 6 solution
is still optimal.
Marcia Ascher. A river-crossing problem in cultural perspective. MM 63 (1990) 26-28.
Describes many appearances in folklore of many cultures. Discusses African variants
of the wolf, goat and cabbage problem in which the man can take two of the items in the
boat. This is much easier, requiring only three crossings, but some versions say that the
man cannot control the items in the boat, so he cannot have the wolf and goat or the
goat and cabbage in the boat with him. This still only takes three crossings. Various
forms of these problems are mentioned: fox, fowl and corn; tiger, sheep and reeds;
jackal, goat and hay; caged cheetah, fowl and rice; leopard, goat and leaves -- see
below for more details.
She also discusses an Ila (Zambia) version with leopard, goat, rat and corn which
is unsolvable!
Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Section 4.8, pp. 109-116 &
Note 8, pp. 119-121. Good survey of the problem and numerous references to the
folklore and ethnographic literature. Amplifies the above article. A version like the
Wolf, goat and cabbage is found in the Cape Verde Islands, in Cameroon and in
Ethiopia. The African version is found as far apart as Algeria and Zanzibar, but with
some variations. An Algerian version with jackal, goat and hay allows one to carry any
two in the boat, but an inefficient solution is presented first. A Kpelle (Liberia) version
with cheetah, fowl and rice adds that the man cannot keep control while rowing so he
cannot take the fowl with either the cheetah or the rice in the boat. A Zanzibar version
with leopard, goat and leaves adds instead that no two items can be left on either bank
together. (A similar version occurs among African-Americans on the Sea Islands of
South Carolina.) Ascher notes that Zaslavsky's description is based on an ambiguous
report of the Kpelle version and probably should be like the Algerian or Kpelle version
just described.
Liz Allen. Brain Sharpeners. New English Library (Hodder & Stoughton), London, 1991.
Crossing the river, pp. 62 & 125. Three mothers and three sons. The sons are unwilling
to be left with strange mothers, so this is a rephrasing of the jealous husbands.
Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995.
Chap. 1, probs. 4-6: The knights and the pages; More knights and pages; Yet more
knights and pages: no man is an island, pp. 4-5 & 100-102. Equivalent to the jealous
couples. Prob. 4 is three couples, solved in 11 crossings. Prob. 5 is four couples -"There is no solution unless one of the four pages is sacrificed. (In medieval times, this
was not a problem.)" Prob. 6 is four couples with an island in the river, solved in
general by moving all pages to the island, then having the pages go back and
accompany his knight to other side, then return to the island. After the last knight is
moved, the pages then move from the island to the other side. This takes 7n - 6 steps
in general. It satisfies the jealousy conditions used by Pressman & Singmaster, but not
those of Loyd & Dudeney.
John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for
SOURCES - page 70
all Ages. Keystone Agencies, Radnor, Ohio, 1997. P. 16: The missionaries and the
pirates. Politically correct rephrasing of the missionaries and the cannibals version. All
the missionaries, but only one pirate, can row. Solves in 13 crossings.
Prof. Dr. Robert Weismantel, Otto-von-Guericke-Universität Magdeburg, Fakultät für
Mathematik, PSF 3120, D-39016 Magdeburg, Germany; tel: 0391/67-18745; email:
weismantel@imo.math.uni-magdeburg.de; has produced a 45 min. film: "Der Wolf, die
Ziege und Kohlköpfe Transportprobleme von Karl dem Grossen bis heute", suitable for
the final years of school.
5.B.1. LOWERING FROM TOWER PROBLEM
The problem is for a collection of people (and objects or animals) to lower themselves
from a window using a rope over a pulley, with baskets at each end. The complication is that
the baskets cannot contain very different weights, i.e. there is a maximum difference in the
weights, otherwise they go too fast. This is often attributed to Carroll.
Carroll-Collingwood. 1899. P. 318 (Collins: 232-233 (232 is lacking in my copy)).
= Carroll-Wakeling II, prob. 4: The captive queen, pp. 8 & 65-66. 3 people of weights
195, 165, 90 and a weight of 75, with difference at most 15. He also gives a more
complex form. No solutions. Although the text clearly says 165, the prevalence of the
exact same problem with 165 replaced by 105 makes me wonder if this was a
misprint?? Wakeling says there is no explicit evidence that Carroll invented this, and
neither book assigns a date, but Carroll seems a more original source than the following
and he was more active before 1890 than after.
An addition is given in both books: add three animals, weighing 60, 45, 30.
Lemon. 1890. The prisoners in the tower, no. 497, pp. 65 & 116. c= Sphinx, The escape,
no. 113, pp. 19 & 100-101. Three people of weights 195, 105, 90 with a weight of 75.
The difference in weights cannot be more than 15.
Hoffmann. 1893. Chap. IV, no. 28: The captives in the tower, pp. 150 & 196
= Hoffmann-Hordern, p. 123. Same as Lemon.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 3: The
captives in the tower. Same as Lemon. Identical to Hoffmann. With colour picture.
No solution.
Loyd. The fire escape puzzle. Cyclopedia, 1914, pp. 71 & 348. c= MPSL2, prob. 140, pp.
98-99 & 165. = SLAHP: Saving the family, pp. 59 & 108. Simplified form of Carroll's
problem. Man, wife, baby & dog, weighing a total of 390.
Williams. Home Entertainments. 1914. The escaping prisoners, pp. 126-127. Same as
Lemon.
Rudin. 1936. No. 92, pp. 31-32 & 94. Same as Lemon.
Haldeman-Julius. 1937. No. 150: Fairy tale, pp. 17 & 28. Same as Lemon, except the largest
weight is printed as 196, possibly an error.
Kinnaird. Op. cit. in 1 -- Loyd. 1946. Pp. 388-389 & 394. Same as Lemon.
Simon Dresner. Science World Book of Brain Teasers. Scholastic Book Services, NY, 1962.
Prob. 61: Escape from the tower, pp. 29 & 99-100. Same as Lemon.
Robert Harbin [pseud. of Ned Williams]. Party Lines. Oldbourne, London, 1963. Escape,
p. 29. As in Lemon.
Howard P. Dinesman. Superior Mathematical Puzzles. Allen & Unwin, London, 1968.
No. 60: The tower escape, pp. 78 & 118. Same as Carroll. Answer in 15 stages. He
cites Carroll, noting that Carroll did not give a solution and he asks if a shorter solution
can be found.
F. Geoffrey Hartswick. In: H. O. Ripley & F. G. Hartswick; Detectograms and Other
Puzzles; Scholastic Book Services, NY, 1969. No. 15: Stolen treasure puzzle, pp.
54-55 & 87. Same as Lemon.
5.B.2. CROSSING A BRIDGE WITH A TORCH
New section.
Four people have to get across a bridge which is dark and needs to be lit with the torch.
The torch can serve for at most two people and the gap is too wide to throw the torch across,
so the torch has to be carried back and forth. The various people are of different ages and
require 5, 10, 20, 25 minutes to cross and when two cross, they have to go at the speed of the
SOURCES - page 71
slower. But the torch (= flashlight) battery will only last an hour. Can it be done? I heard this
about 1997, when it was claimed to be used by Microsoft in interviewing candidates. I never
found any history of it, until I recently found a discussion on Torsten Sillke's site: Crossing the
bridge in an hour (www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/crossing-bridge),
starting in Jun 1997 and last updated in Sep 2001. This cites the 1981 source and the other
references below. Denote the problem with speeds a, b, c, d and total time t by
(a, b, c, d; t), etc. t is sometimes given, sometimes not.
Saul X. Levmore & Elizabeth Early Cook. Super Strategies for Puzzles and Games.
Doubleday, 1981, p. 3 -- ??NYS. (5, 10, 20, 25; 60), as in the introduction to this
section..
Heinrich Hemme. Das Problem des Zwölf-Elfs. Vandenhoeck & Ruprecht, 1998. Prob. 81:
Die Flucht, pp. 40 & 105-106, citing a web posting by Gunther Lientschnig on 4 Dec
1996. (2, 4, 8, 10; t).
Dick Hess. Puzzles from Around the World. Apr 1997. Prob. 107: The Bridge.
(1, 2, 5, 10; 17). Poses versions with more people: (1, 3, 4, 6, 8, 9; 31) and, with a
three-person bridge, (1, 2, 6, 7, 8, 19, 10; 25).
Quantum (May/Jun 1997) 13. Brainteaser B 205: Family planning. Problem (1, 3, 8, 10; 20).
Karen Lingel. Email of 17 Sep 1997 to rec.puzzles. Careful analysis, showing that the 'trick'
solution is better than the 'direct' solution if and only if a + c > 2b. [Indeed, a + c - 2b
is the time saved by the 'trick' solution.] She cites (2, 3, 5, 8; 19) and (2, 2, 3, 3; 11)
to Sillke and (1, 3, 6, 8, 12; 30), from an undated website. Expressing the solution for
more people seems to remain an open question.
5.C. FALSE COINS WITH A BALANCE
See 5.D.3 for use of a weighing scale.
There are several related forms of this problem. Almost all of the items below deal with
12 coins with one false, either heavy or light, and its generalizations, but some other forms
occur, including the following.
8 coins, 1 light: Schell, Dresner
26 coins, 1 light: Schell
8 coins, 1 light: Bath (1959)
9 coins, 1 light: Karapetoff, Meyer (1946), Meyer (1948), M. Adams, Rice
I have been sent an article by Jack Sieburg; Problem Solving by Computer Logic; Data
Processing Magazine, but the date is cut off -- ??
E. D. Schell, proposer; M. Dernham, solver. Problem E651 -- Weighed and found wanting.
AMM 52:1 (Jan 1945) 42 & 7 (Aug/Sep 1945) 397. 8 coins, at most one light -determine the light one in two weighings.
Benjamin L. Schwartz. Letter: Truth about false coins. MM 51 (1978) 254. States that
Schell told Michael Goldberg in 1945 that he had originated the problem.
Emil D. Schell. Letter of 17 Jul 1978 to Paul J. Campbell. Says he did NOT originate the
problem, nor did he submit the version published. He first heard of it from Walter
W. Jacobs about Thanksgiving 1944 in the form of finding at most one light coin
among 26 good coins in three weighings. He submitted this to the AMM, with a note
disclaiming originality. The AMM problem editor published the simpler version
described above, under Schell's name. Schell says he has heard Eilenberg describe the
puzzle as being earlier than Sep 1939. Campbell wrote Eilenberg, but had no response.
Schell's letter is making it appear that the problem derives from the use of
1, 3, 9, ... as weights. This usage leads one to discover that a light coin can be found in
3n coins using n weighings. This is the problem mentioned by Karapetoff. If there is
at most one light coin, then n weighings will determine it among 3n - 1 coins, which
is the form described by Schell. The problem seems to have been almost immediately
converted into the case with one false coin, either heavy or light.
Walter W. Jacobs. Letter of 15 Aug 1978 to Paul J. Campbell. Says he heard of the problem
in 1943 (not 1944) and will try to contact the two people who might have told it to him.
However, Campbell has had no further word.
V. Karapetoff. The nine coin problem and the mathematics of sorting. SM 11 (1945)
SOURCES - page 72
186-187. Discusses 9 coins, one light, and asks for a mathematical approach to the
general problem. (?? -- Cites AMM 52, p. 314, but I cannot find anything relevant in
the whole volume, except the Schell problem. Try again??)
Dwight A. Stewart, proposer; D. B. Parkinson & Lester H. Green, solvers. The counterfeit
coin. In: L. A. Graham, ed.; Ingenious Mathematical Problems and Methods; Dover,
1959; pp. 37-38 & 196-198. 12 coins. First appeared in Oct 1945. Original only asks
for the counterfeit, but second solver shows how to tell if it is heavy or light.
R. L. Goodstein. Note 1845: Find the penny. MG 29 (No. 287) (Dec 1945) 227-229.
Non-optimal solution of general problem.
Editorial Note. Note 1930: Addenda to Note 1845. Ibid. 30 (No. 291) (Oct 1946) 231.
Comments on how to extend to optimal solution.
Howard D. Grossman. The twelve-coin problem. SM 11:3/4 (Sep/Dec 1945) 360-361. Finds
counterfeit and extends to 36 coins.
Lothrop Withington, Jr. Another solution of the 12-coin problem. Ibid., 361-362. Finds also
whether heavy or light.
Donald Eves, proposer; E. D. Schell & Joseph Rosenbaum, solvers. Problem E712 -- The
extended coin problem. AMM 53:3 (Mar 1946) 156 & 54:1 (Jan 1947) 46-48.
12 coins.
Jerome S. Meyer. Puzzle Paradise. Crown, NY, 1946. Prob. 132: The nine pearls, pp. 94 &
132. Nine pearls, one light, in two weighings.
N. J. Fine, proposer & solver. Problem 4203 -- The generalized coin problem. AMM 53:5
(May 1946) 278 & 54:8 (Oct 1947) 489-491. General problem.
H. D. Grossman. Generalization of the twelve-coin problem. SM 12 (1946) 291-292.
Discusses Goodstein's results.
F. J. Dyson. Note 1931: The Problem of the Pennies. MG 30 (No. 291) (Oct 1946) 231-234.
General solution.
C. A. B. Smith. The Counterfeit Coin Problem. MG 31 (No. 293) (Feb 1947) 31-39.
C. W. Raine. Another approach to the twelve-coin problem. SM 14 (1948) 66-67. 12 coins
only.
K. Itkin. A generalization of the twelve-coin problem. SM 14 (1948) 67-68. General
solution.
Howard D. Grossman. Ternary epitaph on coin problems. SM 14 (1948) 69-71. Ternary
solution of Dyson & Smith.
Jerome S. Meyer. Fun-to-do. A Book of Home Entertainment. Dutton, NY, 1948. Prob. 40:
Nine pearls, pp. 41 & 188. Nine pearls, one light, in two weighings.
Blanche Descartes [pseud. of Cedric A. B. Smith]. The twelve coin problem. Eureka 13 (Oct
1950) 7 & 20. Proposal and solution in verse.
J. S. Robertson. Those twelve coins again. SM 16 (1950) 111-115. Article indicates there
will be a continuation, but Schaaf I 32 doesn't cite it and I haven't found it yet.
E. V. Newberry. Note 2342: The penny problem. MG 37 (No. 320) (May 1953) 130. Says
he has made a rug showing the 120 coins problems and makes comments similar to
Littlewood's, below.
J. E. Littlewood. A Mathematician's Miscellany. Methuen, London, 1953; reprinted with
minor corrections, 1957 (& 1960). [All the material cited is also in the later version:
Littlewood's Miscellany, ed. by B. Bollobás, CUP, 1986, but on different pages. Since
the 1953 ed. is scarce, I will also cite the 1986 pages in ( ).] Pp. 9 & 135 (31 & 114).
"It was said that the 'weighing-pennies' problem wasted 10,000 scientist-hours of
war-work, and that there was a proposal to drop it over Germany."
John Paul Adams. We Dare You to Solve This! Berkley Publishing, NY, nd [1957?]. [This
is apparently a collection of problems used in newspapers. The copyright is given as
1955, 1956, 1957.] Prob. 18: Weighty problem, pp. 13 & 46. 9 equal diamonds but one
is light, to be found in 2 weighings.
Hubert Phillips. Something to Think About. Revised ed., Max Parrish, London, 1958.
Foreword, p. 6 & prob. 115: Twelve coins, pp. 81 & 127-128. Foreword says prob.
115 has been added to this edition and "was in oral circulation during the war. So far as
I know, it has only appeared in print in the Law Journal, where I published both the
problem and its solution." This may be an early appearance, so I should try and track
this down. ??NYS
Dan Pedoe. The Gentle Art of Mathematics. (English Universities Press, 1958); Pelican
(Penguin), 1963. P. 30: "We now come to a problem which is said to have been
planted over here during the war by enemy agents, since Operational Research spent so
SOURCES - page 73
many man-hours on its solution."
Philip E. Bath. Fun with Figures. The Epworth Press, London, 1959. No. 7: No weights -no guessing, pp. 8 & 40. 8 balls, including one light, to be determined in two
weighings. Method actually works for  1 light.
M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 9. Five boxes of sugar, but some has been taken from one box and put in
another. Determine which in least number of weighings. Does by weighing each
division of A, B, C, D into two pairs.
Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. The
"False Coin" problem, pp. 178-182. Sketches history and solution.
Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 46:
Dud reckoning, pp. 21 & 94. Find one light among eight in two weighings.
Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965.
Prob. 55, pp. 57 & 98. Six identical appearing coins, three of which are identically
heavy. In two weighings, identify two of the heavy coins.
Charlie Rice. Challenge! Hallmark Editions, Kansas City, Missouri, 1968. Prob. 7, pp. 22 &
54-55. 9 pearls, one light.
Jonathan Always. Puzzling You Again. Tandem, London, 1969. Prob. 86: Light-weight
contest, pp. 51-52 & 106-107. 27 weights of sizes 1, 2, ..., 27, except one is light. Find
it in 3 weighings. He divides into 9 sets of three having equal weights. Using two
weighings, one locates the light weight in a set of three and then weighing two of these
with good weights reveals the light one. [3 weights 1, 2, 3 cannot be done in one
weighing, but 9 weights 1, 2, ..., 9 can be done in two weighings.]
Robert H. Thouless. The 12-balls problem as an illustration of the application of information
theory. MG 54 (No. 389) (Oct 1970) 246-249. Uses information theory to show that
the solution process is essentially determined.
Ron Denyer. Letter. G&P, No. 37 (Jun 1975) 23. Asks for a mnemonic for the 12 coins
puzzles. He notes that one can use three predetermined weighings and find the coin
from the three answers.
Basil Mager & E. Asher. Letters: Coining a mnemonic. G&P, No. 40 (Sep 1975) 26. One
mnemonic for a variable method, another for a predetermined method.
N. J. Maclean. Letter: The twelve coins. G&P, No. 45 (Feb 1976) 28-29. Exposits a ternary
method for predetermined weighings for (3n-3)/2 in n weighings. Each weighing
determines one ternary digit and the resulting ternary number gives both the coin and
whether it is heavy or light.
Tim Sole. The Ticket to Heaven and Other Superior Puzzles. Penguin, 1988. Weighty
problems -- (iii), pp. 124 & 147. Nine equal pies, except someone has removed some
filling from one and inserted it in a pie, possibly the same one. Determine which, if
any, are the heavy and light ones in 4 balancings.
Calvin T. Long. Magic in base 3. MG 76 (No. 477) (Nov 1992) 371-376. Good exposition
of the base 3 method for 12 coins.
Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Problems for an equal-arm
balance, pp. 137-141.
1. Six balls, two of each of three colours. One of each colour is lighter than
normal and all light weights are equal. Determine the light balls in three weighings.
2. Five balls, three normal, one heavy, one light, with the differences being
equal, i.e. the heavy and the light weigh as much as two normals. Determine the heavy
and light in three weighings.
3. Same problem with nine balls and seven normals, done in four weighings.
5.C.1 RANKING COINS WITH A BALANCE
If one weighs only one coin against another, this is the problem of sorting except that
we don't actually put the objects in order. If one weighs pairs, etc., this is a more complex
problem.
J. Schreier. Mathesis Polska 7 (1932) 154-160. ??NYS -- cited by Steinhaus.
Hugo Steinhaus. Mathematical Snapshots. Not in Stechert, NY, 1938, ed. OUP, NY: 1950:
pp. 36-40 & 258; 1960: pp. 51-55 & 322; 1969 (1983): pp. 53-56 & 300. Shows n
objects can be ranked in M(n) = 1 + kn - 2k steps where k = 1 + [log2 n]. Gets M(5) =
8.
SOURCES - page 74
Lester R. Ford Jr. & Selmer M. Johnson. A tournament problem. AMM 66:5 (May 1959)
387-389. Note that log2 n! = L(n) is a lower bound from information theory. Obtain
a better upper bound than Steinhaus, denoted U(n), which is too complex to state here.
For convenience, I give the table of these values here.
n
M(n)
U(n)
L(n)
1
0
0
0
2
1
1
1
3
3
3
3
4
5
5
5
5 6 7 8 9 10 11
8 11 14 17 21 25 29
7 10 13 16 19 22 26
7 10 13 16 19 22 26
12
33
30
29
13
37
34
33
U(n) = L(n) also holds at n = 20 and 21.
Roland Sprague. Unterhaltsame Mathematik. Op. cit. in 4.A.1. 1961. Prob. 22: Ein noch
ungelöstes Problem, pp. 16 & 42-43. (= A still unsolved problem, pp. 17 & 48-49.)
Sketches Steinhaus's method, then does 5 objects in 7 steps. Gives the lower bound
L(n) and says the case n = 12 is still unsolved.
Kobon Fujimura, proposer; editorial comment. Another balance scale problem. RMM 10
(Aug 1962) 34 & 11 (Oct 1962) 42. Eight coins of different weights and a balance.
How many weighings are needed to rank the coins? In No. 11, it says the solution will
appear in No. 13, but it doesn't appear there or in the last issue, No. 14. It also doesn't
appear in the proposer's Tokyo Puzzles.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 6: In the
balance, pp. 18 & 85-86. Rank five balls in order in seven weighings.
John Cameron. Establishing a pecking order. MG 55 (No. 394) (Dec 1971) 391-395.
Reduces Steinhaus's M(n) by 1 for n  5, but this is not as good as Ford & Johnson.
W. Antony Broomhead. Letter: Progress in congress? MG 56 (No. 398) (Dec 1972) 331.
Comments on Cameron's article and says Cameron can be improved. States the values
U(9) and U(10), but says he doesn't know how to do 9 in 19 steps. Cites Sprague
for numerical values, but these don't appear in Sprague -- so Broomhead presumably
computed L(9) and L(10). He gets 10 in 23 steps, which is better than Cameron.
Stanley Collings. Letter: More progress in congress. MG 57 (No. 401) (Oct 1973) 212-213.
Notes the ambiguity in Broomhead's reference to Sprague. Improves Cameron by 1
(or more??) for n  10, but still not as good as Ford & Johnson.
L. J. Upton, proposer; Leroy J. Myers, solver. Problem 1138. CM 12 (1986) 79 & 13
(1987) 230-231. Rank coins weighing 1, 2, 3, 4 with a balance in four weighings.
5.D. MEASURING PROBLEMS
5.D.1.JUGS & BOTTLES
See MUS I 105-124, Tropfke 659.
NOTATION: I-(a, b, c) means we have three jugs of sizes a, b, c with a full and we
want to divide a in half using b and c. We normally assume a  b  c and
GCD(a, b, c) = 1. Halving a is clearly impossible if GCD(b, c) does not divide a/2 or if
b+c < a/2, unless one has a further jug or one can drink some. If a  b+c  a/2 and
GCD(b, c) divides a/2, then the problem is solvable.
More generally, the question is to determine what amounts can be produced, i.e. given
a, b, c as above, can one measure out an amount d? We denote this by II-(a, b, c; d). Since
this also produces a-d, we can assume that d  a/2. Then we must have d  b+c for a
solution. When a  b+c  d, the condition GCD(b, c)  d guarantees that d can be
produced. This also holds for a = b+c-1 and a = b+c-2. The simplest impossible cases are
I-(4, 4, 3) = II-(4, 4, 3; 2) and II-(5, 5, 3; 1). Case I-(a, b, c) is the same as II-(a, b, c; a/2).
If a is a large source, e.g. a stream or a big barrel, we have the problem of measuring d
using b and c without any constraint on a and we denote this II-(, b, c; d). However, the
solution may not use the infiniteness of the source and such a problem may be the same as
II-(b+c, b, c; d).
The general situation when a < b+c is more complex and really requires us to consider
the most general three jug problem: III-(A; a, b, c; d) means we have three jugs of sizes
a, b, c, containing a total amount of liquid A (in some initial configuration) and we wish to
measure out d. In our previous problems, we had A = a. Clearly we must have a+b+c  A.
Again, producing d also produces A-d, so we can assume d  A/2. By considering the
SOURCES - page 75
amounts of empty space in the containers, the problem III-(A; a, b, c; d) is isomorphic to
III-(a+b+c-A; a, b, c; d') for several possible d'.
NOTES. I have been re-examining this problem and I am not sure if I have reached a
final interpretation and formulation. Also, I have recently changed to the above notation and I
may have made some errors in so doing. I have long had the problem in my list of projects for
students, but no one looked at it until 1995-1996 when Nahid Erfani chose it. She has
examined many cases and we have have discovered a number of properties which I do not
recall seeing. E.g. in case I-(a,b,c) with a  b  c and GCD(b,c) = 1, there are two ways to
obtain a/2. If we start by pouring into b, it takes b + c - 1 pourings; if we start by pouring
into c, it takes b + c pourings; so it is always best to start pouring into the larger jug. A
number of situations II-(a,b,c;d) are solvable for all values of d, except a/2. E.g.
II-(a,b,c;a/2) with b+c > a and c > a/2 is unsolvable.
From about the mid 19C, I have not recorded simple problems.
I-( 8,
I-(10,
I-(10,
I-(12,
5, 3):
almost all the entries below
6, 4):
Pacioli, Court
7, 3):
Yoshida
7, 5):
Pacioli, van Etten/Henrion, Ozanam, Bestelmeier, Jackson,
Manuel des Sorciers, Boy's Own Conjuring Book
I-(12, 8, 4):
Pacioli
I-(12, 8, 5):
Bachet, Arago
I-(16, 9, 7):
Bachet-Labosne
I-(16,11, 6):
Bachet-Labosne
I-(16,12, 7):
Bachet-Labosne
I-(20,13, 9):
Bachet-Labosne
I-(42,27,12):
Bachet-Labosne
II-(10,3,2;6)
= II(10,3,2;4)
II-(11,4,3;9):
= II(11,4,3;2)
II-( ,5,3;1):
II-( ,5,3;4):
II-( ,7,4;5):
II-( ,8,5;11):
III-(20;19,13,7;10):
Leacock
McKay
Wood, Serebriakoff, Diagram Group
Chuquet, Wood, Fireside Amusements,
Meyer, Stein, Brandes
Young World,
Devi
General problem, usually form I, sometimes form II: Bachet-Labosne, Schubert, Ahrens,
Cowley, Tweedie, Grossman, Buker, Goodstein, Browne, Scott, Currie, Sawyer,
Court, O'Beirne, Lawrence, McDiarmid & Alfonsin.
Versions with 4 or more jugs: Tartaglia, Anon: Problems drive (1958), Anon (1961),
O'Beirne.
Impossible versions: Pacioli, Bachet, Anon: Problems drive (1958).
Abbot Albert. c1240. Prob. 4, p. 333. I-(8,5,3) -- one solution.
Columbia Algorism. c1350. Chap. 123: I-(8,5,3). Cowley 402-403 & plate opposite 403.
The plate shows the text and three jars. I have a colour slide of the three jars from the
MS.
Munich 14684. 14C. Prob. XVIII & XXIX, pp. 80 & 83. I-(8,5,3).
Folkerts. Aufgabensammlungen. 13-15C. 16 sources with I-(8,5,3).
Pseudo-dell'Abbaco. c1440. Prob. 66, p.62. I-(8,5,3) -- one solution. "This problem is of
little utility ...." I have a colour slide of this.
Chuquet. 1484. Prob. 165. Measure 4 from a cask using 5 and 3. You can pour back into the
cask, i.e. this is II-(,5,3;4). FHM 233 calls this the tavern-keeper's problem.
HB.XI.22. 1488. P. 55 (= Rath 248). Same as Abbot Albert.
Pacioli. De Viribus. c1500.
Ff. 97r - 97v. LIII. C(apitolo). apartire una botte de vino fra doi (To divide a bottle of
wine between two). = Peirani 137-138. I-(8,5,3). One solution.
Ff. 97v - 98v. LIIII. C(apitolo). a partire unaltra botte fra doi (to divide another bottle
SOURCES - page 76
between two). = Peirani 138-139. I-(12,7,5). Dario Uri points out that the
solution is confused and he repeats himself so it takes him 18 pourings instead
of the usual 11. He then says one can divide 18 among three brothers who have
containers of sizes 5, 6, 7, which he does by filling the 6 and then the problem
is reduced to the previous problem. [He could do it rather more easily by pouring
the 6 into the 7 and then refilling the 6!]
Ff. 98v - 99r. LV. (Capitolo) de doi altri sotili divisioni. de botti co'me se dira (Of two
other subtle divisions of bottles as described). = Peirani 139-140. I-(10,6,4) and
I-(12,8,4). Pacioli suggests giving these to idiots.
Ghaligai. Practica D'Arithmetica. 1521. Prob. 20, ff. 64v-65r. I-(8,5,3). One solution.
Cardan. Practica Arithmetice. 1539. Chap. 66, section 33, f. DD.iiii.v (p. 145). I-(8,5,3).
Gives one solution and says one can go the other way.
H&S 51 says I-(8,5,3) case is also in Trenchant (1566). ??NYS
Tartaglia. General Trattato, 1556, art. 132 & 133, p. 255v-256r.
Art. 132: I-(8,5,3).
Art. 133: divide 24 in thirds, using 5, 11, 13.
Buteo. Logistica. 1559. Prob. 73, pp. 282-283. I-(8,5,3).
Gori. Libro di arimetricha. 1571. Ff. 71r-71v (p. 76). I-(8,5,3).
Bachet. Problemes. 1612. Addl. prob. III: Deux bons compagnons ont 8 pintes de vin à
partager entre eux également, ..., 1612: 134-139; 1624: 206-211; 1884: 138-147.
I-(8,5,3) -- both solutions; I-(12,8,5) (omitted by Labosne). Labosne adds I-(16,9,7);
I-(16,11,6); I-(42,27,12); I-(20,13,9); I-(16,12,7) (an impossible case!) and discusses
general case. (This seems to be the first discussion of the general case.)
van Etten. 1624. Prob. 9 (9), pp. 11 & fig. opp. p. 1 (pp. 22-23). I-(8,5,3) -- one solution.
Henrion's Nottes, 1630, pp. 11-13, gives the second solution and poses and solves
I-(12,7,5).
Hunt. 1631 (1651). P. 270 (262). I-(8,5,3). One solution.
Yoshida (Shichibei) Kōyū (= Mitsuyoshi Yoshida) (1598-1672). Jinkō-ki. 2nd ed., 1634 or
1641??. ??NYS The recreational problems are discussed in Kazuo Shimodaira;
Recreative Problems on "Jingōki", a 15 pp booklet sent by Shigeo Takagi. [This has no
details, but Takagi says it is a paper that Shimodaira read at the 15th International
Conference for the History of Science, Edinburgh, Aug 1977 and that it appeared in
Japanese Studies in the History of Science 16 (1977) 95-103. I suspect this is a copy of
a preprint.] This gives both Jingōki and Jinkōki as English versions of the title and says
the recreational problems did not appear in the first edition, 4 vols., 1627, but did
appear in the second edition of 5 vols. (which may be the first use of coloured wood
cuts in Japan), with the recreational problems occurring in vol. 5. He doesn't give a
date, but Mikami, p. 179, indicates that it is 1634, with further editions in 1641, 1675,
though an earlier work by Mikami (1910) says 2nd ed. is 1641. Yoshida (or
Suminokura) is the family name. Shimodaira refers to the current year as the 350th
anniversary of the edition and says copies of it were published then. I have a recent
transcription of some of Yoshida into modern Japanese and a more recent translation
into English, ??NYR, but I don't know if it is the work mentioned by Shimodaira.
Shimodaira discusses a jug problem on p. 14: I-(10,7,3) -- solution in 10 moves.
Shimodaira thinks Yoshida heard about such puzzles from European contacts, but
without numerical values, then made up the numbers. I certainly can see no other
example of these numbers. The recent transcription includes this material as prob. 7 on
pp. 69-70.
Wingate/Kersey. 1678?. Prob. 7, pp. 543-544. I-(8,5,3). Says there is a second way to do it.
Witgeest. Het Natuurlyk Tover-Boek. 1686. Prob. 38, p. 308. I-(8,5,3).
Ozanam. 1694.
Prob. 36, 1696: 91-92; 1708: 82-83. Prob. 42, 1725: 238-240. Prob. 21, 1778:
175-177; 1803: 174-176; 1814: 153-154. Prob. 20, 1840: 79. I-(8,5,3) -- both
solutions.
Prob. 43, 1725: 240-241. Prob. 22, 1778: 177-178; 1803: 176-177; 1814: 154-155.
Prob. 21, 1840: 79-80. I-(12,7,5) -- one solution.
Dilworth. Schoolmaster's Assistant. 1743. Part IV: Questions: A short Collection of pleasant
and diverting Questions, p. 168. Problem 8. I-(8,5,3). (Dilworth cites Wingate for this
-- cf in 5.B.) = D. Adams; Scholar's Arithmetic; 1801, p. 200, no. 10.
Les Amusemens. 1749. Prob. 17, p. 139: Partages égaux avec des Vases inégaux. I-(8,5,3) - both solutions.
SOURCES - page 77
Bestelmeier. 1801. Item 416: Die 3 Maas-Gefäss. I-(12,7,5).
Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 48-49, no. 75:
How to part an eight gallon bottle of wine, equally between two persons, using only two
other bottles, one of five gallons, and the other of three. Gives both solutions.
Jackson. Rational Amusement. 1821. Arithmetical Puzzles.
No. 14, pp. 4 & 54. I-( 8,5,3). One solution.
No. 52, pp. 12 & 67. I-(12,7,5). One solution.
Rational Recreations. 1824. Exer. 10, p. 55. I-(8,5,3) one way.
Manuel des Sorciers. 1825. ??NX
Pp. 55-56, art. 27-28. I-(8,5,3) two ways.
P. 56, art. 29. I-(12,7,5).
Endless Amusement II. 1826? Prob. 7, pp. 193-194. I-(8,5,3). One solution. = New Sphinx,
c1840, p. 133.
Nuts to Crack III (1834), no. 212. I-(8,5,3). 8 gallons of spirits.
Young Man's Book. 1839. Pp. 43-44. I-(8,5,3). Identical to Wingate/Kersey.
The New Sphinx. c1840. P. 133. I-(8,5,3). One solution.
Boy's Own Book. 1843 (Paris): 436 & 441, no. 7. The can of ale: 1855: 395; 1868: 432.
I-(8,5,3). One solution. The 1843 (Paris) reads as though the owners of the 3 and 5
kegs both want to get 4, which would be a problem for the owner of the 3. = Boy's
Treasury, 1844, pp. 425 & 429.
Fireside Amusements. 1850. Prob. 9, pp. 132 & 184. II-(,5,3;4). One solution.
Arago. [Biographie de] Poisson (16 Dec 1850). Oeuvres, Gide & Baudry, Paris, vol. 2, 1854,
pp. 593-??? P. 596 gives the story of Poisson's being fascinated by the problem
I-(12,8,5). "Poisson résolut à l'instant cette question et d'autres dont on lui donna
l'énoncé. Il venait de trouver sa véritable vocation." No solution given by Arago.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868.
Arithmetical puzzles, no. 8, pp. 174-175 (1868: 185-186). I-(8,5,3). Milkmaid with
eight quarts of milk.
Magician's Own Book. 1857.
P. 223-224: Dividing the beer: I-(8,5,3).
P. 224: The difficult case of wine: I-(12,7,5).
Pp. 235-236: The two travellers: I-(8,5,3) posed in verse.
Each problem gives just one solution.
Boy's Own Conjuring Book. 1860.
P. 193: Dividing the beer: I-(8,5,3).
P. 194: The difficult case of wine: I-(12,7,5).
Pp. 202-203: The two travellers: I-(8,5,3) posed in verse.
Each problem gives just one solution.
Illustrated Boy's Own Treasury. 1860. Prob. 21, pp. 428-429 & 433. I-(8,5,3). "A man
coming from the Lochrin distillery with an 8-pint jar full of spirits, ...."
Vinot. 1860. Art. XXXVIII: Les cadeaux difficiles, pp. 57-58. I-(8,5,3). Two solutions.
The Secret Out (UK). c1860. To divide equally eight pints of wine ..., pp. 12-13.
Bachet-Labosne. 1874. For details, see Bachet, 1612. Labosne adds a consideration of the
general case which seems to be the first such.
Kamp. Op. cit. in 5.B. 1877. No. 17, p. 326: I-(8,5,3).
Mittenzwey. 1880. Prob. 106, pp. 22 & 73-74; 1895?: 123, pp. 26 & 75-76; 1917: 123,
pp. 24 & 73-74. I-(8,5,3). One solution.
Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 135,
no. 1. I-(8,5,3). No solution.
Loyd. Problem 11: "Two thieves of Damascus". Tit-Bits 31 (19 Dec 1896 & 16 Jan 1897)
211 & 287. Thieves found with 2 & 2 quarts in pails of size 3 & 5. They claim the
merchant measured the amounts out from a fresh hogshead. Solution is that this could
only be done if the merchant drained the hogshead, which is unreasonable!
Loyd. Problem 13: The Oriental problem. Tit-Bits 31 (19 Jan, 30 Jan & 6 Feb 1897) 269,
325 & 343. = Cyclopedia, 1914, pp. 188 & 364: The merchant of Bagdad. Complex
problem with hogshead of water, barrel of honey, three 10 gallon jugs to be filled with 3
gallons of water, of honey and of half and half honey & water. There are a 2 and a 4
gallon measure and also 13 camels to receive 3 gallons of water each. Solution takes
521 steps. 6 Feb reports solutions in 516 and 513 steps. Cyclopedia gives solution
in 506 steps.
Dudeney. The host's puzzle. London Magazine 8 (No. 46) (May 1902) 370 & 8 (No. 47)
SOURCES - page 78
(Jun 1902) 481-482 (= CP, prob. 6, pp. 28-29 & 166-167). Use 5 and 3 to obtain 1
and 1 from a cask. One must drink some!
H. Schubert. Mathematische Mussestunden, 3rd ed., Göschen, Leipzig, 1907. Vol. 1, chap. 6,
Umfüllungs-Aufgaben, pp. 48-56. Studies general case and obtains some results. (The
material appeared earlier in Zwölf Geduldspiele, 1895, op. cit. in 5.A, Chap. IX,
pp. 110-119. The 13th ed. (De Gruyter, Berlin, 1967), Chap. 9, pp. 62-70, seems to be a
bit more general (??re-read).)
Ahrens. MUS I, 1910, chap. 4, Umfüllungsaufgaben, pp. 105-124. Pp. 106-107 is Arago's
story of Poisson and this problem. He also extends and corrects Schubert's work.
Dudeney. Perplexities: No. 141: New measuring puzzle. Strand Magazine 45 (Jun 1913) 710
& 46 (Jul 1913) 110. (= AM, prob. 365, pp. 110 & 235.) Two 10 quart vessels of
wine with 5 and 4 quart measures. He wants 3 quarts in each measure. (Dudeney gives
numerous other versions in AM.)
Loyd. Cyclopedia. 1914. Milkman's puzzle, pp. 52 & 345. (= MPSL2, prob. 23, pp. 17 &
127-128 = SLAHP: Honest John, the milkman, pp. 21 & 90.) Milkman has two full 40
quart containers and two customers with 5 and 4 quart pails, but both want 2 quarts.
(Loyd Jr. says "I first published [this] in 1900...")
Williams. Home Entertainments. 1914. The measures puzzle, p. 125. I-(8,5,3).
Hummerston. Fun, Mirth & Mystery. 1924. A shortage of milk, Puzzle no. 75, pp. 164 &
183. I-(8,5,3), one solution.
Elizabeth B. Cowley. Note on a linear diophantine equation. AMM 33 (1926) 379-381.
Presents a technique for resolving I-(a,b,c), which gives the result when a = b+c. If
a < b+c, she only seems to determine whether the method gets to a point with A
empty and neither B nor C full and it is not clear to me that this implies impossibility.
She mentions a graphical method of Laisant (Assoc. Franç. Avance. Sci, 1887, pp. 218235) ??NYS.
Wood. Oddities. 1927.
Prob. 15: A problem in pints, pp. 16-17. Small cask and measures of size 5 and 3,
measure out 1 in each measure. Starts by filling the 5 and the 3 and then
emptying the cask, so this becomes a variant of II-(,5,3;1).
Prob. 26: The water-boy's problem, pp. 28-29. II-(;,5,3;4).
Ernest K. Chapin. Scientific Problems and Puzzles. In: S. Loyd Jr.; Tricks and Puzzles,
Vol. 1 (only volume to appear); Experimenter Publishing Co., NY, nd [1927] and
Answers to Sam Loyd's Tricks and Puzzles, nd [1927]. [This book is a selection of
pages from the Cyclopedia, supplemented with about 20 pages by Chapin and some
other material.] P. 89 & Answers p. 8. You have a tablet that has to be dissolved in
7½ quarts of water, though you only need 5 quarts of the resulting mixture. You have 3
and 5 quart measures and a tap.
Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane,
The Bodley Head, 1939, p. 298. "He's trying to think how a farmer with a ten-gallon
can and a three-gallon can and a two-gallon can, manages to measure out six gallons of
milk." II-(10,3,2;6) = II-(10,3,2;4).
M. C. K. Tweedie. A graphical method of solving Tartaglian measuring puzzles. MG 23
(1939) 278-282. The elegant solution method using triangular coordinates.
H. D. Grossman. A generalization of the water-fetching puzzle. AMM 47 (1940) 374-375.
Shows II-(,b,c;d) with GCD(b,c) = 1 is solvable.
McKay. Party Night. 1940.
No. 18, p. 179. II-(11,4,3;9).
No. 19, pp. 179-180. I-(8,5,3).
Meyer. Big Fun Book. 1940. No. 10, pp. 165 & 753. II-(,7,4,5).
W. E. Buker, proposer. Problem E451. AMM 48 (1941) 65. ??NX. General problem of
what amounts are obtainable using three jugs, one full to start with, i.e. I-(a,b,c). See
Browne, Scott, Currie below.
Eric Goodstein. Note 153: The measuring problem. MG 25 (No. 263) (Feb 1941) 49-51.
Shows II-(,b,c;d) with GCD(b,c) = 1 is solvable.
D. H. Browne & Editors. Partial solution of Problem E451. AMM 49 (1942) 125-127.
W. Scott. Partial solution of E451 -- The generalized water-fetching puzzle. AMM 51 (1944)
592. Counterexample to conjecture in previous entry.
J. C. Currie. Partial solution of Problem E451. AMM 53 (1946) 36-40. Technical and not
complete.
W. W. Sawyer. On a well known puzzle. SM 16 (1950) 107-110. Shows that I-(b+c,b,c) is
SOURCES - page 79
solvable if b & c are relatively prime.
David Stein. Party and Indoor Games. Op. cit. in 5.B. c1950. Prob. 13, pp. 79-80. Obtain 5
from a spring using measures 7 and 4, i.e. II-(,7,4,5).
Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14-16 & 30. No. 8. Given an
infinite source, use: 6, 10, 15 to obtain 1, 6, 7 simultaneously; 4, 6, 9, 12 to obtain
1, 2, 3, 4 simultaneously; 6, 9, 12, 15, 21 to obtain 1, 3, 6, 8, 9 simultaneously.
Answer simply says the first two are possible (the second being easy) and the third is
impossible.
Young World. c1960. P. 58: The 11 pint problem. II-(,8,5;11). This is the same as
II-(13,8,5;11) or II-(13,8,5,2).
Anonymous. Moonshine sharing. RMM 2 (Apr 1961) 31 & 3 (Jun 1961) 46. Divide 24 in
thirds using cylindrical containers holding 10, 11, 13. Solution in No. 3 uses the
cylindricity of a container to get it half full.
Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961.
"Pouring" problems -- The "robot" method. General description of the problem.
Attributes Tweedie's triangular 'bouncing ball' method to Perelman, with no reference.
Does I-(8,5,3) two ways, also I-(12,7,5) and I-(16,9,7), then considers type II
questions. Considers the problem with II-(10,6,4;d) and extends to II-(a,6,4;d) for
a > 10, leaving it to the reader to "try to formulate some rule about the results." He
then considers II-(7,6,4;d), noting that the parallelogram has a corner trimmed off.
Then considers II-(12,9,7;d) and II-(9,6,3;d).
Lloyd Jim Steiger. Letter. RMM 4 (Aug 1961) 62. Solves the RMM 2 problem by putting
the 10 inside the 13 to measure 3.
Irving & Peggy Adler. The Adler Book of Puzzles and Riddles. Or Sam Loyd Up-To-Date.
John Day, NY, 1962. Pp. 32 & 46. Farmer has two full 10-gallon cans. Girls come
with 5-quart and 4-quart cans and each wants 2 quarts.
Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965.
Prob. 80, pp. 81 & 109. Tavern has a barrel with 15 pints of beer. Two customers, with
3 pint and 5 pint jugs appear and ask for 1 pint in each jug. Bartender finds it necessary
to drink the other 13 pints!
T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Chap. 4: Jug and bottle department,
pp. 49-75. This gives an extensive discussion of Tweedie's method and various
extensions to four containers, a barrel of unknown size, etc.
P. M. Lawrence. An algebraic approach to some pouring problems. MG 56 (No. 395) (Feb
1972) 13-14. Shows II-(,b,c,d) with d  b+c and GCD(b,c) = 1 is possible and
extends to more jugs.
Louis Grant Brandes. The Math. Wizard. revised ed., J. Weston Walch, Portland, Maine,
1975. Prob. 5: Getting five gallons of water: II-(,7,4,5).
Shakuntala Devi. Puzzles to Puzzle You. Orient Paperbacks (Vision Press), Delhi, 1976.
Prob. 53: The three containers, pp. 57 & 110. III-(20;19,13,7;10). Solution in 15 steps.
Looking at the triangular coordinates diagram of this, one sees that it is actually
isomorphic to II-(19,13,7;10) and this can be seen by considering the amounts
of empty space in the containers.
Prob. 132: Mr. Portchester's problem, pp. 82 & 132. Same as Dudeney (1913).
Victor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A
Second Mensa Puzzle Book, 1985, Muller, London, as: The Mensa Puzzle Book,
Treasure Press, London, 1991.) Problem T.16: Pouring puttonos, part b, pp. 19-20
(1991: 37-38) & Answer 19, pp. 102-103 (1991: 118-119). II-( ,5,3;1).
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley,
Middlesex, 1984. Problem 161, with Solution at the back of the book. II-(,5,3;1),
which can be done as II-(8,5,3;1).
D. St. P. Barnard. 50 Daily Telegraph Brain Twisters. 1985. Op. cit. in 4.A.4. Prob. 4:
Measure for measure, pp. 15, 79-80, 103. Given 10 pints of milk, an 8 pint bowl, a jug
and a flask. He describes how he divides the milk in halves and you must deduce the
size of the jug and the flask.
Colin J. H. McDiarmid & Jorge Ramirez Alfonsin. Sharing jugs of wine. Discrete
Mathematics 125 (1994) 279-287. Solves I-(b+c,b,c) and discusses the problem of
getting from one state of the problem to another in a given number of steps, showing
that GCD(b,c) = 1 guarantees the graph is connected. indeed essentially cyclic.
Considers GCD(b,c)  1. Notes that the work done easily extends to a > b + c. Says
the second author's PhD at Oxford, 1993, deals with more cases.
SOURCES - page 80
John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for
all Ages. Keystone Agencies, Radnor, Ohio, 1997. P. 11: The spoon and the bottle.
Given a 160 ml bottle and a 30 ml spoon, measure 230 ml into a bucket.
5.D.2.RULER WITH MINIMAL NUMBER OF MARKS
Dudeney. Problem 518: The damaged measure. Strand Mag. (Sep 1920) ??NX. Wants a
minimal ruler for 33 inches total length. (=? MP 180)
Dudeney. Problem 530: The six cottagers. Strand Mag. (Jan 1921) ??NX. Wants 6 points
on a circle to give all arc distances 1, 2, ..., 20. (=? MP 181)
Percy Alexander MacMahon. The prime numbers of measurement on a scale. Proc. Camb.
Philos. Soc. 21 (1922-23) 651-654. He considers the infinite case, i.e. a(0) = 0,
a(i+1) = a(i) + least integer which is not yet measurable. This gives:
0, 1, 3, 7, 12, 20, 30, 44, ....
Dudeney. MP. 1926.
Prob. 180: The damaged measure, pp. 77 & 167. (= 536, prob. 453, pp. 173, 383-384.)
Mark a ruler of length 33 with 8 (internal) marks. Gives 16 solutions.
Prob. 181: The six cottagers, pp. 77-78 & 167. = 536, prob. 454, pp. 174 & 384.
A. Brauer. A problem of additive number theory and its application in electrical engineering.
J. Elisha Mitchell Sci. Soc. 61 (1945) 55-56. Problem arises in designing a resistance
box.
Л. Редеи & А. Реньи [L. Redei & A. Ren'i (Rényi)]. О представленин чисел 1, 2,
..., N лосредством разностей [O predstavlenin chisel 1, 2, ... , N losredstvom
raznosteĭ (On the representation of 1, 2, ..., N by differences)]. Мат. Сборник
[Mat. Sbornik] 66 (NS 24) (1949) 385-389.
Anonymous. An unsolved problem. Eureka 11 (Jan 1949) 11 & 30. Place as few marks as
possible to permit measuring integers up to n. For n = 13, an example is: 0, 1, 2, 6,
10, 13. Mentions some general results for a circle.
John Leech. On the representation of 1, 2, ..., n by differences. J. London Math. Soc. 31
(1956) 160-169. Improves Redei & Rényi's results. Gives best examples for small n.
Anon. Puzzle column: What's your potential? MTg 19 (1962) 35 & 20 (1962) 43. Problem
posed in terms of transformer outputs -- can we arrange 6 outputs to give every
integral voltage up through 15? Problem also asks for the general case. Solution
asserts, without real proof, that the optimum occurs with 0, 1, 4, 7, 10, ..., n-11, n-8,
n-5, n-2 or its complement.
T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Chap. 6 discusses several versions of
the problem.
Gardner. SA (Jan 1965) c= Magic Numbers, chap. 6. Describes 1, 2, 6, 10 on a ruler 13
long. Says 3 marks are sufficient on 9 and 4 marks on 12 and asks for proof of the
latter and for the maximum number of distances that 3 marks on 12 can produce.
How can you mark a ruler 36 long? Says Dudeney, MP prob. 180, believed that 9
marks were needed for a ruler longer than 33, but Leech managed to show 8 was
sufficient up to 36.
C. J. Cooke. Differences. MTg 47 (1969) 16. Says the problem in MTg 19 (1962) appears in
H. L. Dorwart's The Geometry of Incidence (1966) related to perfect difference sets but
with an erroneous definition which is corrected by references to H. J. Ryser's
Combinatorial Mathematics. However, this doesn't prove the assertions made in
MTg 20.
Jonathan Always. Puzzles for Puzzlers. Tandem, 1971. Prob. 22: Starting and stopping,
pp. 18 & 66. Circular track, 1900 yards around. How can one place marker posts so
every multiple of 100 yards up to 1900 can be run. Answer: at 0, 1, 3, 9, 15.
Gardner. SA (Mar 1972) = Wheels, Chap. 15.
5.D.3 FALSE COINS WITH A WEIGHING SCALE
H. S. Shapiro, proposer; N. J. Fine, solver. Problem E1399 -- Counterfeit coins. AMM 67
(1960) 82 & 697-698. Genuines weigh 10, counterfeits weigh 9. Given 5 coins
and a scale, how many weighings are needed to find the counterfeits? Answer is 4.
Fine conjectures that the ratio of weighings to coins decreases to 0.
Kobon Fujimura & J. A. H. Hunter, proposers; editorial solution. There's always a way.
RMM 6 (Dec 1961) 47 & 7 (Feb 1962) 53. (c= Fujimura's The Tokyo Puzzles
SOURCES - page 81
(Muller, London, 1979), prob. 29: Pachinko balls, pp. 35 & 131.) Six coins, one false.
Determine which is false and whether it is heavy or light in three weighings on a scale.
In fact one also finds the actual weights.
K. Fujimura, proposer; editorial solution. The 15-coin puzzle. RMM 9 (Jun 1962) & 10
(Aug 1962) 40-41. Same problem with fifteen coins and four weighings.
5.D.4.TIMING WITH HOURGLASSES
I have just started these and they are undoubtedly older than the examples here. I don't
recall ever seeing a general approach to these problems.
Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 17:
Two-minute eggs, pp. 9 & 87. Time 2 minutes with 3 & 5 minute timers.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 21: The
sands of time, pp. 35 & 93. Time 9 minutes with 4 & 7 minute timers.
David B. Lewis. Eureka! Perigee (Putnam), NY, 1983. Pp. 73-74. Time 9 minutes with
4 & 7 minute timers.
Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995.
Chap. 1, prob. 8: Grandfather's breakfast, pp. 6 & 102. Time 15 minutes with 7 & 11
minute timers.
5.D.5.MEASURE HALF A BARREL
I have just started this and there must be much older examples.
Benson. 1904. The water-glass puzzle, p. 254.
Dudeney. AM. 1917. Prob. 364: The barrel puzzle, pp. 109-110 & 235.
King. Best 100. 1927. No. 1, pp. 7 & 38.
Collins. Fun with Figures. 1928. The dairymaid's problem, pp. 29-30.
William A. Bagley. Puzzle Pie. Vawser & Wiles, London, nd [BMC gives 1944]. [There is a
revised edition, but it only affects material on angle trisection.] No. 14: 'Arf an' 'arf,
p. 15.
Anon. The Little Puzzle Book. Peter Pauper Press, Mount Vernon, NY, 1955. P. 52: The
cider barrel.
Jonathan Always. Puzzles for Puzzlers. Tandem, London, 1971. Prob. 87: But me no butts,
pp. 42 & 88.
Richard I. Hess. Email Christmas message to NOBNET, 24 Nov 2000. Solution sent by Nick
Baxter on the same day. You have aquaria (assumed cuboidal) which hold 7 and 12
gallons and a water supply. The 12 gallon aquarium has dots accurately placed in the
centre of each side face. How many steps are required to get 8 gallons into the 12
gallon aquarium? Fill the 12 gallon aquarium and tilt it on one corner so the water level
passes through the centres of the two opposite faces. This leaves 8 gallons! Nick says
this is two steps.
5.E. EULER CIRCUITS AND MAZES
Euler circuits have been used in primitive art, often as symbols of the passage of the
soul to the land of the dead. [MTg 110 (Mar 1985) 55] shows examples from Angola and
New Hebrides. See Ascher (1988 & 1991) for many other examples from other cultures.
┌────┬─────────┬────┐
│
│
│
│
├────┴────┬────┴────┤
│
│
│
└─────────┴─────────┘
Above is the 'five-brick pattern'. See: Clausen, Listing, Kamp, White, Dudeney,
Loyd Jr, Ripley, Meyer, Leeming, Adams, Anon., Ascher. Prior to Loyd Jr, the problem
asked for the edges to be drawn in three paths, but about 1920 the problem changed to
drawing a path across every wall.
Trick solutions: Tom Tit, Dudeney (1913), Houdini, Loyd Jr, Ripley, Meyer,
SOURCES - page 82
Leeming, Adams, Gibson, Anon. (1986).
Non-crossing Euler circuits: Endless Amusement II, Bellew, Carroll 1869,
Mittenzwey, Bile Beans, Meyer, Gardner (1964), Willson, Scott, Singmaster.
Kn denotes the complete graph on n vertices.
Matthäus Merian the Elder. Engraved map of Königsberg. Bernhard Wiezorke has sent me a
coloured reproduction of this, dated as 1641. He used an B&W version in his article:
Puzzles und Brainteasers; OR News, Ausgabe 13 (Nov 2001) 52-54. BLW use a B&W
version on their dust jacket and on p. 2 which they attribute to M. Zeiller; Topographia
Prussiae et Pomerelliae; Frankfurt, c1650. I have seen this in a facsimile of the
Cosmographica due to Merian in the volume on Brandenburg and Pomerania, but it was
not coloured. There seem to be at least two versions of this picture --??CHECK.
L. Euler. Solutio problematis ad geometriam situs pertinentis. (Comm. Acad. Sci. Petropol. 8
(1736(1741)) 128-140.) = Opera Omnia (1) 7 (1923) 1-10. English version: Seven
Bridges of Königsberg is in: BLW, 3-8; SA 189 (Jul 1953) 66-70; World of
Mathematics, vol. 1, 573-580; Struik, Source Book, 183-187.
My late colleague Jeremy Wyndham became interested in the seven bridges problem and
made inquiries which turned up several maps of Königsberg and a list of all the bridges
and their dates of construction (though there is some ambiguity about one bridge). The
first bridge was built in 1286 and until the seventh bridge of 1542, an Euler path was
always possible. No further bridge was built until a railway bridge in 1865 which led to
Saalschütz's 1876 paper -- see below. In 1905 and later, several more bridges were
added, reaching a maximum of ten bridges in 1926 (with 4512 paths from the island),
then one was removed in 1933. Then a road bridge was added, but it is so far out that it
does not show on any map I have seen. Bombing and fighting in 1944-1945 apparently
destroyed all the bridges and the Russians have rebuilt six or seven of them. I have
computed the number of paths in each case -- from 1865 until 1935 or 1944, there were
always Euler paths.
L. Poinsot. Sur les polygones et les polyèdres. J. École Polytech. 4 (Cah. 10) (1810) 16-48.
Pp. 28-33 give Euler paths on K2n+1 and Euler's criterion. Discusses square with
diagonals.
Endless Amusement II. 1826? Prob. 34, p. 211. Pattern of two overlapping squares has a
non-crossing Euler circuit.
Th. Clausen. De linearum tertii ordinis propietatibus. Astronomische Nachrichten 21 (No.
494) (1844), col. 209-216. At the very end, he gives the five-brick pattern and says that
its edges cannot be drawn in three paths.
J. B. Listing. Vorstudien zur Topologie. Göttinger Studien 1 (1847) 811-875. ??NYR.
Gives five brick pattern as in Clausen.
?? Nouv. Ann. Math. 8 (1849?) 74. ??NYS. Lucas says this poses the problem of finding
the number of linear arrangements of a set of dominoes. [For a double N set, N = 2n,
this is (2n+1)(n+1) times the number of circular arrangements, which is n2n+1 times
the number of Euler circuits on K2n+1.]
É. Coupy. Solution d'un problème appartenant a la géométrie de situation, par Euler. Nouv.
Ann. Math. 10 (1851) 106-119. Translation of Euler. Translator's note on p. 119
applies it to the bridges of Paris.
The Sociable. 1858. Prob. 7: Puzzle pleasure garden, pp. 288 & 303. Large maze-like
garden and one is to pass over every path just once -- phrased in verse. = Book of 500
Puzzles, 1859, prob. 7, pp. 6 & 21. = Illustrated Boy's Own Treasury, 1860, prob. 49,
pp. 405 & 443. In fact, if one goes straight across every intersection, one finds the path,
so this is really almost a unicursal problem.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 587, pp. 297 & 410: Ariadnerätsel.
Three diagrams to trace with single lines. No attempt to avoid crossings.
Frank Bellew. The Art of Amusing. Carleton, NY (& Sampson Low & Co., London), 1866
[C&B list a 1871]; John Camden Hotten, London, nd [BMC & NUC say 1870] and
John Grant, Edinburgh, nd [c1870 or 1866?], with slightly different pagination. 1866:
pp. 269-270; 1870: p. 266. Two overlapping squares have a non-crossing Euler circuit.
Lewis Carroll. Letter of 22 Aug 1869 to Isabel Standen. Taken from: Stuart Dodgson
Collingwood; The Life and Letters of Lewis Carroll; T. Fisher Unwin, London, (Dec
1898), 2nd ed., Jan 1899, p. 370: "Have you succeeded in drawing the three squares?"
On pp. 369-370, the recipient is identified as Isabel Standen and she is writing
SOURCES - page 83
Collingwood, apparently sending him the letter. Collingwood interpolates: "This
puzzle was, by the way, a great favourite of his; the problem is to draw three interlaced
squares without going over the same lines twice, or taking the pen off the paper". But
no diagram is given.
Dudeney; Some much-discussed puzzles; op. cit. in 2; 1908, quotes Collingwood,
gives the diagram and continues: "This is sometimes ascribed to him [i.e. Carroll] as its
originator, but I have found it in a little book published in 1835." This was probably a
printing of Endless Amusement II, qv above and in Common References, though this
has two interlaced squares. John Fisher; The Magic of Lewis Carroll; op. cit. in 1;
pp. 58-59, says Carroll would ask for a non-crossing Euler circuit, but this is not clearly
stated in Collingwood. Cf Carroll-Wakeling, prob. 29: The three squares, pp. 38 & 72,
which clearly states that a non-crossing circuit is wanted and notes that there is more
than one solution. Cf Gardner (1964). Carroll-Gardner, pp. 52-53.
Mittenzwey. 1880. Prob. 269-279, pp. 47-48 & 98-100; 1895?: 298-308, pp. 51-52 &
100-102; 1917: 298-308, pp. 46-48 & 95-97. Straightforward unicursal patterns. The
first is K5, but one of the diagonals was missing in my copy of the 1st ed. -- the path is
not to use two consecutive outer edges. The third is the 'envelope' pattern. The fourth
is three overlapping squares, where the two outer squares just touch in the middle. The
last is a simple maze with no dead ends and the path is not to cross itself. See also the
entry for Mittenzwey in 5.E.1, below.
M. Reiss. Évaluation du nombre de combinaisons desquelles les 28 dés d'un jeu de dominos
sont susceptibles d'après la règle de ce jeu. Annali di Matematica Pura ed Applicata (2)
5 (1871) 63-120. Determines the number of linear arrangements of a double-6 set of
dominoes, which gives the number of Euler circuits on K7.
L. Saalschütz. [Report of a lecture.] Schriften der Physikalisch-Ökonomischen Gesellschaft
zu Königsberg 16 (1876) 23-24. Sketches Euler's work, listing the seven bridges. Says
that a recent railway bridge, of 1865, connecting regions B and C on Euler's diagram,
can be considered within the walkable region. He shows there are 48 x 2 x 4 = 384
possible paths -- the 48 are the lists of regions visited starting with A; the 2
corresponds to reversing these lists; the 4 (= 2 x 2) corresponds to taking each of the
two pairs of bridges connecting the same regions in either order, He lists the 48
sequences of regions which start at A. I wrote a program to compute Euler paths and I
tested it on this situation. I find that Saalschütz has omitted two cases, leading to four
sequences or 16 paths starting at A or 32 paths considering both directions. That is,
his 48 should be 52 and his 384 should be 416.
Kamp. Op. cit. in 5.B. 1877. Pp. 322-327 show several unicursal problems.
No. 8 is the five-brick pattern as in Clausen.
No. 10 is two overlapping squares.
No. 11 is a diagram from which one must remove some lines to leave an Eulerian
figure.
C. Hierholzer. Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne
Unterbrechung zu umfahren. Math. Annalen 6 (1873) 30-32. (English is in BLW,
11-12.)
G. Tarry. Géométrie de situation: Nombre de manières distinctes de parcourir en une seule
course toutes les allées d'un labyrinthe rentrant, en ne passant qu'une seule fois par
chacune des allées. Comptes Rendus Assoc. Franç. Avance. Sci. 15, part 2 (1886)
49-53 & Plates I & II. General technique for the number of Euler circuits.
Lucas.
RM2. 1883. Le jeu de dominos -- Dispositions rectilignes, pp. 63-77 & Note 1:
Sur le jeu de dominos, p. 229.
RM4. 1894. La géométrie des réseaux et le problème des dominos, pp. 123-151.
Cites Reiss's work and says (in RM4) that it has been confirmed by Jolivald. The
note in RM2 is expanded in RM4 to explain the connection between dominoes and
K2n+1. There are obviously 2 Euler circuits on K3. He sketches Tarry's method and
uses it to compute that K5 has 88 Euler circuits and K7 has 1299 76320. [This gives
28 42582 11840 domino rings for the double-6 set.] He says Tarry has found that K9
has 911 52005 70212 35200.
Tom Tit, vol. 3. 1893. Le rectangle et ses diagonales, pp. 155-156. = K, no. 16: The
rectangle and its diagonals, pp. 46-48. = R&A, The secret of the rectangle, p. 100.
Trick solutions by folding the paper and making an arc on the back.
Hoffmann. 1893. Chap. X, no. 9: Single-stroke figures, pp. 338 & 375 = Hoffmann-Hordern,
SOURCES - page 84
pp. 230-231. Three figures, including the double crescent 'Seal of Mahomet'. Answer
states Euler's condition.
Dudeney. The shipman's puzzle. London Mag. 9 (No. 49) (Aug 1902) 88-89 & 9 (No. 50)
(Sep 1902) 219 (= CP, prob. 18, pp. 40-41 & 173). Number of Euler circuits on K5.
Benson. 1904. A geometrical problem, p. 255. Seal of Mahomet.
William F. White. A Scrap-Book of Elementary Mathematics. Open Court, 1908. [The 4th
ed., 1942, is identical in content and pagination, omitting only the Frontispiece and the
publisher's catalogue.] Bridges and isles, figure tracing, unicursal signatures, labyrinths,
pp. 170-179. On p. 174, he gives the five-brick puzzle, asking for a route along its
edges.
Dudeney. Perplexities: No. 147: An old three-line puzzle. Strand Magazine 46 (Jul 1913)
110 & (Aug 1913) 221. c= AM, prob. 239: A juvenile puzzle, pp. 68-69 & 197.
Five-brick form to be drawn or rubbed out on a board in three strokes. Either way
requires doing two lines at once, either by folding the paper as you draw or using two
fingers to rub out two lines at once. "I believe Houdin, the conjurer, was fond of
showing this to his child friends, but it was invented before his time -- perhaps in the
Stone Age."
Loyd. Problem of the bridges. Cyclopedia, 1914, pp. 155 & 359-360. = MPSL1, prob. 28,
pp. 26-27 & 130-131. Eight bridges. Asks for number of routes.
Loyd. Puzzle of the letter carrier's route. Cyclopedia, 1914, pp 243 & 372. Asks for a circuit
on a 3 x 4 array with a minimal length of repeated path.
Dudeney. AM. 1917.
Prob. 242: The tube inspector's puzzle, pp. 69 & 198. Minimal route on a 3 x 4 array.
Prob. 261: The monk and the bridges, pp. 75-76 & 202-203. River with one island.
Four bridges from island, two to each side of the river, and another bridge over
the river. How many Euler paths from a given side of the river to the other?
Answer: 16.
Collins. Book of Puzzles. 1927. The fly on the octahedron, pp. 105-108. Asserts there are
1488 Euler circuits on the edges of an octahedron. He counts the reverse as a separate
circuit.
Harry Houdini [pseud. of Ehrich Weiss] Houdini's Book of Magic. 1927 (??NYS); Pinnacle
Books, NY, 1976, p. 19: Can you draw this? Take a square inscribed in a circle and
draw both diagonals. "The idea is to draw the figure without taking your pencil off the
paper and without retracing or crossing a line. There is a trick to it, but it can be done.
The trick in drawing the figure is to fold the paper once and draw a straight line
between the folded halves; then, not removing your pencil, unfold the paper. You will
find that you have drawn two straight lines with one stroke. The rest is simple." This
perplexed me for some time, but I believe the idea is that holding the pencil between the
two parts of the folded sheet and moving the pencil parallel to the fold, one can draw a
line, parallel to the fold, on each part.
Loyd Jr. SLAHP. 1928. Pp. 7-8. Discusses what he calls the "Five-brick puzzle", the
common pattern of five rectangles in a rectangle. He says that the object was to draw
the lines in four strokes -- which is easily done -- but that it was commonly misprinted
as three strokes, which he managed to do by folding the paper. He says "a similar
puzzle ... some ten or fifteen years ago" asked for a path crossing each of the 16 walls
once, which is also impossible.
The Bile Beans Puzzle Book. 1933.
No. 32. Draw the triangular array of three on an edge without crossing.
No. 36. Draw the five-brick pattern in three lines. Folds paper and draws two lines at
once.
R. Ripley. Believe It Or Not! Book 2. (Simon & Schuster, 1931); Pocket Books, NY, 1948,
pp. 70-71. = Omnibus Believe It Or Not! Stanley Paul, London, nd [c1935?], p. 270.
Gives the five-brick problem of drawing a path crossing each wall once, with the trick
solution having the path going along a wall. Asserts "This unicursal problem was
solved thus by the great Euler himself." and cites the Euler paper above!!
Meyer. Big Fun Book. 1940.
Tryangle, pp. 98 & 731. Triangle subdivided into triangles, with three small triangles
along each edge. Draw an Euler circuit without crossings.
Cutting the walls, pp. 637 & 794. Five-brick problem. Solution has line crossing
through a vertex.
Ern Shaw. The Pocket Brains Trust - No. 2. W. H. Allen, London, nd but inscribed 1944.
SOURCES - page 85
Prob. 29: Five bricks teaser, pp. 10 & 39.
Leeming. 1946. Chap. 6, prob. 2: Through the walls, pp. 70 & 184. Five-brick puzzle, with
trick solution having the path go through an intersection.
John Paul Adams. We Dare You to Solve This!. Op. cit. in 5.C. 1957? Prob. 49: In just one
line, pp. 30 & 48-49. Five-brick puzzle, with answer having the path going along a
wall, as in Ripley. Asserts Euler invented this solution.
Gibson. Op. cit. in 4.A.1.a. 1963. Pp. 70 & 75: The "impossible" diagram. Same as Tom
Tit.
Gardner. SA (Apr 1964) = 6th Book, chap. 10. Says Carroll knew that a planar Eulerian
graph could be drawn without crossings. Gives a method of O'Beirne for doing this -two colour the regions and then make a path which separates the colours into simply
connected regions.
Ripley's Puzzles and Games. 1966. P. 39. Euler paths on the 'envelope', i.e. a rectangle with
its diagonals drawn and an extra connection between the top corners, looking like an
unfolded envelope. Asserts the envelope has 50 solutions, but it is not clear if the
central crossing is a further vertex. I did this by hand but did not get 50, so I wrote a
program to count Euler paths. If the central crossing is not a vertex, then I find 44 paths
from one of the odd vertices to the other, and of course 44 going the other way -- and I
had found this number by hand. However, if the central crossing is a vertex, then my
hand solution omitted some cases and the computer found 120 paths from one odd
vertex to the other.
Pp. 40-43 give many problems of drawing non-crossing Euler paths or circuits.
W. Wynne Willson. How to abolish cross-roads. MTg 42 (Spring 1968) 56-59. Euler circuit
of a planar graph can be made without crossings.
[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. Tempo (Grosset & Dunlap),
NY, 1973 (& 1978?? -- both dates are given -- I'm presuming the 1978 is a 2nd ptg or a
reissue under a different imprint??). One line/no crossing, pp. 85-86. Non-crossing
Euler circuits on the triangular array of side 3 and non-crossing Euler paths on the
'envelope' -- cf under Ripley's, above. Asserts the envelope has 50 solutions. I adapted
the program mentioned above to count the number of non-crossing Euler paths -- one
must rearrange the first case as a planar graph -- and there are 16 in the first case and
26 in the second case. Taking the reversals doubles these numbers so it is possible that
the Scotts meant the second case and missed one path and its reversal.
David Singmaster, proposer; Jerrold W. Grossman & E. M. Reingold, solvers. Problem
E2897 -- An Eulerian circuit with no crossings. AMM 88:7 (Aug 1981) 537-538 &
90:4 (Apr 1983) 287-288. A planar Eulerian graph can be drawn with no crossings.
Solution cites some previous work.
Anon. [probably Will Shortz ??check with Shortz]. The impossible file. No. 2: In just one
line. Games (Apr 1986) 34 & 64 & (Jul 1986) 64. Five brick pattern -- draw a line
crossing each wall once. Says it appeared in a 1921 newspaper [perhaps by Loyd Jr??].
Gives the 1921 solution where the path crosses a corner, hence two walls at once. Also
gives a solution with the path going along a wall. In the July issue, Mark Kantrowitz
gives a solution by folding over a corner and also a solution on a torus.
Marcia Ascher. Graphs in cultures: A study in ethnomathematics. HM 15 (1988) 201-227.
Discusses the history of Eulerian circuits and non-crossing versions and then exposits
many forms of the idea in many cultures.
Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Chapter Two: Tracing graphs
in the sand, pp. 30-65. Sketches the history of Eulerian graphs with some interesting
references -- ??NYS. Describes graph tracing in three cultures: the Bushoong and the
Tshokwe of central Africa and the Malekula of Vanuatu (ex-New Hebrides). Extensive
references to the ethnographic literature.
5.E.1. MAZES
This section is mainly concerned with the theory. The history of mazes is sketched first,
with references to more detailed sources. There is even a journal, Caerdroia (53 Thundersley
Grove, Thundersley, Essex, SS7 3EB, England), devoted to mazes and labyrinths, mostly
concentrating on the history. It is an annual, began in 1980 and issue 31 appeared in 2000.
Mazes are considered under Euler Circuits, since the method of Euler Circuits is often
used to find an algorithm. However, some mazes are better treated as Hamiltonian Circuits -see 5.F.2.
SOURCES - page 86
A maze can be considered as a graph formed by the nodes and paths -- the path graph.
For the usual planar maze, one can also look at the graph formed by the walls -- the wall
graph, which is a kind of dual to the path graph. In later mazes, the walls do not form a
connected whole, and an isolated part of the wall appears as a region or 'face' in the path
graph. Such isolated bits of walling are sometimes called islands, but they are the same as the
components of the wall graph, with the outer wall being one component, so the number of
components is one more than the number of islands. The 'hand-on-wall' method will solve a
maze if and only if the goals are adjacent to walls in the component of the outer wall.
A 'ring maze' is a plate with holes and raised areas with an open ring which must be
removed by moving it from hole to hole. I have put these in 11.K.5 as they are a kind of
mechanical or topological puzzle, though there are versions with a simple two legged spacer.
HISTORICAL SOURCES
W. H. Matthews. Mazes & Labyrinths: A General Account of
Their History and Developments. Longmans, Green and Co.,
London, 1922. = Mazes and Labyrinths: Their History and
Development. Dover, 1970. (21 pages of references.)
[For more about the book and the author, see: Zeta Estes;
My Father, W. H. Matthews; Caerdroia (1990) 6-8.]
Walter Shepherd. For Amazement Only. Penguin, 1942; Let's go
amazing, pp. 5-12. Revised as: Mazes and Labyrinths -- A
Book of Puzzles. Dover, 1961; Let's go a-mazing,
pp. v-xi. (Only a few minor changes are made in the
text.) Sketch of the history.
Sven Bergling invented the rolling ball labyrinth puzzle/game
and they began being produced in 1946. [Kenneth Wells;
Wooden Puzzles and Games; David & Charles, Newton Abbot,
1983, p. 114.]
Walter Shepherd. Big Book of Mazes and Labyrinths. Dover,
1973, More amazement, pp. vii-x. Extends the historical
sketch in his previous book, arguing that mazes with
multiple choices perhaps derive from Iron Age hill forts
whose entrances were designed to confuse an enemy.
Janet Bord. Mazes and Labyrinths of the World. Latimer,
London, 1976. (Extensively illustrated.)
Nigel Pennick. Mazes and Labyrinths. Robert Hale, London,
1990.
Adrian Fisher [& Georg Gerster (photographer)]. The Art of the
Maze. Weidenfeld and Nicolson, London, 1990. (Also as:
Labyrinth; Solving the Riddle of the Maze; Harmony (Crown
Publishers), NY, 1990.) Origins and History occupies pp.
11-56, but he also describes many recent developments and
innovations. He has convenient tables of early examples.
Adrian Fisher & Diana Kingham. Mazes. Shire Album 264.
Shire, Aylesbury, 1991.
Adrian Fisher & Jeff Saward. The British Maze Guide. Minotaur
Designs, St. Alban's, 1991.
SOURCES - page 87
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HISTORICAL SKETCH
Up to about the 16C, all mazes were unicursal, i.e. with no decision points. The word
labyrinth is sometimes used to distinguish unicursal mazes from others, but this distinction is
not made consistently. Until about 1000, all mazes were of the classical 'Cretan' seven-ring
type shown above. (However, see Shepherd's point in his 1973 book, above.) The oldest
examples are rock carvings, the earliest being perhaps that in the Tomba del Labirinto at
Luzzanas, Sardinia, c-2000 [Fisher, pp. 12, 25, 26, with photo on p. 12]. (In fact, Luzzanas is
a local name for an uninhabited area of fields, so does not appear on any ordinary map. It is
near Benetutti. See my A Mathematical Gazetteer or Mazing in Sardinia (Caerdroia 30 (1999)
17-21). Jeff Saward writes that current archaeological feeling is that the maze is Roman,
though the cave is probably c-2000.) On pottery, there are labyrinths on fragments, c-1300,
from Tell Rif'at, Syria [the first photos of this appeared in [Caerdroia 30 (2000) 54-55]), and
on tablets, c-1200, from Pylos. Fisher [p. 26] lists the early examples. Staffen Lundán; The
labyrinth in the Mediterranean; Caerdroia 27 (1996) 28-54, catalogues all known 'Cretan'
labyrinths from prehistory to the end of antiquity, c250, excluding the Roman 'spoked' form.
All these probably had some mystical significance about the difficulty of reaching a goal,
often with substantial mythology -- e.g. Theseus in the Labyrinth or, later, the Route to
Jerusalem.
Roman mosaics were unicursal but essentially used the Cretan form four times over in
the four corners. Lundán, above, calls these 'spoked'. Most of the extant examples are 2C-4C,
but some BC examples are known -- the earliest seems to be c-110 at Selinunte, Sicily. Fisher
[pp. 36-37] lists all surviving examples. Saward says the earliest Roman example is at
Pompeii, so  79.
In the medieval period, the Christians developed a quite different unicursal maze. See
Fisher [pp. 60-67] for detailed comparison of this form with the Roman and Cretan forms.
The earliest large Christian example is the Chemin de Jerusalem of 1235 on the floor of
Chartres Cathedral. Fisher [pp. 41 & 48] lists early and later Christian examples.
SOURCES - page 88
The legendary Rosamund's Bower was located in Woodstock Park, Oxfordshire, and its
purported site is marked by a well and fountain. It was some sort of maze to conceal
Rosamund Clifford, the mistress of Henry II (1133-1189), from the Queen, Eleanor of
Aquitaine. Legend says that about 1176, Eleanor managed to solve the maze and confronted
Rosamund with the choice of a dagger or poison -- she drank the poison and Henry never
smiled again. [Fisher, p. 105]. Historically, Henry had imprisoned Eleanor for fomenting
rebellion by her sons and Rosamund was his acknowledged mistress. Rosamund probably
spent her last days at a nunnery in Godstow, near Oxford. The legend of the bower dates from
the 14C and her murder is a later addition [Collins, Book of Puzzles, 1927, p. 121.] In the
19C, many puzzle collections had a maze called Rosamund's Bower.
The earliest record of a hedge maze is of one destroyed in a siege of Paris in 1431.
Non-unicursal mazes and islands in the wall graph start to appear in the late 16C.
Matthews [p. 96] says that: "A simple "interrupted-circle" type of labyrinth was adopted as a
heraldic device by Gonzalo Perez, a Spanish ecclesiastic ... and published ... in 1566 ..." in his
translation of the Odyssey. Matthews doesn't show this, but he then [pp. 96-97] describes and
illustrates a simple maze used as a device by Bois-dofin de Laval, Archbishop of Embrum.
He copies it from Claude Paradin; Devises Héroiques et Emblèmes of the early 17C. It has
four entrances and possibly three goals, with walls having 8 components, two being part of the
outer wall. The central goals is accessible from two of the entrances, but the two minor goals
are each accessible from just one of the other entrances. Presumably this sort of thing is what
Matthews meant as an "interrupted circle".
However, Saward has found a mid 15C anonymous English poem, The Assembly of
Ladies, which describes the efforts of a group of ladies to reach the centre of a maze, which,
as he observes, implies there must be some choices involved.
[Matthews, p. 114] has three examples from a book by Androuet du Cerceau; Les Plus
Excellents Bastiments de France of 1576. Fig. 82 was in the gardens at Charleval and has
four entrances, only one of which goes to the central goal. There are four minor goals. The N
entrance connects to the NE and SE goals, with several dead ends. The E entrance is a dead
end. The S entrance goes to the SW goal. The W entrance goes to the central goal, but the
NW goal is on an island, though 'left-hand-on-wall' goes past it. Figs. 83 and 84 are
essentially identical and seem to be corruptions of unicursal examples so that most of the
maze is bypassed. In fig. 84, one has to walk around to the back of the maze to find the
correct entrance to get to the central goal, which is an interesting idea. A small internal
change in both cases and moving the entrances converts them to a standard unicursal pattern.
Matthews' Chap. XIII [pp. 100-109] is on floral mazes and reproduces some from Jan
Vredeman De Vries; Hortorum Viridariorumque Formae; Antwerp, 1583. Fig. 74 is one of
these and has two components and a short dead-end, but the 'hand-on-wall' rule solves it.
Fig. 73 is another of De Vries's, but it is not all shown. It appears to have two entrances and
there is certainly a decision point by the far gate, but one route goes to the apparent exit at the
bottom of the page. There is a small dead end near the central goal. Fig. 78 shows a maze
from a 17C manuscript book in the Harley Manuscripts at the BL, identified on p. 224 as
BM Harl. 5308 (71, a, 12). This has two components with the central goal in the inner
component, so the 'hand-on-wall' rule fails, but it brings you within sight of the centre and
Matthews describes it as unicursal! Fig. 79 is from Adam Islip; The Orchard and the Garden,
compiled from continental sources and published in 1602. It has 5 components, but four of
these are small enclosures which could be considered as minor goals, especially if they had
seats in them. The 'hand-on-wall' rule gets to the central goal. There is a lengthy dead end
which goes to two of the inner islands. Fig. 80 is from a Dutch book: J. Commelyn;
Nederlantze Hesperides of 1676. It has two components, a central goal and four minor goals.
The 'hand-on-wall' gets you to the centre and passes two minor goals. One minor goal is on a
dead end so 'left-hand-on-wall' gets to it, but 'right-hand-on-wall' does not. The fourth minor
goal is on the island.
At Versailles, c1675, André Le Nôtre built a Garden Maze, but the objective was to
visit, in correct order, 40 fountains based on Aesop's Fables. Each node of the maze had at
least one fountain. Some fountains were not at path junctions, but one can consider these as
nodes of degree two. This is an early example of a Hamiltonian problem, except that one
fountain was located at the end of a short dead end. [Fisher, pp. 49, 79, 130 & 144-145, with
contemporary map on p. 144. Fisher says there are 39 fountains, and the map has 40. Close
examination shows that the map counts two statues at the entrance but omits to count a
fountain between numbers 37 and 38. Matthews, pp. 117-121, says it was built by J.
Hardouin-Mansart and his map has 39 fountains.] It has a main entrance and exit but there is
SOURCES - page 89
another exit, so the perimeter wall already has three components, and there are 14 other
components. Sadly, it was destroyed in 1775.
Several other mazes, of increasing complexity, occur in the second half of the 17C
[Matthews, figs. 93-109, opp. p. 120 - p. 127]. Several of these could be from 20C maze
books. Fig. 94, designed for Chantilly by Le Nôtre, is surprisingly modern in that there are
eight paths spiralling to the centre. The entrance path takes you directly to the centre, so the
real problem is getting back out! One of the mazes presently at Longleat has this same
feature.
The Hampton Court Maze, planted c1690, is the oldest extant hedge maze and one of
the earliest puzzle mazes. ([Christopher Turner; Hampton Court, Richmond and Kew Step by
Step; (As part of: Outer London Step by Step, Faber, 1986); Revised and published in
sections, Faber, 1987, p. 16] says the present shape was laid out in 1714, replacing an earlier
circular shape, but I haven't seen this stated elsewhere.) Matthews [p. 128] says it probably
replaced an older maze. It has dead ends and one island, i.e. the graph has two components,
though the 'hand on wall' rule will solve it.
The second Earl Stanhope (1714-1786) is believed to be the first to design mazes with
the goal (at the centre) surrounded by an island, so that the 'hand on wall' rule will not solve it.
It has seven components and only a few short dead ends.. The fourth Earl planted one of
these at Chevening, Kent, in c1820 and it is extant though not open to the public. [Fisher, p.
71, with photo on p. 72 and diagram on p. 73.] However, investigation in Matthews revealed
the earlier examples above. Further Bernhard Wiezorke (below at 2001) has found a hedge
maze in Germany, dating from c1730, which is not solved by the 'hand on wall' rule. This
maze has 12 components.
In 1973, Stuart Landsborough, an Englishman settled at Wanaka, South Island, New
Zealand, began building his Great Maze. This was the first of the board mazes designed by
Landsborough which were immensely popular in Japan. Over 200 were built in 1984-1987,
with 20 designed by Landsborough. Many of these were three dimensional -- see below.
About 60 have been demolished since then. [Fisher, pp. 78-79 & 118-121 has 6 colour
photos, pp. 156-157 lists Landsborough's designs.]
If Minos' labyrinth ever really existed, it may have been three dimensional and there
may have been garden examples with overbridges, but I don't know of any evidence for such
early three dimensional mazes. Lewis Carroll drew mazes which had paths that crossed over
others making a simple three dimensional maze, in his Mischmasch of c1860, see below.
John Fisher [The Magic of Lewis Carroll; (Nelson, 1973), Penguin, 1975, pp. 19-20] gives
this and another example. Are there earlier examples? Boothroyd & Conway, 1959, seems to
be the earliest cubical maze. Much more complex versions were developed by Larry Evans
from about 1970 and published in a series of books, starting with 3-Dimensional Mazes
(Troubador Press, San Francisco, 1976). His 3-Dimensional Maze Art (Troubador, 1980)
sketches some general history of the maze and describes his development of pictures of three
dimensional mazes. The first actual three dimensional maze seems to be Greg Bright's 1978
maze at Longleat House, Warminster. [Fisher, pp. 74, 76, 94-95 & 152-153, with colour
photos on pp. 94-95.] Since then, Greg Bright, Adrian Fisher, Randoll Coate, Stuart
Landsborough and others have made many innovations. Bright seems to have originated the
use of colour in mazes c1980 and Fisher has extensively developed the idea. [Fisher, pp. 7379.]
Abu‘l-Rayhan Al-Biruni (= ’Abû-alraihân [the h should have an underdot] Muhammad ibn
’Ahmad [the h should have an underdot] Albêrûnî). India. c1030. Chapter XXX. IN:
Al-Beruni's India, trans. by E. C. Sachau, 2 vols., London, 1888, vol. 1, pp. 306-307 (=
p. 158 of the Arabic ed., ??NYS). In describing a story from the fifth and sixth books
of the Ramayana, he says that the demon Ravana made a labyrinthine fortress, which in
Muslim countries "is called Yâvana-koti, which has been frequently explained as
Rome." He then gives "the plan of the labyrinthine fortress", which is the classical
Cretan seven-ring form. Sachau's notes do not indicate whether this plan is actually in
the Ramayana, which dates from perhaps -300.
Pliny. Natural History. c77. Book 36, chap. 19. This gives a brief description of boys
playing on a pavement where a thousand steps are contained in a small space. This has
generally been interpreted as referring to a maze, but it is obviously pretty vague. See:
Michael Behrend; Julian and Troy names; Caerdroia 27 (1996) 18-22, esp. note 5 on
p. 22.
Pacioli. De Viribus. c1500. Part II: Cap. (C)XVII. Do(cumento). de saper fare illa berinto
SOURCES - page 90
con diligentia secondo Vergilio, f. 223v = Peirani 307-308. A sheet (or page) of the
MS has been lost. Cites Vergil, Æneid, part six, for the story of Pasiphæ and the
Minotaur, but the rest is then lost.
Sebastiano Serlio. Architettura, 5 books, 1537-1547. The separate books had several editions
before they were first published together in 1584. The material of interest is in Book IV
which shows two unicursal mazes for gardens. I have seen the following.
Tutte l'Opere d'Architetture et Prospetiva, .... Giacomo de'Franceschi, Venice,
1619; facsimile by Gregg Press, Ridgewood, New Jersey, 1964. F. 199r shows the
designs and f. 197v has some text, partly illegible in my photocopy. [Cf Caerdroia 30
(1999) 15.]
Sebastiano Serlio on Architecture Volume One Books I-V of 'Tutte l'Opere
d'Architettura et Prospetiva'. Translated and edited by Vaughan Hart and Peter Hicks.
Yale Univ. Press, New Haven, 1996. P. 388 shows the designs and p. 389 has the text,
saying these 'are for the compartition of gardens'. The sidenotes state that these pages
are ff. LXXVr and LXXIIIIr of the 3rd ed. of 1544 and ff. 198v-199r and 197v-198r of
the 1618/19 ed.
William Shakespeare. A Midsummer Night's Dream. c1610. Act II, scene I, lines 98-100:
"The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton green
For lack of tread are undistinguishable." Fiske 126 opines that the latter two lines may
indicate that the board was made in the turf, though he admits that they may refer just to
dancers' tracks, but to me it clearly refers to turf mazes.
John Cooke. Greene's Tu Quoque; or the Cittie Gallant; a Play of Much Humour. 1614.
??NYS -- quoted by Matthews, p. 135. A challenge to a duel is given by Spendall to
Staines.
Staines. I accept it ; the meeting place?
Spendall. Beyond the maze in Tuttle.
This refers to a maze in Tothill Fields, close to Westminster Abbey.
Lewis Carroll. Untitled maze. In: Mischmasch, the last of his youthful MS magazines, with
entries from 1855 to 1862. Transcribed version in: The Rectory Umbrella and
Mischmasch; Cassell, 1932; Dover, 1971; p. 165 of the Dover ed. John Fisher [The
Magic of Lewis Carroll; (Nelson, 1973), Penguin, 1975, pp. 19-20] gives this and
another example. Cf Carroll-Wakeling, prob. 35: An amazing maze, pp. 46-47 & 75
and Carroll-Gardner, pp. 80-81 for the Mischmasch example. I don't find the other
example elsewhere, but it was for Georgina "Ina" Watson, so probably c1870.
Mittenzwey. 1880. Prob. 281, pp. 50 & 100; 1895?: 310, pp. 53-54 & 102; 1917: 310,
pp. 49 & 97. The garden of a French place has a maze with 31 points to see. Find a
path past all of them with no repeated edges and no crossings. The pattern is clearly
based on the Versailles maze of c1675 mentioned in the Historical Sketch above, but I
don't recall the additional feature of no crossings occurring before.
C. Wiener. Ueber eine Aufgabe aus der Geometria situs. Math. Annalen 6 (1873) 29-30. An
algorithm for solving a maze. BLW asserts this is very complicated, but it doesn't look
too bad.
M. Trémaux. Algorithm. Described in Lucas, RM1, 1891, pp. 47-51. ??check 1882 ed.
BLW assert Lucas' description is faulty. Also described in MRE, 1st ed., 1892,
pp. 130-131; 3rd ed., 1896, pp. 155-156; 4th ed., 1905, pp. 175-176 is vague; 5th-10th
ed., 1911-1922, 183; 11th ed., 1939, pp. 255-256 (taken from Lucas); (12th ed.
describes Tarry's algorithm instead) and in Dudeney, AM, p. 135 (= Mazes, and how to
thread them, Strand Mag. 37 (No. 220) (Apr 1909) 442-448, esp. 446-447).
G. Tarry. Le problème des labyrinthes. Nouv. Annales de Math. (3) 4 (1895) 187-190.
??NYR
Collins. Book of Puzzles. 1927. How to thread any maze, pp. 122-124. Discusses right hand
rule and its failure, then Trémaux's method.
M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 2. 5 x 5 x 5 cubical maze. Get from a corner to an antipodal corner in a
minimal number of steps.
Anneke Treep. Mazes... How to get out! (part I). CFF 37 (Jun 1995) 18-21. Based on her
MSc thesis at Univ. of Twente. Notes that there has been very little systematic study.
Surveys the algorithms of Tarry, Trémaux, Rosenstiehl. Rosenstiehl is greedy on new
edges, Trémaux is greedy on new nodes and Trémaux is a hybrid of these. ??-oopscheck. Studies probabilities of various routes and the expected traversal time. When
the maze graph is a tree, the methods are equivalent and the expected traversal time is
SOURCES - page 91
the number of edges.
Bernhard Wiezorke. Puzzles und Brainteasers. OR News, Ausgabe 13 (Nov 2001) 52-54.
This reports his discovery of a hedge maze in Germany -- the first he knew of. It is in
Altjessnitz, near Dessau in Sachsen-Anhalt. (My atlas doesn't show such a place, but
Jessnitz is about 10km south of Dessau.) This maze dates from 1720 and has 12
components, with the goal completely separated from the outside so that the 'hand on
wall' rule does not solve it. Torsten Silke later told Wiezorke of two other hedge mazes
in Germany. One, in Probststeierhagen, Schleswig-Holstein, about 12km NE of Kiel, is
in the grounds of the restaurant Zum Irrgarten (At the Labyrinth) and is an early 20C
copy of the Altjessnitz example. The other, in Kleinwelka, Sachsen, about 50km NE of
Dresden, was made in 1992 and is private. Though it has 17 components, the 'hand on
wall' method will solve it. He gives plans of both mazes. He discusses the Seven
Bridges of Königsberg, giving a B&W print of the 1641 plan of the city mentioned at
the beginning of Section 5.E -- he has sent me a colour version of it. He also describes
Tremaux's solution method.
5.E.2. MEMORY WHEELS = CHAIN CODES
These are cycles of 2n 0s and 1s such that each n-tuple of 0s and 1s appears just
once. They are sometimes called De Bruijn sequences, but they have now been traced back to
the late 19C. An example for n = 3 is 00010111.
Émile Baudot. 1884. Used the code for 25 in telegraphy. ??NYS -- mentioned by Stein.
A. de Rivière, proposer; C. Flye Sainte-Marie, solver. Question no. 58. L'Intermédiare des
Mathématiciens 1 (1894) 19-20 & 107-110. ??NYS -- described in Ralston and
Fredricksen (but he gives no. 48 at one point). Deals with the general problem of a
cycle of kn symbols such that every n-tuple of the k basic symbols occurs just once.
Gives the graphical method and shows that such cycles always exist and there are k!g(n)/
kn of them, where g(n) = kn-1. This work was unknown to the following authors until
about 1975.
N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch. Proc. 49 (1946)
758-764. ??NYS -- described in Ralston and Fredricksen. Gives the graphical method
for finding examples and finds there are 2f(n) solutions, where f(n) = 2n-1 - n.
I. J. Good. Normal recurring decimals. J. London Math. Soc. 21 (1946) 167-169. ??NYS -described in Ralston and Fredricksen. Shows there are solutions but doesn't get the
number.
R. L. Goodstein. Note 2590: A permutation problem. MG 40 (No. 331) (Feb 1956) 46-47.
Obtains a kind of recurrence for consecutive n-tuples.
Sherman K. Stein. Mathematics: The Man-made Universe. Freeman, 1963. Chap. 9:
Memory wheels. c= The mathematician as explorer, SA (May 1961) 149-158. Surveys
the topic. Cites the c1000 Sanskrit word: yamátárájabhánasalagám used as the
mnemonic for 01110100(01) giving all triples of short and long beats in Sanskrit
poetry and music. Describes the many reinventions, including Baudot (1882), ??NYS,
and the work of Good (1946), ??NYS, and de Bruijn (1946), ??NYS. 15 references.
R. L. Goodstein. A generalized permutation problem. MG 54 (No. 389) (Oct 1970) 266-267.
Extends his 1956 note to find a cycle of an symbols such that the n-tuples are distinct.
Anthony Ralston. De Bruijn sequences -- A model example of the interaction of discrete
mathematics and computer science. MM 55 (1982) 131-143 & cover. Deals with the
general problem of cycles of kn symbols such that every n-tuple of the k basic
symbols occurs just once. Discusses the history and various proofs and algorithms
which show that such cycles always exist. 27 references.
Harold Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM
Review 24:2 (Apr 1982) 195-221. Mostly about their properties and their generation,
but includes a discussion of the door lock connection, a mention of using the 23 case as
a switch for three lights, and gives a good history. The door lock connection is that
certain push button door locks will open when a four digit code is entered, but they
open if the last four buttons pressed are the correct code, so using a chain code reduces
the number of button pushes required by a burglar to 1/4 of the number required if he
tries all four digit combinations. 58 references.
At G4G2, 1996, Persi Diaconis spoke about applications of the chain code in magic and
mentioned uses in repeated measurement designs, random number generators, robot
SOURCES - page 92
location, door locks, DNA comparison.
They were first used in card tricks by Charles T. Jordan in 1910. Diaconis'
example had a deck of cards which were cut and then five consecutive cards were dealt
to five people in a row. He then said he would determine what cards they had, but first
he needed some help so he asked those with red cards to step forward. The position of
the red cards gives the location of the five cards in a cycle of 32 (which was the size of
the deck)! Further, there are simple recurrences for the sequence so it is fairly easy to
determine the location. One can code the binary quintuples to give the suit and value of
the first card and then use the succeeding quintuples for the succeeding cards.
Long versions of the chain code are printed on factory floors so that a robot can
read it and locate itself.
In Jan 2000, I discussed the Sanskrit chain code with a Sanskrit scholar, Dominik Wujastyk,
who said that there is no known Sanskrit source for it. He has asked numerous pandits
who did not know of it and he said there is is a forthcoming paper on it, but that it did
not locate any Sanskrit source.
5.E.2.a.
PANTACTIC SQUARES
Haubrich's 1995-1996 surveys, op. cit. in 5.H.4, include this.
B. Astle. Pantactic squares. MG 49 (No. 368) (May 1965) 144-152. This is a
two-dimensional version of the memory wheel. Take a 5 x 5 array of cells marked 0
or 1 (or Black or White). There are 16 ways to take a 2 x 2 subarray from the 5 x 5
array. If these give all 16 2 x 2 binary patterns, the array is called pantactic. The
author shows a number of properties and some types of such squares.
C. J. Bouwkamp, P. Janssen & A. Koene. Note on pantactic squares. MG 54 (No. 390) (Dec
1970) 348-351. They find 800 such squares, forming 50 classes of 16 forms.
[Surprisingly, neither paper considers a 4 x 4 array viewed toroidally, which is the natural
generalization of the memory wheel. Precisely two of the fifty classes, namely nos. 25
& 41, give such a solution and these are the same pattern on the torus. One can also
look at the 4 x 4 subarrays of a 131 x 131 or a 128 x 128 array, etc., as well as 3 and
higher dimensional arrays. I submitted the question of the existence and numbers of
these as a problem for CM, but it was considered too technical.]
Ivan Moscovich. US Patent 3,677,549 -- Board Game Apparatus. Applied: 14 Jun 1971;
patented: 18 Jul 1972. Front page, 1p diagrams, 2pp text. Reproduced in Haubrich,
About ..., 1996, op. cit. in 5.H.4. 2pp + 2pp diagrams. This uses the 16 2 x 2 binary
patterns as game pieces. He allows the pieces to be rotated, scoring different values
according to the orientation. No mention of reversing pieces or of the use of the pieces
as a puzzle.
John Humphries. Review of Q-Bits. G&P 54 (Nov 1976) 28. This is Moscovich's game
idea, produced by Orda. Though he mentions changing the rules to having nonmatching, there is no mention of two-sidedness.
Pieter van Delft & Jack Botermans. Creative Puzzles of the World. (As: Puzzels uit de hele
wereld; Spectrum Hobby, 1978); Harry N. Abrams, NY, 1978. The colormatch square,
p. 165. See Haubrich,1994, for description.
Jacques Haubrich. Pantactic patterns and puzzles. CFF 34 (Oct 1994) 19-21. Notes the
toroidal property just mentioned. Says Bouwkamp had the idea of making the 16 basic
squares in coloured card and using them as a MacMahon-type puzzle, with the pieces
double-sided and such that when one side had MacMahon matching, the other side had
non-matching. There are two different bijections between matching patterns and nonmatching patterns, so there are also 800 solutions in 50 classes for the non-matching
problem. Bouwkamp's puzzle appeared in van Delft & Botermans, though they did not
know about and hence did not mention the double-sidedness. [In an email of 22 Aug
2000, Haubrich says he believes Bouwkamp did tell van Delft and Botermans about
this, but somehow it did not get into their book.] The idea was copied by two
manufacturers (Set Squares by Peter Pan Playthings and Regev Magnetics) who did not
understand Bouwkamp's ideas -- i.e. they permitted pieces to rotate. Describes
Verbakel's puzzle of 5.H.2.
Jacques Haubrich. Letter: Pantactic Puzzles = Q-Bits. CFF 37 (Jun 1995) 4. Says that Ivan
Moscovich has responded that he invented the version called "Q-Bits" in 1960-1964,
having the same tiles as Bouwkamp's (but only one-sided [clarified by Haubrich in
SOURCES - page 93
above mentioned email]). His US Patent 3,677,549 (see above) is for a game version of
he idea. The version produced by Orda Ltd. was reviewed in G&P 54 (Nov 1976)
(above). So it seems clear that Moscovich had the idea of the pieces before
Bouwkamp's version was published, but Moscovich's application was to use them in a
game where the orientations could be varied.
5.F. HAMILTONIAN CIRCUITS
For queen's, bishop's and rook's tours, see 6.AK.
A tour is a closed path or circuit.
A path has end points and is sometimes called an open tour.
5.F.1. KNIGHT'S TOURS AND PATHS
GENERAL REFERENCES
Antonius van der Linde. Geschichte und Literatur des Schachspiels. (2 vols., Springer,
Berlin, 1874); one vol. reprint, Olms, Zürich, 1981. [There are two other van der Linde
books: Quellenstudien zur Geschichte des Schachspiels, Berlin, 1881, ??NYS; and
Das Erste Jartausend [sic] der Schachlitteratur (850-1880), (Berlin, 1880); reprinted
with some notes and corrections, Caissa Limited Editions, Delaware, 1979, which is
basically a bibliography of little use here.]
Baron Tassilo von Heydebrand und von der Lasa. Zur Geschichte und Literatur des
Schachspiels. Forschungen. Leipzig, 1897. ??NYS.
Ahrens. MUS I. 1910. Pp. 319-398.
Harold James Ruthven Murray. A History of Chess. OUP, 1913; reprinted by Benjamin
Press, Northampton, Massachusetts, nd [c1986]. This has many references to the
problem, which are detailed below.
Reinhard Wieber. Das Schachspiel in der arabischen Literatur von den Anfängen bis zur
zweiten Hälfte des 16.Jahrhunderts. Verlag für Orientkunde Dr. H. Vorndran,
Walldorf-Hessen, 1972.
George P. Jelliss.
Special Issue: Notes on the Knight's Tour. Chessics 22 (Summer 1985) 61-72.
Further notes on the knight's tour. Chessics 25 (Spring 1986) 106-107.
Notes on Chessics 22 continued. Chessics 29 & 30 (1987) 160.
This is a progress report on his forthcoming book on the knight's tour. I
will record some of his comments at the appropriate points below. He also
studies the 3 x n board extensively.
Two problems with knights on a 3 x 3 board are generally treated here, but cf 5.R.6.
The 4 knights problem has two W and two B knights at the corners (same colours at
adjacent corners) and the problem is to exchange them in 16 moves. The graph of knight's
connections is an 8-cycle with the pieces at alternate nodes. [Putting same colours at opposite
corners allows a solution in 8 moves.]
The 7 knights problem is to place 7 knights on a 3 x 3 board in the 4 corners and 3 of
the sides so each is a knight's move from the previously placed one. This is equivalent to the
octagram puzzle of 5.R.6.
4 knights problem -- see: at-Tilimsâni, 1446; Civis Bononiae, c1475;
7 knights problem -- see: King's Library MS.13, A.xviii, c1275; "Bonus Socius",
c1275; at-Tilimsâni, 1446;
Al-Adli (c840) and as-Suli (c880-946) are the first two great Arabic chess players. Although
none of their works survive, they are referred to by many later writers who claim to
have used their material.
Rudraţa: Kāvyālaʼnkāra [NOTE: ţ denotes a t with an underdot and ʼn denotes an n with
an overdot.]. c900. ??NYS -- described in Murray 53-55, from an 1896 paper by
Jacobi, ??NYS. The poet speaks of verses which have the shapes of "wheel, sword,
club, bow, spear, trident, and plough, which are to be read according to the chessboard
SOURCES - page 94
squares of the chariot [= rook], horse [= knight], elephant [c= bishop], &c." According
to Jacobi, the poet placed syllables in the cells of a half chessboard so that it reads the
same straight across as when following a piece's path. With help from the commentator
Nami, of 1069, the rook's and knight's path's are reconstructed, and are given on Murray
54. Both are readily extended to full board paths, but not tours. The elephant's path is
confused.
Kitâb ash-shatranj mimma’l-lafahu’l-‘Adli waş-Şûlî wa ghair-huma [Book of the Chess;
extracts from the works of al-'Adlî, aş-Şûlî and others]. [NOTE: ş, Ş denote s, S
with underdot.] Copied by Abû Ishâq [the h should have an underdot] Ibrâhîm ibn
al-Mubârak ibn ‘Alî al-Mudhahhab al-Baghdâdî. Murray 171-172 says it is
MS ‘Abd-al-Hamid [the H should have an underdot] I, no. 560, of 1140, and denotes it
AH. Wieber 12-15 says it is now MS Lala Ismail Efendi 560, dates it July-August
1141, and denotes it L. Both cite van der Linde, Quellenstudien, no. xviii, p. 331+,
??NYS. The author is unknown. This MS was discovered in 1880. Catalogues in
Istanbul listed it as Risâla fi’sh-shaţranj by Abû’l-‘Abbâs Ahmad [the h should have
an underdot] al-‘Adlî. It is sometimes attributed to al-Lajlâj who wrote one short
section of this book. Murray, van der Linde and Wieber (p. 41) cite another version:
MS Khedivial Lib., Cairo, Mustafa Pasha, no. 8201, copied c1370, which Murray
denotes as C and Wieber lists as unseen.
Murray 336 gives two distinct tours: AH91 & AH92. The solution of AH91 is a
numbered diagram, but AH92 is 'solved' four times by acrostic poems, where the initial
letters of the lines give the tour in an algebraic notation. Wieber 479-480 gives 2 tours
from ff. 74a-75b: L74a = AH91 and L74b = reflection of AH92. [Since the 'solutions'
of AH92 are poetic, it is not unreasonable to consider the reflection as different.] Also
AH94 = L75b is a knight/bishop tour, where moves of the two types alternate. These
tours may be due to as-Suli. AH196 is a knight/queen tour.
Arabic MS Atif Efendi 2234 (formerly Vefa (‘Atîq Efendî) 2234), Eyyub, Istanbul. Copied by
Muhammad [the h should have an underdot] ibn Hawâ (or Rahwâr -- the MS is
obscure) ibn ‘Othmân al-Mu’addib in 1221. Murray 174-175 describes it as mostly
taken from the above book and denotes it V. A tour is shown on p. 336 as V93 =
AH92. Wieber 20-24 denotes it A. On p. 479, he shows the tour from f. 68b which is
the same as L74b, the reflection of AH92.
King's Library MS.13, A.xviii, British Museum, in French, c1275. Described in van der
Linde I 305-306. Described and transcribed in Murray 579-582 & 588-600, where it is
denoted as K. Van der Linde discusses the knight's path on I 295, with diagram no. 244
on p. 245. Murray 589 gives the text and a numbered diagram of a knight's path as K1.
The path splits into two half board paths: a1 to d1 and e3 to h1, so the first half and the
whole are corner to corner. The first half is also shown as diagram K2 with the half
board covered with pieces and the path described by taking of pieces. K3 is the 7
knights problem
"Bonus Socius" [perhaps Nicolas de Nicolaï]. This is the common name of a collection of
chess problems, assembled c1275, which was copied and translated many times. See
Murray 618-642 for about 11 MSS. Some of these are given below. Fiske 104 &
110-111 discusses some MSS of this collection.
MS Lat. 10286, Nat. Lib., Paris. c1350. Van der Linde I 293-295 describes this
but gives the number as 10287 (formerly 7390). Murray 621 describes it and denotes it
PL. Van der Linde describes a half board knight's path, with a diagram no. 243 shown
on p. 245. The description indicates a gap in the path which can only be filled in one
way. This is a path from a8 to h8 which cannot be extended to the full board.
Murray 641 says that PL275 is the same as problems in two similar MSS and as CB244,
diagrammed on p. 674. However, this is not the same as van der Linde's no. 243,
though cells 1-19 and 31-32 are the same in both paths, so this is also an a8 to h8 path
which does not extend to a full board.
Murray 620 mentions a path in a late Italian MS version of c1530 (Florence, Nat.
Lib. XIX.7.51, which he denotes It) which may be the MS described by van der Linde
I 284 as no. 4 and the half board path described on I 295 with diagram no. 245 on I 245.
Fiske 210-211 describes this and says von der Lasa 163-165 (??NYS) describes it as
early 16C, but Murray does not mention von der Lasa. Fiske says it contains a tour on
f. 28b, which von der Lasa claims is "das älteste beispiel eines vollkommenen
rösselsprunges", but Murray does not detail the problems so I cannot compare these
citations. Fiske also says it also contains the 7 knights problem.
SOURCES - page 95
Dresden MS 0/59, in French, c1400. Murray describes this on pp. 607-613 and denotes it D.
On p. 609, Murray describes D57 which asks for a knight's path on a 4 x 4 board. No
solution is given -- indeed this is impossible, cf Persian MS 211 in the RAS. Ibid. is
D62 which asks for a half board tour, but no answer is provided.
Persian MS 211 in Royal Asiatic Society. Early 15C. ??NYS.
Extensively described as MS 250 bequeathed by Major David Price in: N. Bland;
On the Persian game of chess; J. Royal Asiatic Soc. 13 (1852) 1-70. He dates it as 'at
least 500 years old' and doesn't mention the knight's tour.
Described, as MS No. 260, and partially translated in Duncan Forbes; The History
of Chess; Wm. H. Allen, London, 1860. Forbes says Bland's description is "very
detailed but unsatisfactory". On p. 82 is the end of the translation of the preface:
'"Finally I will show you how to move a Knight from any individual square on the
board, so that he may cover each of the remaining squares in as many moves and finally
come to rest on that square whence he started. I will also show how the same thing may
be done by limiting yourself only to one half, or even to one quarter (1) of the board." -Here the preface abruptly terminates, the following leaf being lost.' Forbes's footnote
(1) correctly doubts that a knight's tour (or even a knight's path) is possible on the 4 x 4
board.
Murray 177 cites it as MS no. 211 and denotes it RAS. He says that it has been
suggested that this MS may be the work of ‘Alâ'addîn Tabrîzî = ‘Alî ash-Shatranjî, late
14C, described on Murray 171. Murray mentions the knight's tour passage on p. 335.
This may be in van der Linde, ??NX. Wieber 45 mentions the MS.
Abû Zakarîyâ Yahya [the h should have an underdot] ibn Ibrâhîm al-Hakîm[the H should
have an underdot]. Nuzhat al-arbâb al-‘aqûl fî’sh-shaţranj [NOTE: ţ denotes a t with
an underdot] al-manqûl (The delight of the intelligent, a description of chess). Arabic
MS 766, John Rylands Library, Manchester.
Bland, loc. cit., pp. 27-28, describes this as no. 146 of Dr. Lee's catalogue and no.
76 of the new catalogue. Forbes, loc. cit., says that Dr. Lee had loaned his two MSS to
someone who had not yet returned them, so Forbes copies Bland's descriptions (on
pp. 27-31) as his Appendix C, with some clarifying notes. (The other of Dr. Lee's MSS
is described below.) Van der Linde I 107ff (??NX) seems to copy Bland & Forbes.
Murray 175-176 describes it as Arab. 59 at John Rylands Library and denotes it
H. He says it was Bland who had borrowed the MSS from Dr. Lee and Murray traces
their route to Dr. Lee and to Manchester. Murray says it is late 15C, is based on al-Adli
and as-Suli and he also describes a later version, denoted Z, late 18C. Wieber 32-35
cites it as MS 766(86) at John Rylands, dates it 1430 and denotes it Y1.
Murray 336 gives three paths. H73 = H75 are the same tour, but with different
keys, one poetic as in Rudraţa [NOTE: ţ denotes a t with an underdot.], one numeric.
H74 is a path attributed to Ali Mani with similar poetic solution. Wieber 480 shows
two diagrams. Y1-39a, Y1-39b, Y1-41b are the same tour as H73, but with different
descriptions, the latter two being attributed to al-Adli. Y1-39a (second diagram) = H74
is attributed to ‘Ali ibn Mani.
Shihâbaddîn Abû’l-‘Abbâs Ahmad [the h should have an underdot] ibn Yahya [the h should
have an underdot] ibn Abî Hajala [the H should have an underdot] at-Tilimsâni
alH-anbalî [the H should have an underdot]. Kitâb ’anmûdhaj al-qitâl fi la‘b
ash-shaţranj [NOTE: ţ denotes a t with an underdot] (Book of the examples of
warfare in the game of chess). Copied by Muhammed ibn ‘Ali ibn Muhammed
al-Arzagî in 1446.
Bland, loc. cit., pp. 28-31, describes this as the second of Dr. Lee's MSS, old
no. 147, new no. 77. Forbes copies this and adds notes. Van der Linde I 105-107
seems to copy from Bland and Forbes. Murray 176-177 says the author died in 1375, so
this might be c1370. He says it is Dr. Lee's on 175-176, that it is MS Arab. 93 at the
John Rylands Library and denotes it Man. Wieber 29-32 cites it as MS 767(59) at the
Rylands Library and denotes it H. On p. 481, he shows a half-board path which cannot
be extended to the full board.
This MS also gives the 4 knights and 7 knights problems. Murray 337, 673
(CB236) & 690 and Wieber 481 show these problems.
Risâlahĭ Shatranj. Persian poem of unknown date and authorship. A copy was sent to Bland
by Dr. Sprenger of Delhi. See Bland, loc. cit., pp. 43-44. [Bland uses á for â.] Bland
says it has the problem of the knight's tour or path. [I think this is the poem mentioned
on Murray 182-183 and hence on Wieber 42.]
SOURCES - page 96
Şifat mal ‘ûb al-faras fî gamî abyât aš-šaţranğ [NOTE: Ş, ţ. denote S, t with underdot.] MS
Gotha 10, Teil 6; ar. 366; Stz. Hal. 408. Date unknown. Wieber 37 & 480 describes
this and gives a path from h8 to e4 which occurs on ff. 70 & 68.
Civis Bononiae [Citizen of Bologna]. Like Bonus Socius, this is a collection of chess
problems, from c1475, which exists in several MSS and printings. All are in Latin,
from Italy, and give essentially the same 288 problems. See Murray 643-703 for
description of about 10 texts and transcription of the problems. Many of the texts are
not in van der Linde. Murray 643 cites MS Lasa, in the library of Baron von der Lasa,
c1475, as the most accurate and complete of the texts. Two other well known versions
are described below.
Paulo Guarino (di Forli) (= Paulus Guarinus). No real title, but the end has
'Explicit liber de partitis scacorum' with the writer's name and the date 4 Jan 1512. This
MS was in the Franz Collection and is now (1913) in the John G. White Collection in
Cleveland, Ohio. This version only contains 76 problems. Van der Linde I 295-297
describes the MS and on p. 294 he describes a half board path and says Guarino's 74 is a
reflection of his no. 243. Murray 645 describes the MS but doesn't list the individual
problems. He implies that CB244, on p. 674, is the tour that appears in all of the Civis
Bononiae texts, but this is not the same as van der Linde's no. 243. CB236, pp. 673 &
690, is the 4 knights problem, which is Guarino's 42 [according to Lucas, RM4, p. 207],
but I don't have a copy of van der Linde's no. 215 to check this, ??NX.
Anon. Sensuit Jeux Partis des eschez: composez nouvellement Pour recreer tous
nobles cueurs et pour eviter oysivete a ceulx qui ont voulente: desir et affection de le
scavoir et apprendre et est appelle ce Livre le jeu des princes et damoiselles. Published
by Denis Janot, Paris, c1535, 12 ff. ??NYS. (This is the item described by von der
Lasa as 'bei Janot gedrucktes Quartbändchen' (MUS #195).) This a late text of 21
problems, mostly taken from Civis Bononiae. Only one copy is known, now (1913) in
Vienna. See van der Linde I 306-307 and Murray 707-708 which identify no. 18 as van
der Linde's no. 243 and with CB244, as with the Guarino work. I can't tell but van der
Linde may identify no. 11 as the 4 knights problem (??NX).
Murray 730 gives another half board path, C92, of c1500 which goes from a8 to
g5. Murray 732 notes that a small rearrangement makes it extendable to the whole
board.
Horatio Gianutio della Mantia. Libro nel quale si tratta della Maniera di giuocar' à Scacchi,
Con alcuni sottilissimi Partiti. Antonio de' Bianchi, Torino, 1597. ??NYS. Gives half
board tours which can be assembled into to a full tour. (Not in the English translation:
The Works of Gianutio and Gustavus Selenus, on the game of Chess, Translated and
arranged by J. H. Sarratt; J. Ebers, London, 1817, vol. 1. -- though the copy I saw didn't
say vol. 1. Van der Linde, Erste Jartausend ... says there are two volumes.)
Bhaţţa Nīlakaņţha. [NOTE: ţ, ņ denote t, n with underdot.] Bhagavantabhāskara. 17C.
End of 5th book. ??NYS, described by Murray 63-66. The author gives three tours, in
the poetic form of Rudraţa [NOTE: ţ denotes a t with an underdot.], which are the
same tour starting at different points. The tour has 180 degree rotational symmetry.
Ozanam. 1725. Prob. 52, 1725: 260-269. Gives solutions due to Pierre Rémond de
Montmort, Abraham de Moivre, Jean-Jacques d'Ortous de Mairan (1678-1771).
Surprisingly, these are all distinct and different from the earlier examples. Ozanam says
he had the problem and the solution from de Mairan in 1722. Says the de Moivre is the
simplest. Kraitchik, Math. des Jeux, op. cit. in 4.A.2, p. 359, dates the de Montmort as
1708 and the de Moivre as 1722, but gives no source for these. Montmort died in 1719.
Ozanam died in 1717 and this edition was edited by Grandin. Van der Linde and
Ahrens say they can find no trace of these solutions prior to Ozanam (1725). See
Ozanam-Montucla, 1778.
Ball, MRE, 1st ed., 1892, p. 139, says the earliest examples he knows are the De
Montmort & De Moivre of the late 17C, but he only cites them from Ozanam-Hutton,
1803, & Ozanam-Riddle, 1840. In the 5th ed., 1911, p. 123, he adds that "They were
sent by their authors to Brook Taylor who seems to have previously suggested the
problem." He gives no reference for the connection to Taylor and I have not seen it
mentioned elsewhere. This note is never changed and may be the source of the
common misconception that knight's tours originated c1700!
Les Amusemens. 1749. Prob. 181, p. 354. Gives de Moivre's tour. Says one can imagine
other methods, but this is the simplest and most interesting.
L. Euler. Letter to C. Goldbach, 26 Apr 1757. In: P.-H. Fuss, ed.; Correspondance
SOURCES - page 97
Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIème Siècle;
(Acad. Imp. des Sciences, St. Pétersbourg, 1843) = Johnson Reprint, NY, 1968, vol. 1,
pp. 654-655. Gives a 180o symmetric tour.
L. Euler. Solution d'une question curieuse qui ne paroit soumise à aucune analyse. (Mém. de
l'Académie des Sciences de Berlin, 15 (1759 (1766)), 310-337.) = Opera Omnia (1) 7
(1923) 26-56. (= Comm. Arithm. Coll., 1849, vol. 1, pp. 337-355.) Produces many
solutions; studies 180o symmetry, two halves, and other size boards.
[Petronio dalla Volpe]. Corsa del Cavallo per tutt'i scacchi dello scacchiere. Lelio della
Volpe, Bologna, 1766. 12pp, of which 2 and 12 are blanks. [Lelio della Volpe is
sometimes given as the author, but he died c1749 and was succeeded by his son
Petronio.] Photographed and printed by Dario Uri from the example in the Libreria
Comunale Archiginnasio di Bologna, no. 17 CAPS XVI 13. The booklet is briefly
described in: Adriano Chicco; Note bibliografiche su gli studi di matematica applicata
agli scacchi, publicati in Italia; Atti del Convegno Nazionale sui Giochi Creative, Siena,
11-14 Jun 1981, ed. by Roberto Magari; Tipografia Senese for GIOCREA (Società
Italiana Giochi Creativi), 1981; p. 155.
The Introduction by the publisher cites Ozanam as the originator of this 'most
ingenious' idea and says he gives examples due to Montmort, Moivre and Mairan. He
also says this material has 'come to hand' but doesn't give any source, so it is generally
thought he was the author. He gives ten paths, starting from each of the 10 essentially
distinct cells. He then gives the three cited paths from Ozanam. He then gives six
tours. Each path is given as a numbered board and a line diagram of the path, which led
Chicco to say there were 38 paths. The line drawing of the first tour is also reproduced
on the cover/title page.
Ozanam-Montucla. 1778. Prob. 23, 1778: 178-182; 1803: 177-180; 1814: 155-157. Prob.
22, 1840: 80-81. Drops the reference to de Mairan as the source of the problem and
adds a fourth tour due to "M. de W***, capitaine au régiment de Kinski". All of these
have a misprint of 22 for 42 in the right hand column of De Moivre's solution.
H. C. von Warnsdorff. Des Rösselsprunges einfachste und allgemeinste Lösung. Th. G. Fr.
Varnhagenschen Buchhandlung, Schmalkalden, 1823, 68pp. ??NYS -- details from
Walker. Rule to make the next move to the cell with the fewest remaining neighbours.
Lucas, L'Arithmétique Amusante, p. 241, gives the place of publication as Berlin.
Boy's Own Book. Not in 1828. 1828-2: 318 states a knight's tour can be made.
George Walker. The Art of Chess-Play: A New Treatise on the Game of Chess. (1832, 80pp.
2nd ed., Sherwood & Co, London, 1833, 160pp. 3rd ed., Sherwood & Co., London,
1841, 300pp. All ??NYS -- details from 4th ed.) 4th ed., Sherwood, Gilbert & Piper,
London, 1846, 375pp. Chap. V -- section: On the knight, p. 37. "The problem
respecting the Knight's covering each square of the board consecutively, has attracted,
in all ages, the attention of the first mathematicians." States Warnsdorff's rule, without
credit, but gives the book in his bibliography on p. 375, and asserts the rule will always
give a tour. No diagram.
Family Friend 2 (1850) 88 & 119, with note on 209. Practical Puzzle -- No. III. Find a
knight's path. Gives one answer. Note says it has been studied since 'an early period'
and cites Hutton, who copies some from Montucla, an article by Walker in Frasers
Magazine (??NYS) which gives Warnsdorff's rule and an article by Roget in
Philosophical Magazine (??NYS) which shows one can start and end on any two
squares of opposite colours. Describes using a pegged board and a string to make pretty
patterns.
Boy's Own Book. Moving the knight over all the squares alternately. 1855: 511-512;
1868: 573; 1881 (NY): 346-347. 1855 says the problem interested Euler, Ozanam, De
Montmart [sic], De Moivre, De Majron [sic] and then gives Warnsdorff's rule, citing
George Walker's 'Treatise on Chess' for it -- presumably 'A New Treatise', London,
1832, with 2nd ed., 1833 & 3rd ed., 1841, ??NYS. Walker also wrote On Moving the
Knight, London, 1840, ??NYS. 1868 drops all the names, but the NY ed. of 1881 is the
same as the 1855. Gives a circuit due to Euler.
Magician's Own Book. 1857. Art. 46: Moving the knight over all the squares alternately,
pp. 283-287. Identical to Boy's Own Book, 1855, but adds Another Method. = Book of
500 Puzzles; 1859, art. 46, pp. 97-101. = Boy's Own Conjuring Book, 1860, prob. 45,
pp. 246-251.
Landells. Boy's Own Toy-Maker. 1858. Moving the knight over all the squares alternately,
p. 143. This is the Another Method of Magician's Own Book, 1857. Cf Illustrated
SOURCES - page 98
Boy's Own Treasury, 1860.
Illustrated Boy's Own Treasury. 1860. Prob. 47: Practical chess puzzle, pp. 404 & 443.
Knight's tour. This is the Another Method of Magician's Own Book.
C. F. de Jaenisch. Traité des Applications de l'Analyse Mathématiques au Jeu des Échecs.
3 vols., no publisher, Saint-Pétersbourg. 1862-1863. Vol. 1: Livre I: Section III: De la
marche du cavalier, pp. 186-259 & Plate III. Vol. 2: Livre II: Problème du Cavalier,
pp. 1-296 & 31 plates (some parts ??NYS). Vol. 3: Addition au Livre II, pp. 239-243
(This Addition ??NYS). This contains a vast amount of miscellaneous material and I
have not yet read it carefully. ??NYR
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 323, pp. 153-154 & 393:
Rösselsprung-Aufgaben. Three arrays of syllables and one must find a poetic riddle by
following a knight's tour. Arrays are 8 x 8, 8 x 8, 6 x 4.
C. Flye Sainte-Marie. Bull. Soc. Math. de France (1876) 144-150. ??NYS -- described by
Jelliss. Shows there is no tour on a 4 x n board and describes what a path must look
like.
Mittenzwey. 1880. Prob. 222-223, pp. 40 & 91; 1895?: 247-248, pp. 44 & 93; 1917:
247-248, pp. 40-41 & 89. First is a knight's path. Second is a board with word
fragments and one has to make a poem, which uses the same path as in the first
problem.
Paul de Hijo [= Abbé Jolivald]. Le Problème du Cavalier des Échecs. Metz, 1882. ??NYS -described by Jelliss and quoted by Lucas. Jelliss notes the BL copy of de Hijo was
destroyed in the war, but he has since told me there are copies in The Hague and
Nijmegen. First determination of the five 6 x 6 tours with 4-fold rotational symmetry,
the 150 ways to cover the 8 x 8 with two circuits of length 32 giving a pattern with
2-fold rotational symmetry, the 378 ways giving reflectional symmetry in a median, the
140 ways with four circuits giving 4-fold rotational symmetry and the 301 ways giving
symmetry in both medians (quoted in Lucas, L'Arithmétique Amusante, pp. 238-241).
Lucas. Nouveaux jeux scientifiques ..., 1889, op. cit. in 4.B.3. (Described on p. 302, figure
on p. 301.) 'La Fasioulette' is an 8 x 8 board with 64 links of length 5 to form
knight's tours.
Knight's move puzzles. The Boy's Own Paper 11 (Nos. 557 & 558) (14 & 21 Sep 1889)
799 & 814. Four Shakespearean quotations concealed as knight's tours on a 8 x 8
board. Beginnings not indicated!
Hoffmann. 1893. Chap. X, no. 6: The knight's tour, pp. 335-336 & 367-373
= Hoffmann-Hordern, pp. 225-229. Gives knight's paths due to Euler and Du Malabare,
a knight's tour due to Monneron, and four other unattributed tours. Gives Warnsdorff's
rule, citing Walker's A New Treatise on Chess, 1832.
Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904, pp. 1080-1093.
Pp. 1084-1086 gives many references to 19C work, including estimates of the number
of tours and results on 'semi-magic tours'.
C. Planck. Chess Amateur (Dec 1908) 83. ??NYS -- described by Jelliss. Shows there are
1728 paths on the 5 x 5 board. Jelliss notes that this counts each path in both
directions and there are only 112 inequivalent tours.
Ahrens. 1910. MUS I 325. Use of knight's tours as a secret code.
Dudeney. AM. 1917. Prob. 339: The four knight's tours, pp. 103 & 229. Quadrisect the
board into four congruent pieces such that there is a knight's tour on the piece. Jelliss
asserts that the solution is unique and says this may be what Persian MS 260 (i.e. 211)
intended. He notes that the four tours can be joined to give a tour with four fold
rotational symmetry.
W. H. Cozens. Cyclically symmetric knight's tours. MG 24 (No. 262) (Dec 1940) 315-323.
Finds symmetric tours on various odd-shaped boards.
H. J. R. Murray. The Knight's Tour. ??NYS. MS of 1942 described by G. P. Jelliss, G&PJ 2
(No. 17) (Oct 1999) 315. Observes that a knight can move from the (0, 0) cell to the
(2, 1) and (1, 2) cella and that the angle between these lines is the smaller angle of a
3, 4, 5 triangle. One can see this by extending the lines to (8, 4) and (5, 10) and
seeing these points form a 3, 4, 5 triangle with (0, 0).
W. H. Cozens. Note 2761: On note 2592. MG 42 (No. 340) (May 1958) 124-125. Note
2592 tried to find the cyclically symmetric tours on the 6 x 6 board and found 4.
Cozens notes two are reflections of the other two and that three such tours were
omitted. He found all these in his 1940 paper.
R. C. Read. Constructing open knight's tours blindfold! Eureka 22 (Oct 1959) 5-9. Describes
SOURCES - page 99
how to construct easily a tour between given cells of opposite colours, correcting a
method of Roget described by Ball (MRE 11th ed, p. 181). Says he can do it blindfold.
W. H. Cozens. Note 2884: On note 2592. MG 44 (No. 348) (May 1960) 117. Estimates
there are 200,000 cyclically symmetric tours on the 10 x 10 board.
Roger F. Wheeler. Note 3059: The KNIGHT's tour on 42 and other boards. MG 47
(No. 360) (May 1963) 136-141. KNIGHT means a knight on a toroidal board. He finds
2688 tours of 19 types on the 42 toroid. (Cf Tylor, 1982??)
J. J. Duby. Un algorithme graphique trouvant tous les circuits Hamiltoniens d'un graphe.
Etude No. 8, IBM France, Paris, 22 Oct 1964. [In English with French title and
summary.] Finds there are 9862 knight's tours on the 6 x 6 board, where the tours all
start at a fixed corner and then go to a fixed one of the two cells reachable from the
corner. He also finds 75,000 tours on the 8 x 8 board which have the same first 35
moves. He believes there may be over a million tours.
Karl Fabel. Wanderungen von Schachfiguren. IN: Eero Bonsdorff, Karl Fabel & Olavi
Riihimaa; Schach und Zahl; Walter Rau Verlag, Düsseldorf, 1966, pp. 40-50. On p. 50,
he says that there are 122,802,512 tours where the knight does two joined half-board
paths. He also says there are upper bounds, determined by several authors, and he gives
1.5 x 1026 as an example.
Gardner. SA (Oct 1967) = Magic Show, chap. 14. Surveys results of which boards have tours
or paths.
D. J. W. Stone. On the Knight's Tour Problem and Its Solution by Graph-Theoretic and Other
Methods. M.Sc. Thesis, Dept. of Computing Science, Univ. of Glasgow, Jan. 1969.
Confirms Duby's 9862 tours on the 6 x 6 board.
David Singmaster. Enumerating unlabelled Hamiltonian circuits. International Series on
Numerical Mathematics, No. 29. Birkhäuser, Basel, 1975, pp. 117-130. Discusses the
work of Duby and Stone and gives an estimate, which Stone endorses, that there are
1023±3 tours on the 8 x 8 board.
C. M. B. Tylor. 2-by-2 tours. Chessics 14 (Jul-Dec 1982) 14. Says there are 17 knight's tours
on a 2 x 2 torus and gives them. Doesn't mention Wheeler, 1963.
Robert Cannon & Stan Dolan. The knight's tour. MG 70 (No. 452) (Jun 1986) 91-100. A
rectangular board is tourable if it has a knight's path between any two cells of opposite
colours. They prove that m x n is tourable if and only if mn is even and m  6, n  6.
They also prove that m x n has a knight's tour if and only if mn is even and
[(m  5, n  5) or (m = 3, n  10)] and that when mn is even, m x n has a knight's path
if and only if m  3, n  3, except for the 3 x 6 and 4 x 4 boards. (These later results
are well known -- see Gardner. The authors only cite Ball's MRE.)
George Jelliss. Figured tours. MS 25:1 (1992/93) 16-20. Exposition of paths and tours
where certain stages of the path form an interesting geometric figure. E.g. Euler's first
paper has a path on the 5 x 5 such that the points on one diagonal are in arithmetic
progression: 1, 7, 13, 19, 25.
Martin Loebbing & Ingo Wegener. The number of knight's tours equals 33,439,123,484,294
--- Counting with binary decision diagrams. Electronic Journal of Combinatorics 3
(1996) article R5. A somewhat vague description of a method for counting knight's
tours -- they speak of directed knight's tours, but it is not clear if they have properly
accounted for the symmetries of a tour or of the board. Several people immediately
pointed out that the number is incorrect because it has to be divisible by four. Two
comments have appeared, ibid. On 15 May 1996, the authors admitted this and said
they would redo the problem, but they have submitted no further comment as of Jan
2001. On 18 Feb 1997, Brendan McKay announced that he had done the computation
another way and found 13,267,364,410,532.
In view of the difference between this and my 1975 estimate of 1023±3 tours, it
might be worth explaining my reasoning. In 1964, Duby found 75,000 tours with the
same first 35 moves. The average valence for a knight on an 8 x 8 board is 5.25, but
one cannot exit from a cell in the same direction as one entered, so we might estimate
the number of ways that the first 35 moves can be made as 4.2535 = 9.9 x 1021.
Multiplying by 75,000 then gives 7.4 x 1026. I think I assumed that some of the first
moves had already been made, e.g. we only allow one move from the starting cell, and
factored by 8 for the symmetries of the square, to get 2.2 x 1025. I can't find my
original calculations, and I find the estimate 1025 in later papers, so I suppose I tried to
reduce the effect of the 4.2535 some more. In retrospect, I had no knowledge of how
many of these had already been tried. If about half of all moves from a cell had already
SOURCES - page 100
been tried before any circuit was found, then the estimate would be more like
2.2534 x 75,000 = 7.1 x 1016. If we divide the given number of circuits by 75,000 and
take the 34th root, we get an average valence of 1.78 remaining, far less than I would
have guessed.
I am grateful to Don Knuth for this reference. Neither he nor I expected to ever
see this number calculated!
5.F.2. OTHER HAMILTONIAN CIRCUITS
For circuits on the n-cube, see also 5.F.4 and 7.M.1,2,3.
For circuits on the chessboard, see also 6.AK.
Le Nôtre. Le Labyrinte de Versailles, c1675. This was a hedge or garden maze, but the
objective was to visit, in correct order, 40 fountains based on Aesop's Fables. Each
node of the maze had at least one fountain. Some fountains were not at path junctions,
but one can consider these as nodes of degree two. This is an early example of a
Hamiltonian problem, except that one fountain was located at the end of a short dead
end. [Fisher, op. cit. in 5.E.1, pp. 49, 79, 130 & 144-145, with contemporary diagram
on p. 144. He says there are 39 fountains, but the diagram has 40.]
T. P. Kirkman. On the partitions of the R-pyramid, being the first class of R-gonous X-edra.
Philos. Trans. Roy. Soc. 148 (1858) 145-161.
W. R. Hamilton. The Icosian Game. 4pp instructions for the board game. J. Jaques and Son,
London, 1859. (Reproduced in BLW, pp. 32-35, with frontispiece photo of the board at
the Royal Irish Academy.)
For a long time, the only known example of the game, produced by Jaques, was at the Royal
Irish Academy in Dublin. This example is inscribed on the back as a present from
Hamilton to his friend, J. T. Graves. It is complete, with pegs and instructions. None
of the obvious museums have an example. Diligent searching in the antique trade failed
to turn up an example in twenty years, but in Feb 1996, James Dalgety found and
acquired an example of the board -- sadly the pegs and instructions were lacking.
Dalgety obtained another board in 1998, again without the pegs and instructions, but in
1999 he obtained another example, with the pegs.
Mittenzwey. 1880. Prob. 281, pp. 50 & 100; 1895?: 310, pp. 53-54 & 102; 1917: 310,
pp. 49 & 97. The garden of a French palace has a maze with 31 points to see. Find a
path past all of them with no repeated edges and no crossings. The pattern is clearly
based on the Versailles maze of c1675 mentioned above, but I don't recall the additional
feature of no crossings occurring before.
T. P. Kirkman. Solution of problem 6610, proposed by himself in verse. Math. Quest. Educ.
Times 35 (1881) 112-116. On p. 115, he says Hamilton told him, upon occasion of
Hamilton presenting him 'with his handsomest copy of the puzzle', that Hamilton got
the idea for the Icosian Game from p. 160 of Kirkman's 1858 article,
Lucas. RM2, 1883, pp. 208-210. First? mention of the solid version. The 2nd ed., 1893, has
a footnote referring to Kirkman, 1858.
John Jaques & Son. The Traveller's Dodecahedron; or, A Voyage Round the World. A New
Puzzle. "This amusing puzzle, exercising considerable skill in its solution, forms a
popular illustration of Sir William Hamilton's Icosian Game. A wood dodecahedron
with the base pentagon stretched so that when it sits on the base, all vertices are visible.
With ivory? pegs at the vertices, a handle that screws into the base, a string with rings
at the ends and one page of instructions, all in a box. No date. The only known
example was obtained by James Dalgety in 2002.
Pearson. 1907. Part III, no. 60: The open door, pp. 60 & 130. Prisoner in one corner of an
8 x 8 array is allowed to exit from from the other corner provided he visits every cell
once. This requires him to enter and leave a cell by the same door.
Ahrens. Mathematische Spiele. 2nd ed., Teubner, Leipzig, 1911. P. 44, note, says that a
Dodekaederspiel is available from Firma Paul Joschkowitz -- Magdeburg for .65 mark.
This is not in the 1st ed. of 1907 and the whole Chapter is dropped in the 3rd ed. of
1916 and the later editions.
Anonymous. The problems drive. Eureka 12 (Oct 1949) 7-8 & 15. No. 3. How many
Hamiltonian circuits are there on a cube, starting from a given point? Reflections and
reversals count as different tours. Answer is 12, but this assumes also that rotations are
SOURCES - page 101
different. See Singmaster, 1975, for careful definitions of how to count. There are 96
labelled circuits, of which 12 start at a given vertex. But if one takes all the 48
symmetries of the cube as equivalences (six of which fix the given vertex), there are just
2 circuits from a given starting point. However, these are actually the same circuit
started at different points. Presumably Kirkman and Hamilton knew of this.
C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26-29. 2nd ed.,
Gollancz, London, 1971, pp. 24-25. Roman knobbed dodecahedra -- an ancient solid
version??
R. E. Ingram. Appendix 2: The Icosian Calculus. In: The Mathematical Papers of Sir
William Rowan Hamilton. Vol. III: Algebra. Ed. by H. Halberstam & R. E. Ingram.
CUP, 1967, pp. 645-647. [Halberstam told me that this Appendix is due to Ingram.]
Discusses the method and asserts that the tetrahedron, cube and dodecahedron have
only one unlabelled circuit, the octahedron has two and the icosahedron has 17.
David Singmaster. Hamiltonian circuits on the regular polyhedra. Notices Amer. Math. Soc.
20 (1973) A-476, no. 73T-A199. Confirms Ingram's results and gives the number of
labelled circuits.
David Singmaster. Op. cit. in 5.F.1. 1975. Carefully defines labelled and unlabelled circuits.
Discusses results on regular polyhedra in 3 and higher dimensions.
David Singmaster. Hamiltonian circuits on the n-dimensional octahedron. J. Combinatorial
Theory (B) 18 (1975) 1-4. Obtains an explicit formula for the number of labelled
circuits on the n-dimensional octahedron and shows it is  (2n)!/e. Gives numbers for
n  8. In unpublished work, it is shown that the number of unlabelled circuits is
asymptotic to the above divided by n!2n4n.
Angus Lavery. The Puzzle Box. G&P 2 (May 1994) 34-35. Alternative solitaire, p. 34.
Asks for a knight's tour on the 33-hole solitaire board. Says he hasn't been able to do it
and offers a prize for a solution. In Solutions, G&P 3 (Jun 1994) 44, he says it cannot
be done and the proof will be given in a future issue, but I never saw it.
5.F.3.
KNIGHT'S TOURS IN HIGHER DIMENSIONS
A.-T. Vandermonde. Remarques sur les problèmes de situation. Hist. de l'Acad. des Sci. avec
les Mémoires (Paris) (1771 (1774)) Mémoires: pp. 566-574 & Plates I & II. ??NYS.
First? mention of cubical problem. (Not given in BLW excerpt.)
F. Maack. Mitt. über Raumschak. 1909, No. 2, p. 31. ??NYS -- cited by Gibbins, below.
Knight's tour on 4 x 4 x 4 board.
Dudeney. AM. 1917. Prob. 340: The cubic knight's tour, pp. 103 & 229. Says
Vandermonde asked for a tour on the faces of a 8 x 8 x 8 cube. He gives it as a
problem with a solution.
N. M. Gibbins. Chess in three and four dimensions. MG 28 (No. 279) (1944) 46-50. Gives
knight's tour on 3 x 3 x 4 board -- an unpublished result due to E. Hubar-Stockar of
Geneva. This is the smallest 3-D board with a tour. Gives Maack's tour on 4 x 4 x 4
board.
Ian Stewart. Solid knight's tours. JRM 4:1 (Jan 1971) 1. Cites Dudeney. Gives a tour
through the entire 8 x 8 x 8 cube by stacking 8 knight's paths.
T. W. Marlow. Closed knight tour of a 4 x 4 x 4 board. Chessics 29 & 30 (1987) 162.
Inspired by Stewart.
5.F.4.
OTHER CIRCUITS IN AND ON A CUBE
The number of Hamiltonian Circuits on the n-dimensional cube is the same as the
number of Gray codes (see 7.M.3) and has been the subject of considerable research. I will
not try to cover this in detail.
D. W. Crowe. The n-dimensional cube and the Tower of Hanoi. AMM 63:1 (Jan 1956)
29-30.
E. N. Gilbert. Gray codes and paths on the n-cube. Bell System Technical Journal 37 (1958)
815-826. Shows there are 9 inequivalent circuits on the 4-cube and 1 on the n-cube for
n = 1, 2, 3. The latter cases are sufficiently easy that they may have been known before
this.
Allen F. Dreyer. US Patent 3,222,072 -- Block Puzzle. Filed: 11 Jun 1965; patented: 7 Dec
1965. 4pp + 2pp diagrams. 27 cubes on an elastic. The holes are straight or diagonal
SOURCES - page 102
so that three consecutive cubes are either in a line or form a right angle. A solution is a
Hamiltonian path through the 27 cells. Such puzzles were made in Germany and I was
given one about 1980 (see Singmaster and Haubrich & Bordewijk below). Dreyer gives
two forms.
Gardner. The binary Gray code. SA (Aug 1972) c= Knotted, chap. 2. Notes that the number
of circuits on the n-cube, n > 4, is not known. SA (Apr 1973) reports that three (or
four) groups had found the number of circuits on the 4-cube -- this material is included
in the Addendum in Knotted, chap. 2, but none of the groups ever seem to have
published their results elsewhere. Unfortunately, none of these found the number of
inequivalent circuits since they failed to take all the equivalences into account -- e.g. for
n = 1, 2, 3, 4, 5, their enumerations give: 2, 8, 96, 43008, 5 80189 28640 for the
numbers of labelled circuits. Gardner's Addendum describes some further work
including some statistical work which estimates the number on the 6-cube is about
2.4 x 1025.
David Singmaster. A cubical path puzzle. Written in 1980 and submitted to JRM, but never
published. For the 3 x 3 x 3 problem, the number, S, of straight through pieces
(ignoring the ends) satisfies 2  S  11.
Mel A. Scott. Computer output, Jun 1986, 66pp. Determines there are 3599 circuits through
the 3 x 3 x 3 cube such that the resulting string of 27 cubes can be made into a cube in
just one way. But cf the next article which gives a different number??
Jacques Haubrich & Nanco Bordewijk. Cube chains. CFF 34 (Oct 1994) 12-15. Erratum,
CFF 35 (Dec 1994) 29. Says Dreyer is the first known reference to the idea and that
they were sold 'from about 1970' Reproduces the first page of diagrams from Dreyer's
patent. Says his first version has a unique solution, but the second has 38 solutions.
They have redone previous work and get new numbers. First, they consider all possible
strings of 27 cubes with at most three in a line (i.e. with at most a single 'straight' piece
between two 'bend' pieces and they find there are 98,515 of these. Only 11,487 of
these can be folded into a 3 x 3 x 3 cube. Of these, 3654 can be folded up in only one
way. The chain with the most solutions had 142 different solutions. They refer to Mel
Scott's tables and indicate that the results correspond -- perhaps I miscounted Scott's
solutions??
5.G. CONNECTION PROBLEMS
5.G.1.
GAS, WATER AND ELECTRICITY
Dudeney. Problem 146 -- Water, gas, and electricity. Strand Mag. 46 (No. 271) (Jul 1913)
110 & (No. 272) (Aug 1913) 221 (c= AM, prob. 251, pp. 73 & 200-201). Earlier
version is slightly more interesting, saying the problem 'that I have called "Water, Gas,
and Electricity" ... is as old as the hills'. Gives trick solution with pipe under one house.
A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton
& Co., London, nd [1927 -- BMC]. No. 96: The "three houses" problem, pp. 89-90 &
114. "Were all the houses connected up with all three supplies or not?" Answer is no -one connection cannot be made.
Loyd, Jr. SLAHP. 1928. The three houses and three wells, pp. 6 & 87-88. "A puzzle ...
which I first brought out in 1900 ..." The drawing is much less polished than
Dudeney's. Trick solution with a pipe under one house, a bit differently laid out than
Dudeney.
The Bile Beans Puzzle Book. 1933. No. 46: Water, gas & electric light. Trick solution
almost identical to Dudeney.
Philip Franklin. The four color problem. In: Galois Lectures; Scripta Mathematica Library
No. 5; Scripta Mathematica, Yeshiva College, NY, 1941, pp. 49-85. On p. 74, he refers
to the graph as "the basis of a familiar puzzle, to join each of three houses with each of
three wells (or in a modern version to a gas, water, and electricity plant)".
Leeming. 1946. Chap. 6, prob. 4: Water, gas and electricity, pp. 71 & 185. Dudeney's trick
solution.
H. ApSimon. Note 2312: All modern conveniences. MG 36 (No. 318) (Dec 1952) 287-288.
Given m houses and n utilities, the maximum number of non-crossing connections is
2(m+n-2) and this occurs when all the resulting regions are 4-sided. He extends to
p-partite graphs in general and a special case.
John Paul Adams. We Dare You to Solve This! Op. cit. in 5.C. 1957? Prob. 50: Another
SOURCES - page 103
enduring favorite appears below, pp. 30 & 49. Electricity, gas, water. Dudeney's trick
solution.
Young World. c1960. P. 4: Crossed lines. Electricity, TV and public address lines. Trick
solution with a line passing under a house.
T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961)
751-753. Shows you can join 4 utilities to 4 houses on a torus without crossing.
5.H. COLOURED SQUARES AND CUBES, ETC.
5.H.1.
INSTANT INSANITY = THE TANTALIZER
Note. Often the diagrams do not show all sides of the pieces so I cannot tell if one
version is the same as another.
Frederick A. Schossow. US Patent 646,463 -- Puzzle. Applied: 19 May 1899; patented:
3 Apr 1900. 1p + 1p diagrams. Described in S&B, p. 38, which also says it is
described in O'Beirne, but I don't find it there?? Four cubes with suit patterns. The net
of each cube is shown. The fourth cube has three clubs.
George Duncan Moffat. UK Patent 9810 -- Improvements in or relating to Puzzle-apparatus.
Applied: 28 May 1900; accepted: 30 Jun 1900. 2pp + 1p diagrams. For a six cube
version with "letters R, K, B, W, F and B-P, the initials of the names of General
Officers of the South African Field Force."
Joseph Meek. UK Patent 2775 -- Improved Puzzle Game. Applied: 5 Feb 1909; complete
specification: 16 Jun 1909; accepted: 3 Feb 1910. 2pp + 1p diagrams. A four cube
version with suit patterns. His discussion seems to describe the pieces drawn by
Schossow.
Slocum. Compendium. Shows: The Great Four Ace Puzzle (Gamage's, 1913); Allies Flag
Puzzle (Gamage's, c1915); Katzenjammer Puzzle (Johnson Smith, 1919).
Edwin F. Silkman. US Patent 2,024,541 -- Puzzle. Applied: 9 Sep 1932; patented: 17 Dec
1935. 2pp + 1 p diagrams. Four cubes marked with suits. The net of each cube is
shown. The third cube has three hearts. This is just a relabelling of Schossow's pattern,
though two cubes have to be reflected which makes no difference to the solution
process.
E. M. Wyatt. The bewitching cubes. Puzzles in Wood. (Bruce Publishing, Co., Milwaukee,
1928) = Woodcraft Supply Corp., Woburn, Mass., 1980, p. 13. A six cube, six way
version.
Abraham. 1933. Prob. 303 -- The four cubes, p. 141 (100). 4 cube version "sold ... in 1932".
A. S. Filipiak. Four ace cube puzzle. 100 Puzzles, How do Make and How to Solve Them.
A. S. Barnes, NY, (1942) = Mathematical Puzzles, and Other Brain Twisters;
A. S. Barnes, NY, 1966; Bell, NY, 1978; p. 108.
Leeming. 1946. Chap. 10, prob. 9: The six cube puzzle, pp. 128-129 & 212. Identical to
Wyatt.
F. de Carteblanche [pseud. of Cedric A. B. Smith]. The coloured cubes problem. Eureka 9
(1947) 9-11. General graphical solution method, now the standard method.
T. H. O'Beirne. Note 2736: Coloured cubes: A new "Tantalizer". MG 41 (No. 338)
(Dec 1957) 292-293. Cites Carteblanche, but says the current version is different.
Gives a nicer version.
T. H. O'Beirne. Note 2787: Coloured cubes: a correction to Note 2736. MG 42 (No. 342)
(Dec 1958) 284. Finds more solutions than he had previously stated.
Norman T. Gridgeman. The 23 colored cubes. MM 44:5 (Nov 1971) 243-252. The 23
colored cubes are the equivalence classes of ways of coloring the faces with 1 to 6
colors. He cites and describes some later methods for attacking Instant Insanity
problems.
Jozsef Bognár. UK Patent Application 2,076,663 A -- Spatial Logical Puzzle. Filed 28 May
1981; published 9 Dec 1981. Cover page + 8pp + 3pp diagrams. Not clear if the patent
was ever granted. Describes Bognár's Planets, which is a four piece instant insanity
where the pieces are spherical and held in a plastic tube. This was called Bolygok in
Hungarian and there is a reference to an earlier Hungarian patent. Also describes his
version with eight pieces held at the corners of a plastic cube.
5.H.2.
MACMAHON PIECES
SOURCES - page 104
Haubrich's 1995-1996 surveys, op. cit. in 5.H.4, include MacMahon puzzles as one
class.
I have just added the Carroll result that there are 30 six-coloured cubes, but this must be
older??
Frank H. Richards. US Patent 331,652 -- Domino. Applied: 13 Jun 1885; patented: 1 Dec
1885. 2pp + 2pp diagrams. Cited by Gardner in Magic Show, but with date 1895.
Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. For triangular matching
games, specifically showing the MacMahon 5-coloured triangles, but considering
reflections as equivalences, so he has 35 pieces. [One of the colours is blank and hence
Gardner said it was a 4-colouring.]
Carroll-Wakeling. c1890? Prob. 15: Painting cubes, pp. 18-19 & 67. This is one of the
problems on undated sheets of paper that Carroll sent to Bartholomew Price. How
many ways can one six-colour a cube? Wakeling gives a solution, but this apparently is
not on Carroll's MS.
Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 3927 A.D. 1892 -Appliances to be used in Playing a New Class of Games. Applied: 29 Feb 1892;
Complete Specification Left: 28 Nov 1892; Accepted: 28 Jan 1893. 5pp + 2pp
diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. Describes the 24
triangles with four types of edge and mentions other numbers of edge types. Describes
various games and puzzles.
Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 8275 A.D. 1892 -Appliances for New Games of Puzzles. Applied: 2 May 1892; Complete Specification
Left: 31 Jan 1893; Accepted: 4 Mar 1893. 2pp. 27 cubes with three colours, opposite
faces having the same colour. Similar sets of 8, 27, 64, etc. cubes. Various matching
games suggested. Using six colours and all six on each cube gives 30 cubes -- the
MacMahon Cubes. Gives a complex matching problem of making two 2 x 2 x 2
cubes. Paul Garcia (email of 15 Nov 2002) commented: "8275 describes 2 different
sets of blocks, using either three colours or six colours. The three colour blocks form a
set of 27 that can be assembled into a large cube with single coloured faces and internal
contact faces matching. For the six colour cubes, the puzzle suggested is to pick out
two associated cubes, and find the sixteen cubes that can be assembled to make a copy
of each. Not quite Mayblox, although using the same colouring system."
James Dalgety. R. Journet & Company A Brief History of the Company & its Puzzles.
Published by the author, North Barrow, Somerset, 1989. On p. 13, he says Mayblox
was patented in 1892. In an email on 12 Nov 2002, he cited UK Patent 8275.
Anon. Report: "Mathematical Society, February 9". Nature 47 (No. 1217) (23 Feb 1893)
406. Report of MacMahon's talk: The group of thirty cubes composed by six
differently coloured squares.
See: Au Bon Marché, 1907, in 5.P.2, for a puzzle of hexagons with matching edges.
Manson. 1911. Likoh, pp. 171-172. MacMahon's 24 four-coloured isosceles right triangles,
attributed to MacMahon.
"Toymaker". The Cubes of Mahomet Puzzle. Work, No. 1447 (9 Dec 1916) 168. 8 sixcoloured cubes to be assembled into a cube with singly-coloured faces and internal
faces to have matching colours.
P. A. MacMahon. New Mathematical Pastimes. CUP, 1921. The whole book deals with
variations of the problem and calculates the numbers of pieces of various types. In
particular, he describes the 24 4-coloured triangles, the 24 3-coloured squares, the
MacMahon cubes, some right-triangular and hexagonal sets and various subsets of
these. With n colours, there are n(n2+2)/3 triangles, n(n+1)(n2-n+2)/4 squares and
n(n+1)(n4-n3+n2+2)/6 hexagons. [If one allows reflectional equivalence, one gets
n(n+1)(n+2)/6 triangles, n(n+1)(n2+n+2)/8 squares and n(n+1)(n4-n3+4n2+2)/12
hexagons. Problem -- is there an easy proof that the number of triangles is BC(n+2,
3)?] On p. 44, he says that Col. Julian R. Jocelyn told him some years ago that one
could duplicate any cube with 8 other cubes such that the internal faces matched.
Slocum. Compendium. Shows Mayblox made by R. Journet from Will Goldston's 1928
catalogue.
F. Winter. Das Spiel der 30 bunten Würfel MacMahon's Problem. Teubner, Leipzig, 1934,
128pp. ??NYR.
Clifford Montrose. Games to play by Yourself. Universal Publications, London, nd [1930s?].
The coloured squares, pp. 78-80. Makes 16 squares with four-coloured edges, using
SOURCES - page 105
five colours, but there is no pattern to the choice. Uses them to make a 4 x 4 array
with matching edges, but seems to require the orientations to be fixed.
M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 6. There are twelve ways to colour the edges of a pentagon, when rotations
and reflections are considered as equivalences. Can you colour the edges of a
dodecahedron so each of these pentagonal colourings occurs once? [If one uses tiles,
one has to have reversible tiles.] Solution says there are three distinct solutions and
describes them by describing contacts between 10 pentagons forming a ring around the
equator.
Richard K. Guy. Some mathematical recreations I & II. Nabla [= Bull. Malayan Math. Soc.]
7 (Oct & Dec 1960) 97-106 & 144-153. Pp. 101-104 discusses MacMahon triangles,
squares and hexagons.
T. H. O'Beirne. Puzzles and paradoxes 5: MacMahon's three-colour set of squares. New
Scientist 9 (No. 220) (2 Feb 1961) 288-289. Finds 18 of the 20 possible monochrome
border patterns.
Gardner. SA (Mar 1961) = New MD, Chap. 16. MacMahon's 3-coloured squares and his
cubes. Addendum in New MD cites Feldman, below.
Gary Feldman. Documentation of the MacMahon Squares Problem. Stanford Artificial
Intelligence Project Memo No. 12, Stanford Computation Center, 16 Jan 1964. ??NYS
Finds 12,261 solutions for the 6 x 4 rectangle with monochrome border -- but see
Philpott, 1982, for 13,328 solutions!!
Gardner. SA (Oct 1968) = Magic Show, Chap. 16. MacMahon's four-coloured triangles and
numerous variants.
Wade E. Philpott. MacMahon's three-color squares. JRM 2:2 (1969) 67-78. Surveys the
topic and repeats Feldman's result.
N. T. Gridgeman, loc. cit. in 5.H.1, 1971, covers some ideas on the MacMahon cubes.
J. J. M. Verbakel. Digitale tegels (Digital tiles). Niet piekeren maar puzzelen (name of a
puzzle column). Trouw (a Dutch newspaper) (1 Feb 1975). ??NYS -- described by
Jacques Haubrich; Pantactic patterns and puzzles; CFF 34 (Oct 1994) 19-21. There are
16 ways to 2-colour the edges of a square if one does not allow them to rotate.
Assemble these into a 4 x 4 square with matching edges. There are 2,765,440
solutions in 172,840 classes of 16. One can add further constraints to yield fewer
solutions -- e.g. assume the 4 x 4 square is on a torus and make all internal lines have a
single colour.
Gardner. Puzzling over a problem-solving matrix, cubes of many colours and
three-dimensional dominoes. SA 239:3 (Sep 1978) 20-30 & 242 c= Fractal, chap. 11.
Good review of MacMahon (photo) and his coloured cubes. Bibliography cites recent
work on Mayblox, etc.
Wade E. Philpott. Instructions for Multimatch. Kadon Enterprises, Pasadena, Maryland,
1982. Multimatch is just the 24 MacMahon 3-coloured squares. This surveys the
history, citing several articles ??NYS, up to the determination of the 13,328 solutions
for the 6 x 4 rectangle with monochrome border, by Hilario Fernández Long (1977)
and John W. Harris (1978).
Torsten Sillke. Three 3 x 3 matching puzzles. CFF 34 (Oct 1994) 22-23. He has wanted an
interesting 9 element subset of the MacMahon pieces and finds that of the 24
MacMahon 3-coloured squares, just 9 of them contain all three colours. He considers
both the corner and the edge versions. The editor notes that a 3 x 3 puzzle has
36 x 32/2 = 576 possible edge contacts and that the number of these which match is a
measure of the difficulty of the puzzle, with most 3 x 3 puzzles having 60 to 80
matches. The corner version of Sillke's puzzle has 78 matches and one solution. The
edge version has 189 matches and many solutions, hence Sillke proposes various
further conditions.
5.H.3.
PATH FORMING PUZZLES
Here we have a set of pieces and one has to join them so that some path is formed. This
is often due to a chain or a snake, etc. New section. Again, Haubrich's 1995-1996 surveys,
op. cit. in 5.H.4, include this as one class.
Hoffmann. 1893. Chap. III, No. 18: The endless chain, pp. 99-100 & 131
SOURCES - page 106
= Hoffmann-Hordern, pp. 91-92, with photo. 18 pieces, some with parts of a chain, to
make into an 8 x 8 array with the chain going through 34 of the cells. All the pieces
are rectangles of width one. Photo shows The Endless Chain, by The Reason
Manufacturing Co., 1880-1895. Hordern Collection, p. 62, shows the same and La
Chaine sans fin, 1880-1905.
Loyd. Cyclopedia. 1914. Sam Loyd's endless chain puzzle, pp. 280 & 377. Chain through
all 64 cells of a chessboard, cut into 13 pieces. The chessboard dissection is of type:
13: 02213 131.
Hummerston. Fun, Mirth & Mystery. 1924. The dissected serpent, p. 131. Same pieces as
Hoffmann, and almost the same pattern.
Collins. Book of Puzzles. 1927. The dissected snake puzzle, pp. 126-127. 17 pieces
forming an 8 x 8 square. All the piece are rectangular pieces of width one except for
one L-hexomino -- if this were cut into straight tetromino and domino, the pieces would
be identical to Hoffmann. The pattern is identical to Hummerston.
See Haubrich in 5.H.4.
5.H.4.
OTHER AND GENERAL
These all have coloured edges unless specified. See S&B, p. 36, for examples.
Edwin L[ajette] Thurston. US Patent 487,797 -- Puzzle. Applied: 30 Sep 1890; patented:
13 Dec 1892. 3pp + 3pp diagrams. Reproduced in Haubrich, About ..., 1996, op. cit.
below. 4 x 4 puzzles with 6-coloured corners or edges, but assuming no colour is
repeated on a piece -- indeed he uses the 15 = BC(6,2) ways of choosing 4 out of 6
colours once only and then has a sixteenth with the same colours as another, but in
different order. Also a star-shaped puzzle of six parallelograms.
Edwin L. Thurston. US Patent 487,798 -- Puzzle. Applied: 30 Sep 1890; patented:
13 Dec 1892. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit.
below. As far as I can see, this is the same as the 4 x 4 puzzle with 6-coloured edges
given above, but he seems to be emphasising the 15 pieces.
Edwin L. Thurston. US Patent 490,689 -- Puzzle. Applied: 30 Sep 1890; patented: 31 Jun
1893. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below.
The patent is for 3 x 3 puzzles with 4-coloured corners or edges, but with pieces
having no repeated colours and in a fixed orientation. He selects some 8 of these pieces
for reasons not made clear and mentions moving them "after the manner of the old 13,
14, 15 puzzle." S&B, p. 36, describes the Calumet Puzzle, Calumet Baking Powder
Co., Chicago, which is a 3 x 3 head to tail puzzle, claimed to be covered by this patent.
Le Berger Malin. France, c1900. 3 x 3 head to tail puzzle, but the edges are numbered and
the matching edges must add to 10. ??NYS -- described by K. Takizawa, N. Takashima
& N. Yoshigahara; Vess Puzzle and Its Family -- A Compendium of 3 by 3 Card
Puzzles; published by the authors, Tokyo, 1983. Slocum has this in two different boxes
and dates it to c1900 -- I had c1915 previously. Haubrich has one version, Produced by
GB&O N.K. Atlas.
Angus K. Rankin. US Patent 1,006,878 -- Puzzle. Applied: 3 Feb 1911; patented: 24 Oct
1911. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below.
Described in S&B, p. 36. Grandpa's Wonder Puzzle. 3 x 3 square puzzle. Each piece
has corners coloured, using four colours, and the colours meeting at a corner must
differ. The patent doesn't show the advertiser's name -- Grandpa's Wonder Soap -- but
is otherwise identical to S&B's photo.
Daily Mail World Record Net Sale puzzle. 1920-1921. Instructions and picture of the pieces.
Letter from Whitehouse to me describing its invention. 19 6-coloured hexagons
without repeated colours. Daily Mail articles as follows. There may be others that I
missed and sometimes the page number is a bit unclear. Note that 5 Dec was a Sunday.
9 Nov 1920, p. 5. "Daily Mail" puzzle. To be issued on 7 Dec.
13 Nov 1920, p. 4. Hexagon mystery.
17 Nov 1920, p. 5. New mystery puzzle. Asserts the inventor does not know the
solution -- i.e. the solution has been locked up in a safe.
20 Nov 1920, p. 4. What is it?
23 Nov 1920, p. 5. Fascinating puzzle. The most fascinating puzzle since "Pigs in
Clover".
25 Nov 1920, p. 5. Can you do it?
SOURCES - page 107
29 Nov 1920, p. 5. £250 puzzle.
1 Dec 1920, p. 4. Mystery puzzle clues.
2 Dec 1920, p. 5. £250 puzzle race.
3 Dec 1920, p. 5. The puzzle.
4 Dec 1920, p. 4. The puzzle. Amplifies on the inventor not knowing the solution -after the idea was approved, a new pattern was created by someone else and
locked up.
6 Dec 1920, unnumbered back page. Photo with caption: £250 for solving this.
7 Dec 1920, p. 7. "Daily Mail" Puzzle. Released today. £100 for getting the locked
up solution. £100 for the first alternative solution and £50 for the next alternative
solution. "It is believed that more than one solution is possible."
8 Dec 1920, p. 5. "Daily Mail" puzzle.
9 Dec 1920, p. 5. Can you do it?
10 Dec 1920, p. 4. It can be done.
13 Dec 1920, p. 9. Most popular pastime. "More than 500,000 Daily Mail Puzzles
have been sold."
15 Dec 1920, p. 4. Puzzle king & the 19 hexagons. Dudeney says he does not think it
can be solved "except by trial."
16 Dec 1920, p. 4. Tantalising 19 hexagons.
16 Dec 1920, unnumbered back page. Banner at top has: "The Daily Mail" puzzle.
Middle of page has a cartoon of sailors trying to solve it.
17 Dec 1920, p. 5? The Xmas game.
18 Dec 1920, p. 7. Puzzle Xmas 'card'.
20 Dec 1920, p. 7. Hexagon fun.
22 Dec 1920, p. 3. 3,000,000 fascinated. It is assumed that about 5 people try each
example and so this indicates that nearly 600,000 have been sold.
23 Dec 1920, p. 3. Too many cooks.
23 Dec 1920, unnumbered back page. Cartoon: The hexagonal dawn!
28 Dec 1920, p. 3? Puzzled millions. "On Christmas Eve the sales exceeded 600,000
...."
29 Dec 1920, p. 3? "I will do it."
30 Dec 1920, p. 8. Puzzle fun.
3 Jan 1921, p. 3. The Daily Mail Puzzle. C. Lewis, aged 21, a postal clerk solved it
within two hours of purchase and submitted his solution on 7 Dec. Hundreds of
identical solutions were submitted, but no alternative solutions have yet
appeared. There are two pairs of identical pieces: 1 & 12, 4 & 10.
3 Jan 1921, p. 10 = unnumbered back page. Hexagon Puzzle Solved, with photo of C.
Lewis and diagram of solution.
10 Jan 1921, p. 4. Hexagon puzzle. Since no alternative hexagonal solutions were
received, the other £150 is awarded to those who submitted the most ingenious
other solution -- this was judged to be a butterfly shape, submitted by 11 persons,
who shared the £150.
Horace Hydes & Francis Reginald Beaman Whitehouse. UK Patent 173,588 -- Improvements
in Dominoes. Applied: 29 Sep 1920; complete application: 29 Jun 1921; accepted:
29 Dec 1921. Reproduced in Haubrich, About ..., 1996, op. cit. below. 3pp + 1p
diagrams. This is the patent for the above puzzle, corresponding to provisional patent
27599/20 on the package. The illustration shows a solved puzzle based on 'A stitch in
time saves nine'.
George Henry Haswell. US Patent 1,558,165 -- Puzzle. Applied: 3 Jul 1924; patented: 11
Sep 1925. Reproduced in Haubrich, About ..., 1996, op. cit. below. 2pp + 1p diagrams.
For edge-matching hexagons with further internal markings which have to be aligned.
[E.g. one could draw a diagonal and require all diagonals to be vertical -- this greatly
simplifies the puzzle!] If one numbers the vertices 1, 2, ..., 6, he gives an example
formed by drawing the diagonals 13, 15, 42, 46 which produces six triangles along the
edges and an internal rhombus.
C. Dudley Langford. Note 2829: Dominoes numbered in the corners. MG 43 (No. 344)
(May 1959) 120-122. Considers triangles, squares and hexagons with numbers at the
corners. There are the same number of pieces as with numbers on the edges, but corner
numbering gives many more kinds of edges. E.g. with four numbers, there are 24
triangles, but these have 16 edge patterns instead of 4. The editor (R. L. Goodstein)
tells Langford that he has made cubical dominoes "presumably with faces numbered".
SOURCES - page 108
Langford suggests cubes with numbers at the corners. [I find 23 cubes with two corner
numbers and 333 with three corner numbers. ??check]
Piet Hein. US Patent 4,005,868 -- Puzzle. Applied: 23 Jun 1975; patented: 1 Feb 1977.
Front page + 8pp diagrams + 5pp text. Basically non-matching puzzles using marks at
the corners of faces of the regular polyhedra. He devises boards so the problems can be
treated as planar.
Kiyoshi Takizawa; Naoaki Takashima & Nob. Yoshigahara. Vess Puzzle and Its Family -A Compendium of 3 by 3 Card Puzzles. Published by the authors, Tokyo, Japan,
1983. Studies 32 types (in 48 versions) of 3 x 3 'head to tail' matching puzzles and
4 related types (in 4 versions). All solutions are shown and most puzzles are
illustrated with colour photographs of one solution. (Haubrich counts 51 versions -check??)
Melford D. Clark. US Patent 4,410,180 -- Puzzle. Applied: 16 Nov 1981; patented: 18 Oct
1983. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. 2pp + 2pp diagrams.
Corner matching squares, but with the pieces marked 1, 2, ..., so that the pieces
marked 1 form a 1 x 1 square, the pieces marked 2 allow this to be extended to a 2 x
2 square, etc. There are n2 - (n-1)2 pieces marked n.
Jacques Haubrich. Compendium of Card Matching Puzzles. Printed by the author,
Aeneaslaan 21, NL-5631 LA Eindhoven, Netherlands, 1995. 2 vol., 325pp. describing
over 1050 puzzles. He classifies them by the nine most common matching rules:
Heads and Tails; Edge Matching (i.e. MacMahon); Path Matching; Corner Matching;
Corner Dismatching; Jig-Saw-Like; Continuous Path; Edge Dismatching; Hybrid.
He does not include Jig-Saw-Like puzzles here. Using the number of cards and their
shape, then the matching rules, he has 136 types. 31 different numbers of cards occur:
4, 6-16, 18-21, 23-25, 28, 30, 36, 40, 45, 48, 56, 64, 70, 80, 85, 100. There is an index
of 961 puzzle names. He says Hoffmann is the earliest published example. He notes
that most path puzzles have a global criterion that the result have a single circuit which
slightly removes them from his matching criterion and he does not treat them as
thoroughly. He has developed computer programs to solve each type of puzzle and has
checked them all.
Jacques Haubrich. About, Beyond and Behind Card Matching Puzzles. [= Vol. 3 of above].
Ibid, Apr 1996, 87pp. This is a general discussion of the different kinds of puzzles,
how to solve them and their history, reproducing ten patents and two obituaries.
5.I.
LATIN SQUARES AND EULER SQUARES
This topic ties in with certain tournament problems but I have not covered them. See
also Hoffmann and Loughlin & Flood in 5.A.2 for examples of two orthogonal 3 x 3 Latin
squares. The derangement problems in 5.K.2 give Latin rectangles.
Ahrens-1 & Ahrens-2. Opp. cit. in 7.N. 1917 & 1922. Ahrens-1 discusses and cites early
examples of Latin squares, going back to medieval Islam (c1200), where they were used
on amulets. Ahrens-2 particularly discusses work of al-Buni -- see below.
(Ahmed [the h should have an underdot] ibn ‘Alî ibn Jûsuf) el-Bûni, (Abû'l-‘Abbâs,
el-Qoresî.) = Abu-l‘Abbas al-Buni. (??= Muhyi'l-Dîn Abû’l-‘Abbâs al-Bûnî -- can't
relocate my source of this form.) Sams al-ma‘ârif = Shams al-ma‘ârif al-kubrâ = Šams
al-ma‘ārif. c1200. ??NYS. Ahrens-1 describes this briefly and incorrectly. He
expands and corrects this work in Ahrens-2. See 7.N for more details. Ahrens notes
that a 4 x 4 magic square can be based on the pattern of two orthogonal Latin squares
of order 4, and Al-Buni's work indicates knowledge of such a pattern, exemplified by
the square
8, 11, 14, 1; 13, 2, 7, 12; 3, 16, 9, 6; 10, 5, 4, 15 considered (mod 4). He
also has Latin squares of order 4 using letters from a name of God. He goes on to
show 7 Latin squares of order 7, using the same 7 letters each time -- though four
are corrupted. (Throughout, the Latin squares also have 'Latin' diagonals, i.e. the
diagonals contain all the values.) These are arranged so each has a different letter in the
first place. It is conjectured that these are associated with the days of the week or the
planets.
Tagliente. Libro de Abaco. (1515). 1541. F. 18v. 7 x 7 Latin square with entries 1, 13, 2,
SOURCES - page 109
14, 3, 10, 4 cyclically shifted forward -- i.e. the second row starts 13, 2, .... This is an
elaborate plate which notes that the sum of each file is 47 and has a motto: Sola Virtu la
Fama Volla, but I could find no text or other reason for its appearance!
Inscription on memorial to Hannibal Bassett, d. 1708, in Meneage parish church, St. Mawgan,
Cornwall. I first heard of this from Chris Abbess, who reported it in some newsletter in
c1993. However, [Peter Haining; The Graveyard Wit; Frank Graham, Newcastle, 1973,
p. 133] cites this as being at Cunwallow, near Helstone, Cornwall. [W. H. Howe;
Everybody's Book of Epitaphs Being for the Most Part What the Living Think of the
Dead; Saxon & Co., London, nd [c1895] (facsimile by Pryor Publications, Whitstable,
1995); p. 173] says it is in Gunwallow Churchyard. Spelling and punctuation vary a bit.
The following gives a detailed account.
Alfred Hayman Cummings. The Churches and Antiquities of Cury & Gunwalloe, in the
Lizard District, including Local Traditions. E. Marlborough & Co., London & Truro,
1875, pp. 130-131. ??NX. "It has been said that there once existed ... the curious
epitaph --" and gives a considerable rearrangement of the inscription below. He
continues "But this is in all probability a mistake, as repeated search has been made for
it, not only by the writer, but by a former Vicar of Gunwalloe, and it could nowhere be
found, while there is a plate with an inscription in the church at Mawgan, the next
parish, which might be very easily the one referred to." He gives the following
inscription, saying it is to Hannibal Basset, d. 1708-9. Chris Weeks was kind enough to
actually go to the church of St. Winwaloe, Gunwalloe, where he found nothing, and to
St. Mawgan in Meneage, a few miles away. Chris Weeks sent pictures of Gunwallowe
-- the church is close to the cliff edge and it looks like there could once have been more
churchyard on the other side of the church where the cliff has fallen away. In the
church at St. Mawgan is the brass plate with 'the Acrostic Brass Inscription', but it is not
clearly associated with a grave and I wonder if it may have been moved from
Gunwallowe when a grave was eroded by the sea. It is on the left of the arch by the
pulpit. I reproduce Chris Weeks' copy of the text. He has sent a photograph, but it was
dark and the photo is not very clear, but one can make out the Latin square part.
Hanniball Baet here Inter'd doth lye
Who dying lives to all Eternitye
hee departed this life the 17th of Ian
1709/8 in the 22th year of his age ~
A lover of learning
Shall
Wee
all
dye
wee
shall
dye
all
all
dye
shall
wee
dye
all
wee
shall
The  are old style long esses. The superscript th is actually over the numeral.
The 9 is over the 8 in the year and there is no stroke. This is because it was before
England adopted the Gregorian calendar and so the year began on 25 Mar and was a
year behind the continent between 1 Jan and 25 Mar. Correspondence of the time
commonly would show 1708/9 at this time, and I have used this form for typographic
convenience, but with the 9 over the 8 as on the tomb.
A word game book points out that this inscription is also palindromic!!
Richard Breen. Funny Endings. Penny Publishing, UK, 1999, p. 35. Gives the following
form: Shall we all die? / We shall die all. / All die shall we? / Die all we shall and
notes that it is a word palindrome and says it comes from Gunwallam [sic], near
Helstone.
Joseph Sauveur. Construction générale des quarrés magiques. Mémoires de l'Académie
Royale des Sciences 1710(1711) 92-138. ??NYS -- described in Cammann-4, p. 297,
(see 7.N for details of Cammann) which says Sauveur invented Latin squares and
describes some of his work.
Ozanam. 1725. 1725: vol. IV, prob. 29, p. 434 & fig. 35, plate 10 (12). Two 4 x 4
orthogonal squares, using A, K, Q, J of the 4 suits, but it looks like:
J, A, K, Q; Q, K, A, J; A, J, Q, K; K, Q, J, A; but
SOURCES - page 110
the  and  look very similar. From later versions of the same diagram, it is clear
that the first row should have its  and  reversed. Note the diagonals also contain
all four ranks and suits. (I have a reference for this to the 1723 edition.)
Minguet. 1733. Pp. 146-148 (1864: 142-143; not noticed in other editions). Two 4 x 4
orthogonal squares, using A, K, Q, J (= As, Rey, Caballo (knight), Sota (knave)) of the
4 suits, but the Spanish suits, in descending order, are: Espadas, Bastos, Oros, Copas.
The result is described but not drawn, as:
RO, AE, CC, SB; SC, CB, AO, RE; AB, RC, SE, CO; CE, SO, RB, AC;
which would translate into the more usual cards as:
K, A, Q, J; J, Q, A, K; A, K, J, Q; Q, J, K, A.
However, I'm not sure of the order of the Caballo and Sota; if they were reversed, which
would interchange Q and J in the latter pattern, then both Ozanam and Minguet would
have the property that each row is a cyclic shift or reversal of A, K, Q, J.
Alberti. 1747. Art. 29, p. 203 (108) & fig. 36, plate IX, opp. p. 204 (108). Two 4 x 4
orthogonal squares, figure simplified from the correct form of Ozanam, 1725.
L. Euler. Recherches sur une nouvelle espèce de Quarrés Magiques. (Verhandelingen
uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen
(= Flessingue) 9 (1782) 85-239.) = Opera Omnia (1) 7 (1923) 291-392.
(= Comm. Arithm. 2 (1849) 302-361.)
Manuel des Sorciers. 1825. Pp. 78-79, art. 39. ??NX Correct form of Ozanam.
The Secret Out. 1859. How to Arrange the Twelve Picture Cards and the four Aces of a Pack
in four Rows, so that there will be in Neither Row two Cards of the same Value nor two
of the same Suit, whether counted Horizontally or Perpendicularly, pp. 90-92. Two
4 x 4 orthogonal Latin squares, not the same as in Ozanam.
Bachet-Labosne. Problemes. 3rd ed., 1874. Supp. prob. XI, 1884: 200-202. Two 4 x 4
orthogonal squares.
Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles, No. XVI, pp. 17-18.
Similar to Ozanam.
Hoffmann. 1893. Chap. X, no. 14: Another card puzzle, pp. 342 & 378-379
= Hoffmann-Hordern, pp. 234 & 236. Two orthogonal Latin squares, but the diagonals
do not contain all the suits and ranks.
A, J, Q, K; J, A, K, Q; Q, K, A, J; K, Q, J, A.
G. Tarry. Le probleme de 36 officiers. Comptes Rendus de l'Association Française pour
l'Avancement de Science Naturel 1 (1900) 122-123 & 2 (1901) 170-203. ??NYS
Dudeney. Problem 521. Weekly Dispatch (1 Nov, 15 Nov, 1903) both p. 10.
H. A. Thurston. Latin squares. Eureka 9 (Apr 1947) 19-21. Survey of current knowledge.
T. G. Room. Note 2569: A new type of magic square. MG 39 (No. 330) (Dec 1955) 307.
Introduces 'Room Squares'. Take the 2n(2n-1)/2 combinations from 2n symbols and
insert them in a 2n-1 x 2n-1 grid so that each row and column contains all 2n
symbols. There are n entries and n-1 blanks in each row and column. There is an
easy solution for n = 1. n = 2 and n = 3 are impossible. Gives a solution for n = 4.
This is a design for a round-robin tournament with the additional constraint of 2n-1
sites such that each player plays once at each site.
Parker shows there are two orthogonal Latin squares of order 10 in 1959.
R. C. Bose & S. S. Shrikande. On the falsity of Euler's conjecture about the nonexistence of
two orthogonal Latin squares of order 4t+2. Proc. Nat. Acad. Sci. (USA) 45: 5 (1959)
734-737.
Gardner. SA (Nov 1959) c= New MD, chap. 14. Describes Bose & Shrikande's work. SA
cover shows a 10 x 10 counterexample in colour. Kara Lynn and David Klarner
actually made a quilt of this, thereby producing a counterpane counterexample! They
told me that the hardest part of the task was finding ten sufficiently contrasting colours
of material.
H. Howard Frisinger. Note: The solution of a famous two-centuries-old problem: the
Leonhard Euler-Latin square conjecture. HM 8 (1981) 56-60. Good survey of the
history.
Jacques Bouteloup. Carrés Magiques, Carrés Latins et Eulériens. Éditions du Choix,
Bréançon, 1991. Nice systematic survey of this field, analysing many classic methods.
An Eulerian square is essentially two orthogonal Latin squares.
5.I.1. EIGHT QUEENS PROBLEM
SOURCES - page 111
See MUS I 210-284. S&B 37 shows examples. See also 5.Z. See also 6.T for
examples where no three are in a row.
Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904. Pp. 1082-1084
discusses history and results for the n queens problem, with many references.
Paul J. Campbell. Gauss and the eight queens problem. HM 4:4 (Nov 1977) 397-404.
Detailed history. Demonstrates that Gauss did not obtain a complete solution and traces
how this misconception originated and spread.
"Schachfreund" (Max Bezzel). Berliner Schachzeitung 3 (Sep 1848) 363. ??NYS
Solutions. Ibid. 4 (Jan 1849) 40. ??NYS (Ahrens says this only gives two solutions.
A. C. White says two or three. Jaenisch says a total of 5 solutions were published here
and in 1854.)
Franz Nauck. Eine in das Gebiet der Mathematik fallende Aufgabe von Herrn Dr. Nauck in
Schleusingen. Illustrirte Zeitung (Leipzig) 14 (No. 361) (1 Jun 1850) 352. Reposes
problem. [The papers do not give a first name or initial. The only Nauck in the first six
volumes of Poggendorff is Ernst Friedrich (1819-1875), a geologist. Ahrens gives no
initial. Campbell gives Franz.]
Franz Nauck. Briefwechseln mit Allen für Alle. Illustrirte Zeitung (Leipzig) 15 (No. 377)
(21 Sep 1850) 182. Complete solution.
Editorial comments: Briefwechsel. Illustrirte Zeitung (Leipzig) 15 (No. 378) (28 Sep 1850)
207. Thanks 6 correspondents for the complete solution and says Nauck reports that a
blind person has also found all 92 solutions.
Gauss read the Illustrirte Zeitung and worked on the problem, corresponding with his friend
Schumacher starting on 1 Sep 1850. Campbell discusses the content of the letters,
which were published in: C. A. F. Peters, ed; Briefwechsel zwischen C. F. Gauss und
H. C. Schumacher; vol. 6, Altona, 1865, ??NYS. John Brillhart writes that there is
some material in Gauss' Werke, vol. XII: Varia kleine Notizen verschiednen Inhalts ...
5, pp. 19-28, ??NYS -- not cited by Campbell.
F. J. E. Lionnet. Question 251. Nouvelles Annales de Mathématiques 11 (1852) 114-115.
Reposes problem and gives an abstract version.
Giusto Bellavitis. Terza rivista di alcuni articoli dei Comptes Rendus dell'Accademia delle
Scienze di Francia e di alcuni questioni des Nouvelles Annales des mathématiques. Atti
dell'I. R. Istituto Veneto di Scienze, Lettere ed Arti (3) 6 [= vol. 19] (1860/61) 376-392
& 411-436 (as part of Adunanza del Giorno 17 Marzo 1861 on pp. 347-436). The
material of interest is: Q. 251. Disposizione sullo scacchiere di otto regine, on
pp. 434-435. Gives the 12 essentially different solutions. Lucas (1895) says Bellavitis
was the first to find all solutions, but see above. However this may be the first
appearance of the 12 essentially different solutions.
C. F. de Jaenisch. Op. cit. in 5.F.1. 1862. Vol. 1, pp. 122-135. Gives the 12 basic solutions
and shows they produce 92. Notes that in every solution, 4 queens are on white squares
and 4 are on black.
A. C. Cretaine. Études sur le Problème de la Marche du Cavalier au Jeu des Échecs et
Solution du Problème des Huit Dames. A. Cretaine, Paris, 1865. ??NYS -- cited by
Lucas (1895). Shows it is possible to solve the eight queens problem after placing one
queen arbitrarily.
G. Bellavitis. Algebra N. 72 Lionnet. Atti dell'Istituto Veneto (3) 15 (1869/70) 844-845.
Siegmund Günther. Zur mathematische Theorie des Schachbretts. Grunert's Archiv der
Mathematik und Physik 56 (1874) 281-292. ??NYS. Sketches history of the problem -see Campbell. He gives a theoretical, but not very practical, approach via determinants
which he carries out for 4 x 4 and 5 x 5.
J. W. L. Glaisher. On the problem of the eight queens. Philosophical Magazine (4) 48 (1874)
457-467. Gives a sketch of Günther's history which creates several errors, in particular
attributing the solution to Gauss -- see Campbell, who suggests Glaisher could not read
German well. (However, in 1921 & 1923, Glaisher published two long articles
involving the history of 15-16C German mathematics, showing great familiarity with
the language.) Simplifies and extends Günther's approach and does 6 x 6, 7 x 7, 8 x 8
boards.
Lucas. RM2, 1883. Note V: Additions du Tome premier. Pp. 238-240. Gives the solutions
on the 9 x 9 board, due to P. H. Schoute, in a series of articles titled Wiskundige
Verpoozingen in Eigen Haard. Gives the solutions on the 10 x 10 board, found by
SOURCES - page 112
M. Delannoy.
S&B, p. 37, show an 1886 puzzle version of the six queens problem.
A. Pein. Aufstellung von n Königinnen auf einem Schachbrett von n2 Feldern. Leipzig.
??NYS -- cited by Ball, MRE, 4th ed., 1905 as giving the 92 inequivalent solutions on
the 10 x 10.
Ball. MRE, 1st ed., 1892. The eight queens problem, pp. 85-88. Cites Günther and Glaisher
and repeats the historical errors. Sketches Günther's approach, but only cites Glaisher's
extension of it. He gives the numbers of solutions and of inequivalent solutions up
through 10 x 10 -- see Dudeney below for these numbers, but the two values in ( ) are
not given by Dudeney. He states results for the 9 x 9 and 10 x 10, citing Lucas. Says
that a 6 x 6 version "is sold in the streets of London for a penny".
Hoffmann. 1893.
Chap. VI, pp. 272-273 & 286 = Hoffmann-Hordern, pp. 187-189, with photo.
No. 24: No two in a row. Eight queens. Photo on p. 188 shows Jeu des Sentinelles, by
Watilliaux, dated 1874-1895.
No. 25: The "Simple" Puzzle. Nine queens. Says a version was sold by Messrs.
Feltham, with a notched board but the pieces were allowed to move over the
gaps, so it was really a 9 x 9 board.
Chap. X, No. 18: The Treasure at Medinet, pp. 343-344 & 381 = Hoffmann-Hordern,
pp. 237-239. This is a solution of the eight queens problem, cut into four quadrants and
jumbled. The goal is to reconstruct the solution. Photo on p. 239 shows Jeu des
Manifestants, with box.
Hordern Collection, p. 94, and S&B, p. 37, show a version of this with same box,
but which divides the board into eight 2 x 4 rectangles.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 1: The
famous Italian pin puzzle. 6 queens puzzle. No solution.
Lucas. L'Arithmétique Amusante. 1895. Note IV: Section I: Les huit dames, pp. 210-220.
Asserts Bellavitis was the first to find all solutions. Discusses symmetries and shows
the 12 basic solutions. Correctly describes Jaenisch as obscure. Gives an easy solution
of Cretaine's problem which can be remembered as a trick. Shows there are six
solutions which can be superimposed with no overlap, i.e. six solutions using disjoint
sets of cells.
C. D. Locock, conductor. Chess Column. Knowledge 19 (Jan 1896) 23-24; (Feb 1896)
47-48; (May 1896) 119; (Jul 1896) 167-168. This series begins by saying most players
know there is a solution, "but, possibly, some may be surprised to learn that there are
ninety-two ways of performing the feat, ...." He then enumerates them. Second article
studies various properties of the solutions, particularly looking for examples where one
solution shifts to produce another one. Third article notes some readers' comments.
Fourth article is a long communication from W. J. Ashdown about the number of
distinct solutions, which he gets as 24 rather than the usual 12.
T. B. Sprague. Proc. Edinburgh Math. Soc. 17 (1898-9) 43-68. ??NYS -- cited by Ball,
MRE, 4th ed., 1905, as giving the 341 inequivalent solutions on the 11 x 11.
Benson. 1904. Pins and dots puzzle, p. 253. 6 queens problem, one solution.
Ball. MRE, 4th ed., 1905. The eight queens problem, pp. 114-120. Corrects some history by
citing MUS, 1st ed., 1901. Gives one instance of Glaisher's method -- going from 4 x 4
to 5 x 5 and its results going up to 8 x 8. Says the 92 inequivalent solutions on the
10 x 10 were given by Pein and the 341 inequivalent solutions on the 11 x 11 were
given by Sprague. The 5th ed., pp. 113-119 calls it "One of the classical problems
connected with a chess-board" and adds examples of solutions up to 21 x 21 due to
Mr. Derington.
Pearson. 1907. Part III, no. 59: Stray dots, pp. 59 & 130. Same as Hoffmann's Treasure at
Medinet.
Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY,
1909. The eight provinces, pp. 14-15 & 65. Same as Hoffmann's Treasure at Medinet.
A. C. White. Sam Loyd and His Chess Problems. 1913. Op. cit. in 1. P. 101 says Loyd
discovered that all solutions have a piece at d1 or equivalent.
Williams. Home Entertainments. 1914. A draughtboard puzzle, p. 115. "Arrange eight men
on a draughtboard in such a way that no two are upon the same line in any direction."
This is not well stated!! Gives one solution: 52468317 and says "Work out other
solutions for yourself."
Dudeney. AM. 1917. The guarded chessboard, pp. 95-96. Gives the number of ways of
SOURCES - page 113
placing n queens and the number of inequivalent ways. The values in ( ) are given by
Ball, but not by Dudeney.
n
ways
inequivalent ways
4
2
1
5
10
2
6
4
1
7
8
40 92
6 12
9
(352)
46
10
11
(724)
92 341
12
(1766)
13
(1346)
Ball. MRE, 9th ed., 1920. The eight queens problem, pp. 113-119. Omits references to Pein
and Sprague and adds the number of inequivalent solutions for the 12 x 12 and
13 x 13.
Blyth. Match-Stick Magic. 1921. No pairs allowed, p. 74. 6 queens problem.
Hummerston. Fun, Mirth & Mystery. 1924. No two in a line, p. 48. Chessboard. Place 'so
that no two are upon the same line in any direction along straight or diagonal lines?'
Gives one solution: 47531682, 'but there are hundreds of other ways'. You can let
someone place the first piece.
Rohrbough. Puzzle Craft. 1932. Houdini Puzzle, p. 17. 6 x 6 case.
Rohrbough. Brain Resters and Testers. c1935. Houdini Puzzle, p. 25. 6 x 6 problem. "-From New York World some years ago, credited to Harry Houdini." I have never seen
this attribution elsewhere.
Pál Révész. Mathematik auf dem Schachbrett. In: Endre Hódi, ed. Mathematisches Mosaik.
(As: Matematikai Érdekességek; Gondolat, Budapest, 1969.) Translated by Günther
Eisenreich. Urania-Verlag, Leipzig, 1977. Pp. 20-27. On p. 24, he says that all
solutions have 4 queens on white and 4 on black. He says that one can place at most 5
non-attacking queens on one colour.
Doubleday - 2. 1971. Too easy?, pp. 97-98. The two solutions on the 4 x 4 board are
disjoint.
Dean S. Clark & Oved Shisha. Proof without words: Inductive construction of an infinite
chessboard with maximal placement of nonattacking queens. MM 61:2 (1988) 98.
Consider a 5 x 5 board with queens in cells (1,1), (2,4), (3,2), (4,5), (5,3). 5 such
boards can be similarly placed within a 25 x 25 board viewed as a 5 x 5 array of 5 x
5 boards and this has no queens attacking. Repeating the inflationary process gives a
solution on the board of edge 53, then the board of edge 54, .... They cite their paper:
Invulnerable queens on an infinite chessboard; Annals of the NY Acad. of Sci.: Third
Intern. Conf. on Comb. Math.; to appear. ??NYS.
Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. Squares before your eyes, pp. 21 & 106.
Asks for solutions of the eight queens problem with no piece on either main diagonal.
Two of the 12 basic solutions have this, but one of these is the symmetric case, so there
are 12 solutions of this problem.
Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by
the author. See the discussion in 6.F. He finds the following numbers of solutions for
placing n queens, n = 1, 2, ..., 18.
1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 3 65596, 22 79184,
147 72512, 958 15104, 6660 90624.
5.I.2. COLOURING CHESSBOARD WITH NO REPEATS IN A LINE
New section. I know there is a general result that an n x n board can be n-coloured if
n satisfies some condition like n  1 or 5 (mod 6), but I don't recall any other old examples
of the problem.
Dudeney. Problem 50: A problem in mosaics. Tit-Bits 32 (11 Sep 1897) 439 & 33
(2 Oct 1897) 3. An 8 x 8 board with two adjacent corners omitted can be 8-coloured
with no two in a row, column or diagonal. = Anon. & Dudeney; A chat with the Puzzle
King; The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900)
89.
Dudeney. AM. 1917. Prob. 302: A problem in mosaics, pp. 90 & 215-216. The solution to
the previous problem is given and then it is asked to relay the tiles so that the omitted
squares are the (3,3) and (3,6) cells.
Hummerston. Fun, Mirth & Mystery. 1924. Q.E.D. -- The office boy problem, Puzzle no.
30, pp. 82 & 176. Wants to mark the cells of a 4 x 4 board with no two the same in
any 'straight line ..., either horizontally, vertically, or diagonally.' His answer is:
SOURCES - page 114
ABCD, CDEA, EABC, BCDA, which has no two the same on any short diagonal.
The problem uses coins of values: A, B, C, D, E = 12, 30, 120, 24, 6 and the object is
to maximize the total value of the arrangement. In fact, there are only two ways to 5colour the board and they are mirror images. Four colours are used three times and one
is used four times -- setting the value 120 on the latter cells gives the maximum value of
696.
5.J.
SQUARED SQUARES, ETC.
NOTE. Perfect means no two squares are the same size. Compound means there is a
squared subrectangle. Simple means not compound.
Dudeney. Puzzling Times at Solvamhall Castle: Lady Isabel's casket. London Mag. 7
(No. 42) (Jan 1902) 584 & 8 (No. 43) (Feb 1902) 56. = CP, prob. 40, pp. 67 &
191-193. Square into 12 unequal squares and a rectangle.
Max Dehn. Über die Zerlegung von Rechtecken in Rechtecke. Math. Annalen 57 (1903)
314-332. Long and technical. No examples. Shows sides must be parallel and
commensurable.
Loyd. The patch quilt puzzle. Cyclopedia, 1914, pp. 39 & 344. = MPSL1, prob. 76, pp. 73 &
147-148. c= SLAHP: Building a patchquilt, pp. 30 & 92. 13 x 13 into 11 squares, not
simple nor perfect. (Gardner, in 536, says this appeared in Loyd's "Our Puzzle
Magazine", issue 1 (1907), ??NYS.)
Loyd. The darktown patch quilt party. Cyclopedia, 1914, pp. 65 & 347. 12 x 12 into 11
squares, not simple nor perfect, in two ways.
P. J. Federico. Squaring rectangles and squares -- A historical review with annotated
bibliography. In: Graph Theory and Related Topics; ed. by J. A. Bondy & U. S. R.
Murty; Academic Press, NY, 1979, pp. 173-196. Pp. 189-190 give the background to
Moroń's work. Moroń later found the first example of Sprague but did not publish it.
Z. Moroń. O rozkładach prostokątów na kwadraty (In Polish) (On the dissection of a
rectangle into squares). Przegląd Matematyczno-Fizyczny (Warsaw) 3 (1925) 152-153.
Decomposes rectangles into 9 and 10 unequal squares. (Translation provided by A.
Mąkowski, 1p. Translation also available from M. Goldberg, ??NYS.)
M. Kraitchik. La Mathématique des Jeux, 1930, op. cit. in 4.A.2, p. 272. Gives Loyd's "Patch
quilt puzzle" solution and Lusin's opinion that there is no perfect solution.
A. Schoenflies. Einführung in der analytische Geometrie der Ebene und des Raumes. 2nd
ed., revised and extended by M. Dehn, Springer, Berlin, 1931. Appendix VI: Ungelöste
Probleme der Analytischen Geometrie, pp. 402-411. Same results as in Dehn's 1903
paper.
Michio Abe. On the problem to cover simply and without gap the inside of a square with a
finite number of squares which are all different from one another (in Japanese). Proc.
Phys.-Math. Soc. Japan 4 (1931) 359-366. ??NYS
Michio Abe. Same title (in English). Ibid. (3) 14 (1932) 385-387. Gives 191 x 195
rectangle into 11 squares. Shows there are squared rectangles arbitrarily close to
squares.
Alfred Stöhr. Über Zerlegung von Rechtecken in inkongruente Quadrate. Schr. Math. Inst.
und Inst. angew. Math. Univ. Berlin 4:5 (1939), Teubner, Leipzig, pp. 119-140.
??NYR. (This was his dissertation at the Univ. of Berlin.)
S. Chowla. The division of a rectangle into unequal squares. Math. Student 7 (1939) 69-70.
Reconstructs Moroń's 9 square decomposition.
Minutes of the 203rd Meeting of the Trinity Mathematical Society (Cambridge)
(13 Mar 1939). Minute Books, vol. III, pp. 244-246. Minutes of A. Stone's lecture:
"Squaring the Square". Announces Brooks's example with 39 elements, side 4639,
but containing a perfect subrectangle.
Minutes of the 204th Meeting of the Trinity Mathematical Society (Cambridge)
(24 Apr 1939). Minute Books, vol. III, p. 248. Announcement by C. A. B. Smith that
Tutte had found a perfect squared square with no perfect subrectangle.
R. Sprague. Recreation in Mathematics. Op. cit. in 4.A.1. 1963. The expanded foreword of
the English edition adds comments on Dudeney's "Lady Isabel's Casket", which led to
the following paper.
R. Sprague. Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate. Math.
Zeitschr. 45 (1939) 607-608. First perfect squared square -- 55 elements, side 4205.
SOURCES - page 115
R. Sprague. Zur Abschätzung der Mindestzahl inkongruenter Quadrate, die ein gegebenes
Rechteck ausfüllen. Math. Zeitschrift 46 (1940) 460-471. Tutte's 1979 commentary
says this shows every rectangle with commensurable sides can be dissected into unequal
squares.
A. H. Stone, proposer; M. Goldberg & W. T. Tutte, solvers. Problem E401. AMM 47:1
(Jan 1940) 48 & AMM 47:8 (Oct 1940) 570-572. Perfect squared square -- 28
elements, side 1015.
R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte. The dissection of rectangles into
squares. Duke Math. J. 7 (1940) 312-340. = Selected Papers of W. T. Tutte; Charles
Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 10-38, with commentary by
Tutte on pp. 1-9. Tutte's 1979 commentary says Smith was perplexed by the solution of
Dudeney's "Lady Isabel's Casket" -- see also his 1958 article.
A. H. Stone, proposer; Michael Goldberg, solver. Problem E476. AMM 48 (1941) 405
??NYS & 49 (1942) 198-199. An isosceles right triangle can be dissected into 6
similar figures, all of different sizes. Editorial notes say that Douglas and Starke found
a different solution and that one can replace 6 by any larger number, but it is not
known if 6 is the least such. Stone asks if there is any solution where the smaller
triangles have no common sides.
M. Kraitchik. Mathematical Recreations, op. cit. in 4.A.2, 1943. P. 198. Shows the
compound perfect squared square with 26 elements and side 608 from Brooks, et al.
C. J. Bouwkamp. On the construction of simple perfect squared squares. Konink. Neder.
Akad. van Wetensch. Proc. 50 (1947) 72-78 = Indag. Math. 9 (1947) 57-63. This
criticised the method of Brooks, Smith, Stone & Tutte, but was later retracted.
Brooks, Smith, Stone & Tutte. A simple perfect square. Konink. Neder. Akad. van
Wetensch. Proc. 50 (1947) 1300-1301. = Selected Papers of W. T. Tutte; Charles
Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 99-100, with commentary
by Tutte on p. 98. Bouwkamp had published several notes and was unable to make the
authors' 1940 method work. Here they clarify the situation and give an example. One
writer said they give details of Sprague's first example, but the example is not described
as being the same as in Sprague.
W. T. Tutte. The dissection of equilateral triangles into equilateral triangles. Proc. Camb.
Phil Soc. 44 (1948) 464-482. = Selected Papers of W. T. Tutte; Charles Babbage
Research Institute, St. Pierre, Manitoba, 1979; pp. 106-125, with commentary by Tutte
on pp. 101-105.
T. H. Willcocks, proposer and solver. Problem 7795. Fairy Chess Review 7:1 (Aug 1948) 97
& 106 (misnumberings for 5 & 14). Refers to prob. 7523 -- ??NYS. Finds compound
perfect squares of orders 27, 27, 28 and 24.
T. H. Willcocks. A note on some perfect squares. Canadian J. Math. 3 (1951) 304-308.
Describes the result in Fairy Chess Review prob. 7795.
T. H. Willcocks. Fairy Chess Review (Feb & Jun 1951). Prob. 8972. ??NYS -- cited and
described by G. P. Jelliss; Prob. 44 -- A double squaring, G&PJ 2 (No. 17) (Oct 1999)
318-319. Squares of edges 3, 5, 9, 11, 14, 19, 20, 24, 31, 33, 36, 39, 42 can be formed
into a 75 x 112 rectangle in two different ways. {These are reproduced, without
attribution, as Fig. 21, p. 33 of Joseph S. Madachy; Madachy's Mathematical
Recreations; Dover, 1979 (this is a corrected reprint of Mathematics on Vacation, 1966,
??NYS). The 1979 ed. has an errata slip inserted for p. 33 as the description of Fig. 21
was omitted in the text, but the erratum doesn't cite a source for the result.} The G&PJ
problem then poses a new problem from Willcocks involving 21 squares to be made
into a rectangle in two different ways -- it is not clear if these have to be the same shape.
M. Goldberg. The squaring of developable surfaces. SM 18 (1952) 17-24. Squares cylinder,
Möbius strip, cone.
W. T. Tutte. Squaring the square. Guest column for SA (Nov 1958). c= Gardner's 2nd Book,
pp. 186-209. The latter = Selected Papers of W. T. Tutte; Charles Babbage Research
Institute, St. Pierre, Manitoba, 1979; pp. 244-266, with a note by Tutte on p. 244, but
the references have been omitted. Historical account -- cites Dudeney as the original
inspiration of Smith.
R. L. Hutchings & J. D. Blake. Problems drive 1962. Eureka 25 (Oct 1962) 20-21 & 34-35.
Prob. G. Assemble squares of sides 2, 5, 7, 9, 16, 25, 28, 33, 36 into a rectangle. The
rectangle is 69 x 61 and is not either of Moroń's examples.
W. T. Tutte. The quest of the perfect square. AMM 72:2, part II (Feb 1965) 29-35.
= Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre,
SOURCES - page 116
Manitoba, 1979; pp. 432-438, with brief commentary by Tutte on p. 431. General
survey, updating his 1958 survey.
Blanche Descartes [pseud. of Cedric A. B. Smith]. Division of a square into rectangles.
Eureka 34 (1971) 31-35. Surveys some history and Stone's dissection of an isosceles
right triangle into 6 others of different sizes (see above). Tutte has a dissection of an
equilateral triangle into 15 equilateral triangles -- but some of the pieces must have the
same area so we consider up and down pointing triangles as + and - areas and then all
the areas are different. Author then considers dissecting a square into incongruent but
equiareal rectangles. He finds it can be done in n pieces for any n  7.
A. J. W. Duijvestijn. Simple perfect squared square of lowest order. J. Combinatorial Thy. B
25 (1978) 240-243. Finds a perfect square of minimal order 21.
A. J. W. Duijvestin, P. J. Federico & P. Leeuw. Compound perfect squares. AMM 89 (1982)
15-32. Shows Willcocks' example has the smallest order for a compound perfect square
and is the only example of its order, 24.
5.J.1. MRS PERKINS'S QUILT
This is the problem of cutting a square into smaller squares.
Loyd. Cyclopedia, 1914, pp. 248 & 372, 307 & 380. Cut 3 x 3 into 6 squares: 2 x 2 and
5 1 x 1.
Dudeney. AM. 1917. Prob. 173: Mrs Perkins's quilt, pp. 47 & 180. Same as Loyd's "Patch
quilt puzzle" in 5.J.
Dudeney. PCP. 1932. Prob. 117: Square of Squares, pp. 53 & 148-149. = 536, prob. 343,
pp. 120 & 324-325. c= "Mrs Perkins's quilt".
N. J. Fine & I. Niven, proposers; F. Herzog, solver. Problem E724 -- Admissible Numbers.
AMM 53 (1946) 271 & 54 (1947) 41-42. Cubical version.
J. H. Conway. Mrs Perkins's quilt. Proc. Camb. Phil. Soc. 60 (1964) 363-368.
G. B. Trustrum. Mrs Perkins's quilt. Ibid. 61 (1965) 7-11.
Ripley's Puzzles and Games. 1966. Pp. 16-17, item 7. "Can you divide a square into 6
perfect squares?" Answer as in Loyd.
Nick Lord. Note 72.11: Subdividing hypercubes. MG 72 (No. 459) (Mar 1988) 47-48.
Gives an upper bound for impossible numbers in d dimensions.
David Tall. To prove or not to prove. Mathematics Review 1:3 (Jan 1991) 29-32. Tall
regularly uses the question as an exercise in problem solving. About ten years earlier, a
14 year old girl pointed out that the problem doesn't clearly rule out rejoining pieces.
E.g. by cutting along the diagonals and rejoining, one can make two squares.
5.J.2. CUBING THE CUBE
S. Chowla. Problem 1779. Math. Student 7 (1939) 80. (Solution given in Brooks, et al.,
Duke Math. J., op. cit. in 5.J, section 10.4, but they give no reference to a solution in
Math. Student.)
5.J.3. TILING A SQUARE OF SIDE 70 WITH SQUARES OF SIDES
1, 2, ..., 24
J. R. Bitner. Use of Macros in Backtrack Programming. M.Sc. Thesis, ref.
UIUCDCS-R-74-687, Univ. of Illinois, Urbana-Champaign, 1974, ??NYS. Shows such
a tiling is impossible.
5.K. DERANGEMENTS
Let D(n) = the number of derangements of n things, i.e. permutations leaving no
point fixed.
Eberhard Knobloch. Euler and the history of a problem in probability theory. Gaņita-Bhāratī
[NOTE: ņ denotes an n with an underdot] (Bull. Ind. Soc. Hist. Math.) 6 (1984) 1-12.
Discusses the history, noting that many 19C authors were unaware of Euler's work.
There is some ambiguity in his descriptions due to early confusion of n as the number
of cards and n as the number of the card on which a match first occurs. Describes
SOURCES - page 117
numerous others who worked on the problem up to about 1900: De Moivre, Waring,
Lambert, Laplace, Cantor, etc.
Pierre Rémond de Montmort. Essai d'analyse sur les jeux de hazards. (1708); Seconde
edition revue & augmentee de plusieurs lettres, (Quillau, Paris, 1713 (reprinted by
Chelsea, NY, 1980)); 2nd issue, Jombert & Quillau, 1714. Problèmes divers sur le jeu
du trieze, pp. 54-64. In the original game, one has a deck of 52 cards and counts
1, 2, ..., 13 as one turns over the cards. If a card of rank i occurs at the i-th count,
then the player wins. In general, one simplifies by assuming there are n distinct cards
numbered 1, ..., n and one counts 1, ..., n. One can ask for the probability of winning
at some time and of winning at the k-th draw. In 1708, Montmort already gives tables
of the number of permutations of n cards such that one wins on the k-th draw, for
n = 1, ..., 6. He gives various recurrences and the series expression for the probability
and (more or less) finds its limit. In the 2nd ed., he gives a proof of the series
expression, due to Nicholas Bernoulli, and John Bernoulli says he has found it also.
Nicholas' solution covers the general case with repeated cards. [See: F. N. David;
Games, Gods and Gambling; Griffin, London, 1962, pp. 144-146 & 157.] (Comtet and
David say it is in the 1708 ed. I have seen it on pp. 54-64 of an edition which is
uncertain, but probably 1708, ??NX. Knobloch cites 1713, pp. 130-143, but adds that
Montmort gave the results without proofs in the 1708 ed. and includes several letters
from and to John I and Nicholas I Bernoulli in the 1713 ed., pp. 290-324, and mentions
the problem in his Preface -- ??NYS.)
Abraham de Moivre. The Doctrine of Chances: or, A Method of Calculating the Probability
of Events in Play. W. Pearson for the Author, London, 1718. Prob. XXV, pp. 59-63.
(= 2nd ed, H. Woodfall for the Author, London, 1738. Prob. XXXIV, pp. 95-98.)
States and demonstrates the formula for finding the probability of p items to be correct
and q items to be incorrect out of n items. One of his examples is the probability of
six items being deranged being 53/144.
L. Euler. Calcul de la probabilité dans le jeu de rencontre. Mémoires de l'Académie des
Sciences de Berlin (7) (1751(1753)) 255-270. = Opera Omnia (1) 7 (1923) 11-25.
Obtains the series for the probability and notes it approaches 1/e.
L. Euler. Fragmenta ex Adversariis Mathematicis Deprompta. MS of 1750-1755.
Pp. 287-288: Problema de permutationibus. First published in Opera Omnia (1) 7
(1923) 542-545. Obtains alternating series for D(n).
Ozanam-Montucla. 1778. Prob. 5, 1778: 125-126; 1803: 123-124; 1814: 108-109;
1840: omitted. Describes Jeu du Treize, where a person takes a whole deck and turns
up the cards, counting 1, 2, ..., 13 as he goes. He wins if a card of rank i appears at
the i-th count. Montucla's description is brief and indicates there are several variations
of the game. Hutton gives a lengthier description of one version. Cites Montmort for
the probability of winning as .632..
L. Euler. Solutio quaestionis curiosae ex doctrina combinationum. (Mem. Acad. Sci. St.
Pétersbourg 3 (1809/10(1811)) 57-64.) = Opera Omnia (1) 7 (1923) 435-440. (This
was presented to the Acad. on 18 Oct 1779.) Shows D(n) = (n-1) [D(n-1) + D(n-2)]
and D(n) = nD(n-1) + (-1)n.
Ball. MRE. 1st ed., 1892. Pp. 106-107: The mousetrap and Treize. In the first, one puts out
n cards in a circle and counts out. If the count k occurs on the k-th card, the card is
removed and one starts again. Says Cayley and Steen have studied this. It looks a bit
like a derangement question.
Bill Severn. Packs of Fun. 101 Unusual Things to Do with Playing Cards and To Know
about Them. David McKay, NY, 1967. P. 24: Games for One: Up and down. Using a
deck of 52 cards, count through 1, 2, ..., 13 four times. You lose if a card of rank i
appears when you count i, i.e. you win if the cards are a generalized derangement.
Though a natural extension of the problem, I can't recall seeing it treated, perhaps
because it seems to get very messy. However, a quick investigation reveals that the
probability of such a generalized derangement should approach e-4.
Brian R. Stonebridge. Derangements of a multiset. Bull. Inst. Math. Appl. 28:3 (Mar 1992)
47-49. Gets a reasonable extension to multisets, i.e. sets with repeated elements.
5.K.1.
DERANGED BOXES OF A, B AND A & B
Three boxes contain A or B or A & B, but they have been shifted about so each is in
SOURCES - page 118
one of the other boxes. You can look at one item from one box to determine what is in all of
them. This is just added and is certainly older than the examples below.
Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 84:
Marble garble, pp. 40 & 110. Black and white marbles.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 26:
Mexican jumping beans, pp. 40-41 & 96. Red and black beans in matchboxes. The
problem continues with a Bertrand box paradox -- see 8.H.1.
Doubleday - 3. 1972. Open the box, pp. 147-148. Black and white marbles.
5.K.2.
OTHER LOGIC PUZZLES BASED ON DERANGEMENTS
These typically involve a butcher, a baker and a brewer whose surnames are Butcher,
Baker and Brewer, but no one has the profession of his name. I generally only state the
beginning of the problem.
New section -- there must be older examples. Gardner, in an article: My ten favorite
brainteasers in Games (collected in Games Big Book of Games, 1984, pp. 130-131) says this
is one of his favorite problems. ??locate
I now see these lead to Latin rectangles, cf Section 5.I.
R. Turner, proposer: The sons of the dons; Eureka 2 (May 1939) 9-10. K. Tweedie, solver:
On the problem of the sons of the dons. Eureka 4 (May 1940) 21-23. Six dons, in
analysis, geometry, algebra, dynamics, physics and astronomy, each have a son who
studies one of these subjects, but none studies the subject of his father. Several further
restrictions, e.g., there are no two students who each study the subject of the other's
father.
M. Adams. Puzzle Book. 1939. Prob. B.91: Easter bonnet, pp. 80 & 107. Women named
Green, Black, Brown and White with 4 colours of hats and 4 colours of dresses, but
name, hat and dress are always distinct.
J. B. Parker. Round the table. Eureka 5 (Jan 1941) 20-21 & 6 (May 1941) 11. Seven men,
whose names are colours, with ties, socks and cars, being coloured with three of the
names of other men and all colours used for each item, sitting at a table with eight
places.
Anonymous. The umbrella problem. Eureka 9 (Apr 1947) 22 & 10 (Mar 1948) 25. Six
men 'of negligible honesty' each go away with another's umbrella.
Jonathan Always. Puzzles to Puzzle You. Tandem, London, 1965. No. 30: Something about
ties, pp. 16 & 74-75. Black, Green and Brown are wearing ties, but none has the colour
of his name, remarked the green tie wearer to Mr. Black.
David Singmaster. The deranged secretary. If a secretary puts n letters all in wrong
envelopes, how many envelopes must one open before one knows what is each of the
unopened envelopes?
Problem proposal and solution 71.B. MG 71 (No. 455) (Mar 1987) 65 & 71 (No. 457)
(Oct 1987) 238-239.
Open question. The Weekend Telegraph (11 Jun 1988) XV & (18 Jun 1988) XV.
5.K.3.
CAYLEY'S MOUSETRAP
This is a solitaire (= patience) game developed by Cayley, based on Treize. Take a deck
of cards, numbered 1, 2, ..., n, and shuffle them. Count through them. If a card does not
match its count, put it on bottom and continue. If it matches, set it aside and start counting
again from 1. One wins if all cards are set aside. In this case, pick up the deck and start a
new game.
T. W. O. Richards, proposer; Richard I. Hess, solver. Prob. 1828. CM 19 (1993) 78 & 20
(1994) 77-78. Asks whether there is any arrangement which allows three or more
consecutive wins. No theoretical solution. Searching finds one solution for n = 6 and
n = 8 and 8 solutions for n = 9.
5.L. MÉNAGE PROBLEM
How many ways can n couples be seated, alternating sexes, with no couples adjacent?
SOURCES - page 119
A. Cayley. On a problem of arrangements. Proc. Roy. Soc. Edin. 9 (1878) 338-342. Problem
raised by Tait. Uses inclusion/exclusion to get a closed sum.
T. Muir. On Professor Tait's problem of arrangements. Ibid., 382-387. Uses determinants to
get a simple n-term recurrence.
A. Cayley. Note on Mr. Muir's solution of a problem of arrangement. Ibid., 388-391. Uses
generating function to simplify to a usable form.
T. Muir. Additional note on a problem of arrangement. Ibid., 11 (1882) 187-190. Obtains
Laisant's 2nd order and 4th order recurrences.
É. Lucas. Théorie des Nombres. Gauthier-Villars, Paris, 1891; reprinted by Blanchard, Paris,
1958. Section 123, example II, p. 215 & Note III, pp. 491-495. Lucas appears not to
have known of the work of Cayley and Muir. He describes Laisant's results. The 2nd
order, non-homogeneous recurrence, on pp. 494-495, is attributed to Moreau.
C. Laisant. Sur deux problèmes de permutations. Bull. Soc. Math. de France 19 (1890-91)
105-108. General approach to problems of restricted occupancy. His work yields a 2nd
order non-homogeneous recurrence and homogeneous 3rd and 4th order recurrences.
He cites Lucas, but says Moreau's work is unpublished.
H. M. Taylor. A problem on arrangements. Messenger of Math. 32 (1903) 60-63. Gets
almost to Muir & Laisant's 4th order recurrence.
J. Touchard. Sur un problème de permutations. C. R. Acad. Sci. Paris 198 (1934) 631-633.
Solution in terms of a complicated integral. States the explicit summation.
I. Kaplansky. Solution of the "problème des ménages". Bull. Amer. Math. Soc. 49 (1943)
784-785. Obtains the now usual explicit summation.
I. Kaplansky & J. Riordan. The problème de ménages. SM 12 (1946) 113-124. Gives the
history and a uniform approach.
J. Touchard. Permutations discordant with two given permutations. SM 19 (1953) 109-119.
Says he prepared a 65pp MS developing the results announced in 1934 and
rediscovered in Kaplansky and in Kaplansky & Riordan. Proves Kaplansky's lemma on
selections by finding the generating functions which involve Chebyshev polynomials.
Obtains the explicit summation, as done by Kaplansky. Extends to more general
problems.
M. Wyman & L. Moser. On the 'problème des ménages'. Canadian J. Math. 10 (1958)
468-480. Analytic study. Updates the history -- 26 references. Gives table of values
for n = 0 (1) 65.
Jacques Dutka. On the 'Problème des ménages'. Math. Intell. 8:3 (1986) 18-25 & 33.
Thorough survey & history -- 25 references.
Kenneth P. Bogart & Peter G. Doyle. Non-sexist solution of the ménage problem. AMM 93
(1986) 514-518. 14 references.
5.M. SIX PEOPLE AT A PARTY -- RAMSEY THEORY
In a group of six people, there is a triple who all know each other or there is a triple who
are all strangers. I.e., the Ramsey number R(3,3) = 6. I will not go into the more complex
aspects of this -- see Graham & Spencer for a survey.
P. Erdös & G. Szekeres. A combinatorial problem in geometry. Compositio Math. 2 (1935)
463-470. [= Paul Erdös; The Art of Counting; Ed. by Joel Spencer, MIT Press, 1973,
pp. 5-12.] They prove that if n  BC(a+b-2, a-1), then any two-colouring of Ka
contains a monochromatic Ka or Kb.
William Lowell Putnam Examination, 1953, part I, problem 2. In: L. E. Bush; The William
Lowell Putnam Mathematical Competition; AMM 60 (1953) 539-542. Reprinted in:
A. M. Gleason, R. E. Greenwood & L. M. Kelly; The William Lowell Putnam
Mathematical Competition Problems and Solutions -- 1938-1964; MAA, 1980; pp. 38
& 365-366. The classic six people at a party problem.
R. E. Greenwood & A. M. Gleason. Combinatorial relations and chromatic graphs. Canadian
J. Math. 7 (1955) 1-7. Considers n = n(a,b,...) such that a two colouring of Kn
contains a Ka of the first colour or a Kb of the second colour or .... Thus n(3,3) = 6.
They find the bound and many other results of Erdös & Szekeres.
C. W. Bostwick, proposer; John Rainwater & J. D. Baum, solvers. Problem E1321 -- A
gathering of six people. AMM 65 (1958) 446 & 66 (1959) 141-142.
Gamow & Stern. 1958. Diagonal strings. Pp. 93-95.
SOURCES - page 120
G. J. Simmons. The Game of Sim. JRM 2 (1969) 66.
M. Gardner. SA (Jan 1973) c= Knotted, chap. 9. Exposits Sim. Reports Simmons' result that
it is second person (determined after his 1969 article above). The Addendum in
Knotted reports that several people have shown that Sim on five points is a draw.
Numerous references.
Ronald L. Graham & Joel H. Spencer. Ramsey theory. SA 263:1 (Jul 1990) 80-85. Popular
survey of Ramsey theory beginning from Ramsey and Erdös & Szekeres.
5.N. JEEP OR EXPLORER'S PROBLEM
See Ball for some general discussion and notation.
Alcuin. 9C. Prob. 52: Propositio de homine patrefamilias. Wants to get 90 measures over a
distance of 30 leagues. He is trying to get the most to the other side, so this is different
than the 20C versions. Solution is confusing, but Folkerts rectifies a misprint and this
makes it less confusing. Alcuin's camels only eat when loaded!! (Or else they perish
when their carrying is done??) The camel take a load to a point 20 leagues away and
leaves 10 there, then returns. This results in getting 20 to the destination.
The optimum solution is for the camel to make two return trips and a single trip
to 10 leucas, so he will have consumed 30 measures and he has 60 measures to
carry on. He now makes one return and a single trip of another 15 leucas, so he will
have consumed another 30 measures, leaving 30 to carry on the last 5 leucas, so he
reaches home with 25 measures.
Pacioli. De Viribus. c1500. Probs. 49-52. Agostini only describes Prob. 49 in some detail.
Ff. 94r - 95v. XLIX. (Capitolo) de doi aportare pome ch' piu navanza (Of two ways to
transport as many apples as possible). = Peirani 134-135. One has 90 apples to
transport 30 miles from Borgo [San Sepolcro] to Perosia [Perugia], but one eats
one apple per mile and one can carry at most 30 apples. He carries 30 apples 20
miles and leaves 10 there and returns, without eating on the return trip! (So this
= Alcuin.) Pacioli continues and gives the optimum solution!
F. 95v. L. C(apitolo). de .3. navi per .30. gabelle 90. mesure (Of three ships holding 90
measures, passing 30 customs points). Each ship has to pay one measure at each
customs point. Mathematically the same as the previous.
F. 96r. LI. C(apitolo). de portar .100. perle .10. miglia lontano 10. per volta et ogni
miglio lascia 1a (To carry 100 pearls 10 miles, 10 at a time, leaving one every
mile). = Peirani 136-137. Takes them 2 miles in ten trips, giving 80 there. Then
takes them to the destination in 8 trips, getting 16 to the destination.
Ff. 96v - 97r. LII. C(apitolo). el medesimo con piu avanzo per altro modo (The same
with more carried by another method). Continues the previous problem and takes
them 5 miles in ten trips, giving 50 there. Then takes them to the destination in 5
trips, getting 25 to the destination.
[This is optimal for a single stop -- if one makes the stop at distance a,
then one gets a(10-a) to the destination. One can make more stops, but this is
restricted by the fact that pearls cannot be divided. Assuming that the amount of
pearls accumulated at each depot is a multiple of ten, one can get 28 to the
destination by using depots at 2 and 7 or 5 and 7. One can get 27 to the
destination with depots at 4 and 9 or 5 and 9. These are all the ways one can
put in two depots with integral multiples of 10 at each depot and none of these
can be extended to three such depots. If the material being transported was a
continuous material like grain, then I think the optimal method is to first move 1
mile to get 90 there, then move another 10/9 to get 80 there, then another 10/8
to get 70 there, ..., continuing until we get 40 at 8.4563..., and then make four
trips to the destination. This gets 33.8254 to the destination. Is this the best
method??]
Cardan. Practica Arithmetice. 1539. Chap. 66, section 57, ff. EE.vi.v - EE.vii.v
(pp. 152-153). Complicated problem involving carrying food and material up the
Tower of Babel! Tower is assumed 36 miles high and seems to require 15625 porters.
Mittenzwey. 1880. Prob. 135, pp. 28-29; 1895?: 153, p. 32; 1917: 153, pp. 29. If eight
porters can carry eight full loads from A to B in an hour, how long will it take four
porters? The obvious answer is two hours, but he observes that the porters have to
return from B to A and it will take three hours. [Probably a little less as they should
SOURCES - page 121
return in less time than they go.]
Pearson. 1907. Part II, pp. 139 & 216. Two explorers who can carry food for 12 days. (No
depots, i.e. form A of Ball, below.)
Loyd. A dash for the South Pole. Ladies' Home Journal (15 Dec 1910). ??NYS -- source?? - WS??
Ball. MRE, 5th ed., 1911. Exploration problems, pp. 23-24. He distinguishes two forms of
the problem, with n explorers who can carry food for d days.
A. Without depots, they can get one man nd/(n+1) days into the desert and
back.
B. With depots permitted, they can get a man d/2 (1/1 + 1/2 + ... + 1/n) into the
desert and back. This is the more common form.
Dudeney. Problem 744: Exploring the desert. Strand Mag. (1925). ??NX. (??= MP 49)
Dudeney. MP. 1926. Prob. 49: Exploring the desert, pp. 21 & 111 (= 536, prob. 76, pp. 22 &
240). A version of Ball's form A, with n = 9, d = 10, but replacing days by stages of
length 40 miles.
Abraham. 1933. Prob. 34 -- The explorers, pp. 13 & 25 (9-10 & 112). 4 explorers, each
carrying food for 5 days. Mentions general case. This is Ball's form A.
Haldeman-Julius. 1937. No. 10: The four explorers, pp. 4 & 21. Ball's form A, with n = 4,
d = 5.
Olaf Helmer. Problem in logistics: The Jeep problem. Project Rand Report RA-15015
(1 Dec 1946) 7pp.
N. J. Fine. The jeep problem. AMM 54 (1947) 24-31.
C. G. Phipps. The jeep problem: a more general solution. AMM 54 (1947) 458-462.
G. G. Alway. Note 2707: Crossing the desert. MG 41 (No. 337) (1957) 209. If a jeep can
carry enough fuel to get halfway across, how much fuel is needed to get across? For a
desert of width 2, this leads to the series 1 + 1/3 + 1/5 + 1/7 + .... See Lehmann and
Pyle below.
G. C. S[hephard, ed.] The problems drive. Eureka 11 (Jan 1949) 10-11 & 30. No. 2. Four
explorers, starting from a supply base. Each can carry food for 100 miles and goes 25
miles per day. Two men do the returning to base and bringing out more supplies. the
third man does ferrying to the fourth man. How far can the fourth man get into the
desert and return? Answer is 100 miles. Ball's form B would give 104 1/6.
Gamow & Stern. 1958. Refueling. Pp. 114-115.
Pyle, I. C. The explorer's problem. Eureka 21 (Oct 1958) 5-7. Considers a lorry whose load
of fuel takes it a distance which we assume as the unit. What is the widest desert one
can cross? And how do you do it? This is similar to Alway, above. He starts at the far
side and sees you have to have a load at distance 1 from the far side, then two loads at
distance 1 + 1/3 from the far side, then three loads at 1 + 1/3 + 1/5, .... This
diverges, so any width can be crossed. Does examples with given widths of 2, 3 and 4
units. Editor notes that he is not convinced the method is optimal.
Martin Gardner, SA (May & June 1959) c= 2nd Book, chap. 14, prob. 1. (The book gives
extensive references which were not in SA.)
R. L. Goodstein. Letter: Explorer's problem. Eureka 22 (Oct 1959) 23. Says Alway shows
that Pyle's method is optimal. Editor notes Gardner's article and that Eureka was cited
in the solutions in Jun.
David Gale. The jeep once more or jeeper by the dozen & Correction to "The Jeep once
more or jeeper by the dozen". AMM 77:5 (May 1970) 493-501 & 78:6 (Jun-Jul 1971)
644-645. Gives an elaborate approach via a formula of Banach for path lengths in one
dimension. This formally proves that the various methods used are actually optimal and
that a continuous string of depots cannot help, etc. Notes that the cost for a round trip is
only slightly more than for a one-way trip -- but the Correction points out that this is
wrong and indeed the round trip is nearly four times as expensive as a one-way trip.
Considers sending several jeeps. Says he hasn't been able to do the round trip problem
when there is fuel on both sides of the desert. Comments on use of dynamic
programming, noting that R. E. Bellman [Dynamic Programming; Princeton Univ.
Press, 1955, p. 103, ex. 54-55] gives the problem as exercises without solution and that
he cannot see how to do it!
Birtwistle. Math. Puzzles & Perplexities. 1971.
The expedition, pp. 124-125, 183 & 194. Ball's form A, first with n = 5, d = 6, then
in general.
Second expedition, pp. 125-126. Ball's form B, done in general.
SOURCES - page 122
Third expedition, pp. 126, 183-184 & 194. Three men want to cross a 180 mile wide
desert. They can travel 20 miles per day and can carry food for six days, which
can be stored at depots. Minimize the total distance travelled. Solution seems
erroneous to me.
A. K. Austin. Jeep trips and card stacks. MTg 58 (1972) 24-25. There are n flags located at
distances a1, a1 + a2, a1 + a2 + a3, .... Jeep has to begin at the origin, go to the first
flag, return to the origin, go to the second flag, return, .... He can unload and load fuel
at the flags. Can he do this with F fuel? Author shows this is equivalent to
successfully stacking cards over a cliff with successive overhangs being a1, a2, a3, ....
Doubleday - 3. 1972. Traveller's Tale, pp. 63-64. d = 8 and we want one man to get across
the desert of width 12. How many porters, who return to base, are needed? The
solution implies that no depots are used. Reasoning as in Ball's case A, we see that n
men can support one man crossing a desert of width 2nd/(n+1). If depots are permitted,
this is essentially the jeep problem and n men can support a man getting across a desert
of width d [1 + 1/3 + 1/5 + ... + 1/(2n-1)]
Johannes Lehmann. Kurzweil durch Mathe. Urania Verlag, Leipzig, 1980. No. 13, pp. 27 &
129. d = 4 and we want to get a man across a desert of width 6. Similar to
Doubleday - 3.
Pierre Berloquin. [Le Jardin du Sphinx. Dunod, Paris, 1981.] Translated by Charles Scribner
Jr as: The Garden of the Sphinx. Scribner's, NY, 1985.
Prob. 1: Water in the desert, pp. 3 & 85.
Prob. 40: Less water in the desert, pp. 26 & 111.
Prob. 80: Beyond thirst, pp. 48 & 140.
Prob. 141: The barrier of thirst, pp. 79 & 181.
Prob. 150: No holds barred, pp. 82 & 150.
In all of these, d = 5 and we want to get a man across a desert of width 4, and
sometimes back, which is slightly different than the problem of getting to the maximum
distance and back.
Prob. 1 is Ball's form A, with n = 4 men, using 20 days' water.
Prob. 40 is Ball's form B, but using only whole day trips, using 14 days' water.
Prob. 80 is Ball's form B, optimized for width 4, using 11½ days' water.
Prob. 141 uses depots and bearers who don't return, as in Alcuin?? You can get
one man, who is the only one to return, a distance d (1/2 + 1/3 + ... + 1/(n+1)) into the
desert this way. He gives the optimum form for width 4, using 9½ days' water.
Prob. 150 is like prob. 141, except that no one returns! You can get him
d (1 + 1/2 + ... + 1/n) into the desert this way. The optimum here uses 4 days' water.
D. R. Westbrook. Note 74.7: The desert fox, a variation of the jeep problem. MG 74
(No. 467) (1990) 49-50. A more complex version, posed by A. K. Dewdney in SA
(Jan 1987), is solved here.
Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. Round-trip, pp. 96-97 & 140. Plane
wants to circle the earth, but can only carry fuel to go half-way. Other planes can
accompany and transfer fuel, but must return to base.
Dylan Gow. Flyaway. MS 25:3 (1992/3) 84-86. Considers the standard problem without
return as in Alway, Pyle and Lehmann -- but finds a non-optimal solution.
Wolfram Hinderer. Optimal crossing of a desert. MS 26:4 (1993/4) 100-102. Finds optimal
solutions for Gow's problem and for the case with return -- i.e. Ball's B. Also considers
use of extra jeeps that do not return, i.e. Berloquin's 141 & 150. Notes that extra jeeps
that must return to base do not change the distance that one jeep can reach. [But it
changes the time required.]
Harold Boas. Letter: Crossing deserts. MS 26:4 (1993/4) 122. Notes the problem has a long
history and cites Fine, Phipps, Gale (and correction), Alway.
David Singmaster. Letter: Crossing deserts. MS 27:3 (1994/5) 63. Points out that the
history is far older and sketches the history given above.
Günter Rote & Guochuan Zhang. Optimal Logistics for Expeditions: The Jeep Problem with
Complete Refilling. Karl-Franzens-Universität Graz & Technische Universität Graz.
Bericht 71 (24 Jun 1996). This deals with a variant. "We have n cans of fuel on the
edge of a desert and a jeep with an empty tank whose capacity is just one can. The jeep
can carry one can in addition to the fuel in its tank. Moreover, when a can is opened,
the fuel must immediately be filled into the jeep's tank. The goal is to find the farthest
point in the desert which the jeep can reach by consuming the n cans of fuel. Derick
Wood [1984] treated this problem similarly to the classical problem and gave the first
SOURCES - page 123
solution. Ute and Wilfried Brauer [1989] presented a new strategy and got a better
solution than Wood's. They also conjectured that their solution was optimal for
infinitely many values of n. We give an algorithm which produces a better solution
than Brauers' for all n > 6, and we use a linear programming formulation to derive an
upper bound which shows that our solution is optimal." 14 references, several not given
above.
5.O. TAIT'S COUNTER PUZZLE: BBBBWWWW TO WBWBWBWB
See S&B 125.
The rules are that one can move two counters as an ordered pair, e.g. from
BBBBWWWW to BBB..WWWBW, but not to BBB..WWWWB -- except in Lucas (1895)
and AM prob. 237, where such reversal must be done. Also, moving to BBB..WWW.BW is
sometimes explicitly prohibited, but it is not always clear just where one can move to. It is
also not always specified where the blank spaces are at the beginning and end positions.
Gardner, 1961, requires that the two counters must be BW or WB.
Barbeau, 1995, notes that moving to BWBWBWBW is a different problem, requiring
an extra move. I had not noticed this difference before -- indeed I previously had it the wrong
way round in the heading of this section. I must check to see if this occurs earlier. See
Achugbue & Chin, 1979-80, for this version.
Genjun Nakane (= Hōjiku Nakane). Kanja-otogi-soshi (Book of amusing problems for the
entertainment of thinkers). 1743. ??NYS. (See: T. Hayashi; Tait's problem with
counters in the Japanese mathematics; Bibl. Mathem. (3) 6 (1905) 323, for this and
other Japanese references of 1844 and 1879, ??NYS.)
P. G. Tait. Listing's Topologie. Philosophical Mag. (Ser. 5) 17 (No. 103) (Jan 1884) 30-46 &
plate opp. p. 80. Section 12, pp. 39-40. He says he recently saw it being played on a
train.
George Hope Verney (= Lloyd-Verney). Chess eccentricities. Longmans, 1885. P. 193: The
pawn puzzle. ??NX With 4 & 4.
Lucas. Amusements par les jetons. La Nature 15 (1887, 2nd sem.) 10-11. ??NYS -- cited by
Ahrens, title obtained from Harkin. Probably c= the material in RM3, below.
Ball. MRE, 1st ed., 1892, pp. 48-49.
Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles No. XIV: The eight-card
puzzle, pp. 14-15. Uses cards: BRBRBRBR and asks to bring the colours together,
explicitly requiring the moved cards to be placed in contact with the unmoved cards.
Hoffmann. 1893. Chap. VI, pp. 270-271 & 284-285 = Hoffmann-Hordern, pp. 184-186, with
photo.
No. 19: The "Four and Four" puzzle. Photo on p. 184 shows a version named Monkey
Puzzle advertising Brooke's Soap to go from BBBBBWWWW.. to
..WBWBWBWB .
No. 20: The "Five and Five" puzzle.
No. 21: The "Six and Six" puzzle.
Lucas. RM3. 1893. Amusements par les jetons, pp. 145-151. He gives Delannoy's general
solution for n of each colour in n moves. Remarks that one can reverse the moved
pair.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 11:
The Egyptian disc puzzle. 4 & 4. "Two discs adjoining each other to be moved at a
time; no gaps to be left in the line." -- this seems to prevent one from making any
moves at all!! No solution.
Lucas. L'Arithmétique Amusante. 1895. Pp. 84-108.
Prob. XXI - XXIV and Méthode générale, pp. 84-97. Gives solution for 4, 5, 6, 7 and
the general solution for n & n in n moves due to Delannoy.
Rouges et noires, avec interversion, prob. XXV - XXVIII and Méthode générale, pp.
97-108. Interversion means that the two pieces being moved are reversed or
turned over, e.g. from BBBBWWWW to BBB..WWWWB, but not to
BBB..WWWBW. Gives solutions for 4, 5, 6, 7, 8 pairs and in general in n
moves, but he ends with a gap, e.g. ....BB..BB and it takes an extra move to
close up the gap.
Ball. MRE, 3rd ed., 1896, pp. 65-66. Cites Delannoy's solution as being in La Nature
(Jun 1887) 10. ??NYS.
SOURCES - page 124
Ahrens. MUS I. 1910. Pp. 14-15 & 19-25. Cites Tait and gives Delannoy's general solution,
from Lucas.
Ball. MRE, 5th ed., 1911, pp. 75-77. Adds a citation to Hayashi, but incorrectly gives the
date as 1896.
Loyd. Cyclopedia. 1914. After dinner tricks, pp. 41 & 344. 4 & 4.
Williams. Home Entertainments. 1914. The eight counters puzzle, pp. 116-117. Standard
version, but with black and white reversed, in four moves. Says the moved counters
must be placed in line with and touching the others.
Dudeney. AM. 1917.
Prob. 236: The hat puzzle, pp. 67 & 196-197. BWBWBWBWBW.. to have the Bs
and Ws together and two blanks at an end. Uses 5 moves to get to
..WWWWWBBBBB.
Prob. 237: Boys and girls, pp. 67-68 & 197. ..BWBWBWBW to have the Bs and Ws
together with two blanks at an end, but pairs must be reversed as they are moved.
Solution in 5 moves to WWWWBBBB... = Putnam, no. 2. Cf Lucas, 1895.
Blyth. Match-Stick Magic. 1921. Transferring in twos, pp. 80-81. WBWBWBWB.. to
..BBBBWWWW in four moves.
King. Best 100. 1927. No. 66, pp. 27 & 55. = Foulsham's, no. 9, pp. 9 & 13.
BWBWBWBW.. to ..WWWWBBBB, specifically prescribed.
Rohrbough. Brain Resters and Testers. c1935. Alternate in Four Moves, p. 4.
..BBBBWWWW to WBWBWBWB.. , but he doesn't specify the blanks, showing all
stages as closed up to 8 spaces, except the first two stages have a gap in the middle.
McKay. At Home Tonight. 1940.
Prob. 43: Arranging counters, pp. 73 & 87-88. RBRBRB.... to ....BBBRRR in three
moves. Sketches general solution.
Prob. 45: Triplets, pp. 74 & 88. YRBYRBYRB.. to BBBYYYRRR.. in 5 moves.
McKay. Party Night. 1940. Heads and tails again, p. 151. RBRBR.. to ..BBRRR in three
moves. RBRBRB.. to ..BBBRRR in four moves. RBRBRBRB.. to ..BBBBRRRR
in four moves. Notes that the first move takes coins 2 & 3 to the end and thereafter one
is always filling the spaces just vacated.
Gardner. SA (Jun & Jul 1961) = New MD, chap. 19, no. 1: Collating the coins. BWBWB
to BBBWW, moving pairs of BW or WB only, but the final position may be shifted.
Gardner thanks H. S. Percival for the idea. Solution in 4 moves, using gaps and with
the solution shifted by six spaces to the right. Thanks to Heinrich Hemme for this
reference.
Joseph S. Madachy. Mathematics on Vacation. (Scribners, NY, 1966, ??NYS);
c= Madachy's Mathematical Recreations. Dover, 1979. Prob. 3: Nine-coin move,
pp. 115 & 128-129 (where the solution is headed Eight-coin move). This uses three
types of coin, which I will denote by B, R, W. BRWBRWBRW  WWWRRRBBB
by moving two adjacent unlike coins at a time and not placing the two coins away from
the rest. Eight move solution leaves the coins in the same places, but uses two extra
cells at each end. From the discussion of Bergerson's problem, see below, it is clear that
the earlier book omitted the word unlike and had a nine move solution, which has been
replaced by Bergerson's eight move solution.
Yeong-Wen Hwang. An interlacing transformation problem. AMM 67 (1967) 974-976.
Shows the problem with 2n pieces, n > 2, can be solved in n moves and this is
minimal.
Doubleday - 1. 1969. Prob. 70: Oranges and lemons, pp. 86 & 170. = Doubleday - 4,
pp. 95-96. BWBWBWBWBW.. considered as a cycle. There are two solutions in five
moves: to ..WWWWWBBBBB, which never uses the cycle; and to:
BBWWWWWW..BBB.
Howard W. Bergerson, proposer; Editorial discussion; D. Dobrev, further solver; R. H.
Jones, further solver. JRM 2:2 (Apr 1969) 97; 3:1 (Jan 1970) 47-48; 3:4 (Oct 1970)
233-234; 6:2 (Spring 1973) 158. Gives Madachy's 1966 problem and says there is a
shorter solution. The editor points out that Madachy's book and Bergerson have
omitted unlike. Bergerson has an eight move solution of the intended problem, using
two extra cells at each end, and Leigh James gives a six move solution of the stated
problem, also using two extra cells at each end. Dobrev gives solutions in six and five
steps, using only two extra cells at the right. Jones notes that the problem does not state
that the coins have to be adjacent and produces a four move solution of the stated
problem, going from ....BRWBRWBRW.... to WWW..R..R..R..BBB.
SOURCES - page 125
Jan M. Gombert. Coin strings. MM 42:5 (Nov 1969) 244-247. Notes that BWBWB...... 
......BBBWW can be done in four moves. In general, BWB...BWB, with n Ws and
n+1 Bs alternating can be transformed to BB...BWW...W in n2 moves and this is
minimal. This requires shifting the whole string n(n+1) to the right and a move can go
to places separated from the rest of the pieces. By symmetry, ......BWBWB 
WWBBB...... in the same number of moves.
Doubleday - 2. 1971. Two by two, pp. 107-108. ..BWBWBWBW to WWWWBBBB... He
doesn't specify where the extra spaces are, but says the first two must move to the end
of the row, then two more into the space, and so on. The solution always has two
moving into an internal space after the first move.
Wayne A. Wickelgren. How to Solve Problems. Freeman, 1974. Checker-rearrangement
problem, pp. 144-146. BWBWB to BBBWW by moving two adjacent checkers, of
different colours, at a time. Solves in four moves, but the pattern moves six places to
the left.
Putnam. Puzzle Fun. 1978.
No. 1: Nickles [sic] & dimes, pp. 1 & 25. Usual version with 8 coins. Solution has
blanks at the opposite end to where they began.
No. 2: Nickles [sic] & dimes variation, pp. 1 & 25. Same, except the order of each pair
must be reversed as it moves. Solution in five moves with blanks at opposite end
to where they started. = AM 237. Cf Lucas, 1895.
James O. Achugbue & Francis Y. Chin. Some new results on a shuffling problem. JRM
12:2 (1979-80) 125-129. They demonstrate that any pattern of n & n occupying 2n
consecutive cells can be transformed into any other pattern in the same cells, using only
two extra cells at the right, except for the case n = 3 where 10 cells are used. They
then find an optimal solution for BB...BW...WW  BWBW...BW in n+1 moves
using two extra cells. They seem to leave open the question of whether the number of
moves could be shortened by using more cells.
Walter Gibson. Big Book of Magic for All Ages. Kaye & Ward, Kingswood, Surrey, 1982.
Six cents at a time, p. 117. Uses pennies and nickels. .....PNPNP to NNPPP..... in
four moves.
Tricky turnover, p. 137. HTHTHT to HHHTTT in two moves. This requires turning
over one of the two coins on each move.
Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Pp. 117, 119 & 123-126. He
asks to move BBBWWW to WBWBWB and to BWBWBW and notes that the latter
takes an extra move. He sketches the general solutions.
5.P. GENERAL MOVING PIECE PUZZLES
See also under 5.A.
5.P.1. SHUNTING PUZZLES
See Hordern, op. cit. in 5.A, pp. 167-177, for a survey of these puzzles. The
Chifu-Chemulpo (or Russo-Jap Railway) Puzzle of 1903 is actually not of this type since all
the pieces can move by themselves -- Hordern, pp. 124-125 & plate VIII.
See S&B 124-125.
A 'spur' is a dead-end line. A 'side-line' is a line or siding joined to another at both ends.
Mittenzwey. 1880. Prob. 219-221, pp. 39-40 & 91; 1895?: 244-246, pp. 43-44 & 93; 1917:
244-246, pp. 40 & 89. First two have a canal too narrow to permit boats to pass, with a
'bight', or widening, big enough to hold one boat while another passes. First problem
has two boats meeting one boat; second problem has two boats meeting two boats. The
third problem has a single track railway with a side-line big enough to hold an engine
and 16 wagons on the side-line or on the main line between the switches. Two trains
consisting of an engine and 20 wagons meet.
Lucas. RM2, 1883, pp. 131-133. Passing with a spur and with a side-line.
Alexander Henry Reed. UK Patent 15,051 -- Improvements in Puzzles. Complete
specification: 8 Dec 1885. 4pp + 1p diagrams. Reverse a train using a small turntable
on the line. This has forms with one line and with two crossing lines. One object is to
spell 'Humpty Dumptie'. He also has a circular line with three turntables (equivalent to
the recent Top-Spin Puzzle of F. Lammertinck).
SOURCES - page 126
Pryse Protheroe. US Patent 332,211 -- Puzzle. Applied: 18 Sep 1885; patented: 8 Dec 1885.
3pp + 1p diagrams. Described in Hordern, p. 167. Identical to the Reed patent above!
Both Reed and Protheroe are described as residents of suburban London. The Reed
patent says it was communicated from abroad by an Israel J. Merritt Jr of New York and
it doesn't assert that Reed is the inventor, so perhaps Reed and Merritt were agents for
Protheroe.
Jeffrey & Son (Syracuse, NY). Great Railroad Puzzle. Postcard puzzle produced in 1888.
??NYS. Described in Hordern, pp. 175-176. Passing with a turntable that holds two
wagons.
Arthur G. Farwell. US Patent 437,186 -- Toy or Puzzle. Applied: 20 May 1889; patented:
30 Sep 1890. 1p + 1p diagrams. Described in Hordern, pp. 167-169. Great Northern
Puzzle. This requires interchanging two cars on the legs of a 'delta' switch which is too
short to allow the engine through, but will let the cars through. Hordern lists 6 later
patents on the same basic idea.
Ball. MRE, 1st ed., 1892, pp. 43-44. Great Northern Puzzle "which I bought some eight or
nine years ago." (Hordern, p. 167, erroneously attributes this quote to Ahrens.)
Loyd. Problem 28: A railway puzzle. Tit-Bits 32 (10 Apr & 1 May 1897) 23 & 79. Engine
and 3 cars need to pass 4 cars by means of a 'delta' switch whose branches and tail hold
only one car. Solution with 28 reversals.
Loyd. Problem 31: The turn-table puzzle. Tit-Bits 32 (1 & 22 May 1897) 79 & 135.
Reverse an engine and 9 cars with an 8 track turntable whose lines hold 3 cars. The
turntable is a double curved connection which connects, e.g. track 1 to tracks 4 or 6.
E. Fourrey. Récréations Arithmétiques. Op. cit. in 4.A.1. 1899. Art. 239: Problèmes de
Chemin de fer, pp. 184-189.
I. Three parallel tracks with two switched crossing tracks. Train of 21 wagons on the
first track must leave wagons 9 & 12 on third track.
II. Delta shape with a turntable at the point of the delta, which can only hold the wagons
and not the engine, so this is isomorphic to Farwell.
III. This is a more complex railway problem involving timetables on a circular line.
J. W. B. Shunting! c1900. ??NYS. Described in Hordern, pp. 176-177 & plate XII.
Reversing a train with a turntable that holds three wagons.
Orril L. Hubbard. US Patent 753,266 -- Puzzle. Applied: 21 Apr 1902; patented: 1 Mar
1904. 3pp + 1p diagrams. Great Railroad Puzzle, described in Hordern, pp. 175-176.
Improved version of the Jeffrey & Son puzzle of 1888. Engine & 2 cars to pass engine
& 3 cars, using a turntable that holds two cars, preserving order of each train.
Harry Lionel Hook & George Frederick White. UK Patent 26,645 -- An Improved Puzzle or
Game. Applied: 3 Dec 1902; accepted: 11 Jun 1903. 2pp + 1p diagrams. This is very
cryptic, but appears to be a kind of sliding piece Puzzle using turntables.
Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:1 (May 1903) 50-51 & 10:2
(Jun 1903) 140-141 & 10:4 (Aug 1903) 336-337. A railway puzzle. One north-south
line with a spur heading north which is holding 7 trucks, but cannot hold the engine as
well, so the engine is on the main line heading south. An engine pulling seven trucks
arrives from the north and wants to get past. First solution uses 17 stages; second uses
12 stages.
Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:5 (Sep 1903) 426-427 & 10:6
(Oct 1903) 530-531. A shunting problem. Same as Fourrey - II, hence isomorphic to
Farwell. Solution in 17 stages.
Celluloid Starch Puzzle. c1905. Described in Hordern, pp. 169-170. Cars on the three parts
of a 'delta' switch with an engine approaching. Reverse the engine, leaving all cars on
their original places. More complexly, suppose the tail of the 'delta' only holds one car
or the engine.
Livingston B. Pennell. US Patent 783,589 -- Game Apparatus. Applied: 20 Mar 1902;
patented: 28 Feb 1905. 3pp + 1p diagrams. Described in Hordern, p. 173. Passing
with a side line -- engine & 3 cars to pass engine & 3 cars using a siding which already
contains 3 cars, without couplings, so these three can only be pushed. Also the engines
can move at most three cars at a time.
William Rich & Harry Pritchard. UK Patent 7647 -- Railway Game and Puzzle. Applied:
11 Apr 1905; complete specification: 11 Oct 1905; accepted: 14 Dec 1905. 2pp + 1p
diagrams. Main line with two short and two long spurs.
Ball. MRE, 4th ed., 1905, pp. 61-63, adds a problem with a side-line, "on sale in the streets in
1905“. The 5th ed., 1911, pp. 69-71 & 82, adds the name "Chifu-Chemulpo Puzzle"
SOURCES - page 127
and that the minimum number of moves is 26, in more than one way. P. 82 gives
solutions of both problems.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Great Northern Puzzle. He says the
"Railway puzzle" was very popular "about twenty years ago".
Ahrens. MUS I. 1910. Pp. 3-4. Great Northern. Says it is apparently modern and cites
Fourrey for other examples.
Anon. Prob. 6. Hobbies 32 (No. 814) (20 May 1911) 145 & (No. 817) (10 Jun 1911) 208.
Great Northern Puzzle. Solution asks if readers know any other railway puzzles.
Loyd. The switch problem & Primitive railroading problem. Cyclopedia, 1914, pp. 167 &
361; 89 & 350 (= MPSL2, prob. 24, pp. 18-19; MPSL1, prob. 95, pp. 92 & 155).
Passing with a 'delta' switch & passing with a spur. The first is like Tit-Bits Problem
28, but the engine and 3 cars have to pass 5 cars. Solution in 32 moves. See Hordern,
pp. 170-171.
Hummerston. Fun, Mirth & Mystery. 1924. The Chinese railways, pp. 103 & 188. Imagine
a line of positions: ABCEHGJLMN with single positions D, I, F, K attached to
positions C, H, G, L. You have eight engines at ABCD and KLMN and the object is
to exchange them, preserving the order. He does it in 18 moves, where a move can be
of any length.
King. Best 100. 1927. No. 14, pp. 12 & 41. Side-line with a bridge over it too low for the
engine. Must interchange two wagons on the side-line which are on opposite sides of
the bridge.
B. M. Fairbanks. Railroad switching problems. IN: S. Loyd Jr., ed.; Tricks and Puzzles;
op. cit. in 5.D.1 under Chapin; 1927. P. 85 & Answers p. 7. Three realistic problems
with several spurs and sidelines.
Loyd Jr. SLAHP. 1928. Switching cars, pp. 54 & 106. Great Northern puzzle. See Hordern,
pp. 168-169.
Doubleday - 2. 1971. Traffic jam, pp. 85-86. Version with cars in a narrow lane and a layby. Two cars going each way. Though the lay-by is three cars wide and just over a car
long, he restricts its use so that it acts like it is two cars wide.
5.P.2. TAQUIN
Jacques Haubrich has kindly enlightened me that 'taquin' simply means 'teaser'. So
these items should be re-categorised.
Lucas. RM3. 1893. 3ème Récréation -- Le jeu du caméléon et le jeu des jonctions de points,
pp. 89-103. Pp. 91-97 -- Le taquin de neuf cases avec un seul port. I thought that
taquin was the French generic term for such puzzles, but I find no other usage than that
below, except in referring to the 15 Puzzle -- see references to taquin in 5.A.
Au Bon Marché (the Paris department store). Catalogue of 1907, p. 13. Reproduced in Mary
Hillier; Automata and Mechanical Toys; An Illustrated History; Jupiter Books, London,
1976, p. 179. This shows Le Taquin Japonais Jeu de Patience Casse-tete. This
comprises 16 hexagonal pieces, looking like a corner view of a die, so each has three
rhombic parts containing a pattern of pips. They are to be placed as the corners of four
interlocked hexagons with the numbers on adjacent rhombi matching.
5.Q. NUMBER OF REGIONS DETERMINED BY N LINES OR PLANES
Mittenzwey. 1880. Prob. 200, pp. 37 & 89; 1895?: 225, pp. 41 & 91; 1917: 225, pp. 38 &
88. Family of 4 adults and 4 children. With three cuts, divide a cake so the adults and
the children get equal pieces. He makes two perpendicular diametrical cuts and then a
circular cut around the middle. He seems to mean the adults get equal pieces and the
children get equal pieces, not necessarily the same. But if the circular cut is at 2/2 of
the radius, then the areas are all equal. Not clear where this should go -- also entered in
5.T.
Jakob Steiner. Einige Gesetze über die Theilung der Ebene und des Raumes. (J. reine u.
angew. Math. 1 (1826) 349-364) = Gesam. Werke, 1881, vol. 1, pp. 77-94. Says the
plane problem has been raised before, even in a Pestalozzi school book, but believes he
is first to consider 3-space. Considers division by lines and circles (planes and spheres)
and allows parallel families, but no three coincident.
Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306 & Three puzzles;
SOURCES - page 128
Knowledge 9 (Sep 1886) 336-337. "3. A man marks 6 straight lines on a field in such
a way as to enclose 10 spaces. How does he manage this?" Solution begins: "III. To
inclose ten spaces by six ropes fastened to nine pegs." Take (0,0), (1,0), ..., (n,0), (0,n),
..., (0,1), as 2n+1 points, using n+2 ropes from (0,0) to (n,0) and to (0,n) and from
(i,0) to (0,n+1-i) to enclose n(n+1)/2 areas.
Richard A. Proctor. Our puzzles. Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40.
Describes several ways of solving previous problem and asks for a symmetric version.
G. Chrystal. Algebra -- An Elementary Text-Book. Vol. 2, A. & C. Black, Edinburgh,
1889. [Note -- the 1889 version of vol. 1 is a 2nd ed.] Chap. 23, Exercises IV, p.
34. Several similar problems and the following.
No. 7 -- find number of interior and of exterior intersections of the diagonals of a
convex n-gon.
No. 8 -- n points in general position in space, draw planes through every three and find
number of lines and of points of intersection.
L. Schläfli. Theorie der vielfachen Kontinuität. Neue Denkschriften der allgemeinen
schweizerischen Gesellschaft für die Naturwissenschaften 38:IV, Zürich, 1901, 239 pp.
= Ges. Math. Abh., Birkhäuser, Basel, 1950-1956, vol. 1, pp. 167-392. (Pp. 388-392
are a Nachwort by J. J. Burckhardt.) Material of interest is Art. 16: Über die Zahl der
Teile, ..., pp. 209-212. Obtains formula for k hyperplanes in n space.
Loyd, Dudeney, Pearson & Loyd Jr. give various puzzles based on this topic.
Howard D. Grossman. Plane- and space-dissection. SM 11 (1945) 189-190. Notes Schläfli's
result and observes that the number of regions determined by k+1 hyperspheres in n
space is twice the number of regions determined by k hyperplanes and gives a two to
one correspondence for the case n = 2.
Leo Moser, solver. MM 26 (Mar 1953) 226. ??NYS. Given in: Charles W. Trigg;
Mathematical Quickies; (McGraw-Hill, NY, 1967); corrected ed., Dover, 1985.
Quickie 32: Triangles in a circle, pp. 11 & 90-91. N points on a circle with all
diagonals drawn. Assume no three diagonals are concurrent. How many triangles are
formed whose vertices are internal intersections?
Timothy Murphy. The dissection of a circle by chords. MG 56 (No. 396) (May 1972)
113-115 + Correction (No. 397) (Oct 1972) 235-236. N points on a circle, in a plane
or on a sphere; or N lines in a plane or on a sphere, all simply done, using Euler's
formula.
Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Slicing cakes, pp. 33 & 61.
Cut a circular cake into 12 equal pieces with 4 cuts. [From this, we see that N full
cuts can yield either 2N or 4(N-1) equal pieces. Further, if we make k circular cuts
producing k+1 regions of equal area and then make N-k diametric cuts equally
spaced, we get 2(k+1)(N-k) pieces of the same size.]
Looking at this problem, I see that one can obtain any number of pieces from N+1 up
through the maximum.
5.Q.1.
NUMBER OF INTERSECTIONS DETERMINED BY N LINES
Chrystal. Text Book of Algebra. 2nd ed., vol. 2, 1889, p. 34, ex. 7. See above.
Loyd Jr. SLAHP. 1928. When drummers meet, pp. 74 & 115. Six straight railroads can
meet in 15 points.
Paul Erdös, proposer; Norbert Kaufman & R. H. Koch and Arthur Rosenthal, solvers.
Problem E750. AMM 53 (1946) 591 & 54 (1947) 344. The first solution is given in
Trigg, op. cit. in 5.Q, Quickie 191: Intersections of diagonals, pp. 53 & 166-167. In a
convex n-gon, how many intersections of diagonals are there? This counts a triple
intersection as three ordinary (i.e. double) intersections or assumes no three diagonals
are concurrent. Editorial notes add some extra results and cite Chrystal.
5.R. JUMPING PIECE GAMES
See also 5.O. Some of these are puzzles, but some are games and are described in the
standard works on games -- see the beginning of 4.B.
5.R.1.PEG SOLITAIRE
See MUS I 182-210.
SOURCES - page 129
Ahrens, MUS I 182-183, gives legend associating this with American Indians. Bergholt,
below, and Beasley, below, find this legend in the 1799 Encyclopédie Méthodique:
Dictionnaire des Jeux Mathématique (??*), ??NYS. Ahrens also cites some early 19C
material which has not been located. Bergholt says some maintain the game comes
from China.
Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford,
1694. De Ludo dicto Ufuba wa Hulana, p. 233. This has a 5 x 5 board with each side
having 12 men, but the description is extremely brief. It seems to have two players,
but this may simply refer to the two types of piece. I'm not clear whether it's played like
solitaire (with the jumped pieces being removed) or like frogs & toads. I would be
grateful if someone could read the Latin carefully. The name of the puzzle is clearly
Arabic and Hyde cites an Arabic source, Hanzoanitas (not further identified on the
pages I have) -- I would be grateful to anyone who can track down and translate Arabic
sources.
G. W. Leibniz. Le Jeu du Solitaire. Unpublished MS LH XXXV 3 A 10 f. 1-2, of c1678.
Transcribed in: S. de Mora-Charles; Quelques jeux de hazard selon Leibniz; HM 19
(1992) 125-157. Text is on pp. 152-154. 37 hole board. Says the Germans call it 'Die
Melancholy' and that it is now the mode at the French court.
Claude-Auguste Berey. Engraving: Madame la Princesse de Soubize jouant au Jeu de
Solitaire. 1697(?). Beasley (below) discovered and added this while his book was in
proof. It shows the 37-hole French board. Reproduced in: Pieter van Delft & Jack
Botermans; Creative Puzzles of the World; op. cit. in 5.E.2.a, p. 170.
G. W. Leibniz. Jeu des Productions. Unpublished MS LH XXXV 8,30 f. 4, of 1698.
Transcribed in: de Mora-Charles, loc. cit. above. Text is on pp. 154-155. 37 hole
board. Considers the game in reverse.
Trouvain. Engraving: Dame de Qualité Jouant au Solitaire. 1698(?).
Claude-Auguste Berey. Engraving: Nouveau Jeu de Solitaire. Undated, but Berey was active
c1690-c1730. Reproduced in: R. C. Bell; The Board Game Book; Marshall Cavendish,
London, 1979, pp. 54-55 and in: Jasia Reichardt, ed.; Play Orbit [catalogue of an
exhibition at the ICA, London, and elsewhere in 1969-1970]; Studio International,
1969, p. 38. Beasley's additional notes point out that this engraving is well known, but
he had not realised its date until the earlier Berey engraving was discovered. This
engraving includes the legend associating the game with the American indians -- "son
origine vient de l'amerique ou les Peuples vont seuls à la chasse, et au retour plantent
leurs flèches en des trous de leur cases, ce qui donna idée a un françois de composer ce
jeu ...." Reichardt says the original is in the Bibliothèque Nationale.
The three engravings above are reproduced in: Henri d'Allemagne; Musée rétrospectif de la
classe 100, Jeux, à l'exposition universelle international de 1900 à Paris, Tome II,
pp. 152-158. D'Allemagne says the originals are in the Bibliothèque Nationale, Paris.
He (and de Mora-Charles) also cites Rémond de Montmort, 2nd ed., 1713 -- see below.
G. W. Leibniz. Annotatio de quibusdam Ludis; inprimis de Ludo quodam Sinico,
differentiaque Scachici et Latrunculorum & novo genere Ludi Navalis. Misc.
Berolinensia (= Misc. Soc. Reg., Berlin) 1 (1710) 24. Last para. on p. 24 relates to
solitaire. (English translation on p. xii of Beasley, below.)
Pierre Rémond de Montmort. Essai d'analyse sur les jeux de hazards. (1708); Seconde
edition revue & augmentee de plusieurs lettres, (Quillau, Paris, 1713 (reprinted by
Chelsea, NY, 1980)); 2nd issue, Jombert & Quillau, 1714. Avertissement (to the 2nd
ed.), xli-xl. "J'ai trouvé dans le premier volume de l'Academie Royale de Berlin, ...; il
propose ensuite des Problèmes sur un jeu qui a été à la mode en France il y a douze ou
quinze ans, qui se nomme Le Solitaire."
Edward Hordern's collection has a wooden 37 hole board on the back of which is inscribed
"Invented by Lord Derwentwater when Imprisoned in the Tower". The writing is old, at
least 19C, possibly earlier. However the Encyclopedia Britannica article on
Derwentwater and the DNB article on Radcliffe, James, shows that the relevant Lord
was most likely to have been James Radcliffe (1689-1716), the 3rd Earl from 1705,
who joined the Stuart rising in 1715, was captured at Preston, was imprisoned in the
Tower and was beheaded on 24 Feb 1716, so the implied date of invention is 1715 or
1716. The third Earl became a figure of romance and many stories and books appeared
SOURCES - page 130
about him, so the invention of solitaire could well have been attributed to him.
Though the title was attainted and hence legally extinct, it was claimed by
relatives. Both James's brother Charles (1693-1746), the claimed 5th Earl from 1731,
and Charles's son James Bartholomew (1725-1786), the claimed 6th Earl from 1746,
spent time in prison for their Stuart sympathies. Charles escaped from Newgate Prison
after the 1715 rising, but both were captured on their way to the 1745 rising and taken
to the Tower where Charles was beheaded. If either of these is the Lord Derwentwater
referred to, then the date must be 1745 or 1746. A guide book to Northumberland,
where the family lived at Dyvelston (or Dilston) Castle, near Hexham, asserts the last
Derwentwater was executed in 1745, while the [Blue Guide] says the last was executed
for his part in the 1715 uprising.
In any case, the claim seems unlikely.
G. W. Leibniz. Letter to de Montmort (17 Jan 1716). In: C. J. Gerhardt, ed.; Die
Philosophischen Schriften von Gottfried Wilhelm Leibniz; (Berlin, 1887) = Olms,
Hildesheim, 1960; Vol. 3, pp. 667-669. Relevant passage is on pp. 668-669. (Poinsot,
op. cit. in 5.E, p. 17, quotes this as letter VIII in Leibn. Opera philologica.)
J. C. Wiegleb. Unterricht in der natürlichen Magie. Nicolai, Berlin & Stettin, 1779. Anhang
von dreyen Solitärspielen, pp. 413-416, ??NYS -- cited by Beasley. First known
diagram of the 33-hole board.
Catel. Kunst-Cabinet. 1790. Das Grillenspiel (Solitaire), p. 50 & fig. 167 on plate VI.
33 hole board. (Das Schaaf- und Wolfspiel, p. 52 & fig. 169 on plate VI, is a game on
the 33-hole board.)
Bestelmeier. 1801. Item 511: Ein Solitair, oder Nonnenspiel. 33 hole board.
Strutt. Op. cit. in 4.B.1. The Solitary Game. (1801: Book IV, p. 238. ??NYS -- cited by
Beasley -- may be actually 1791??) 1833: Book IV, chap. II, art. XV, p. 319. c= StruttCox, p. 259. Beasley says this is the first attribution to a prisoner in the Bastille. The
description is vague: "fifty or sixty" holes and "a certain number of pegs". Strutt-Cox
adds a note that "The game of Solitaire, reimported from France, ..., came again into
Fashion in England in the late" 1850s and early 1860s.
Ada Lovelace. Letter of 16 Feb 1840 to Charles Babbage. BM MSS 37191, f. 331. ??NYS -reproduced in Teri Perl; Math Equals; Addison-Wesley, Menlo Park, California, 1978,
pp. 109-110. Discusses the 37 hole board and wonders if there is a mathematical
formula for it.
M. Reiss. Beiträge zur Theorie des Solitär-Spiels. J. reine angew. Math. 54 (1857) 376-379.
St. v. Kosiński & Louis Wolfsberg. German Patent 42919 -- Geduldspiel. Patented:
25 Sep 1877. 1p + 1p diagrams. 33 hole version.
The Sociable. 1858. The game of solitaire, pp. 282-284. 37 hole board. "It is supposed to
have been invented in America, by a Frenchman, to beguile the wearisomeness
attendant upon forest life, and for the amusement of the Indians, who pass much of their
time alone at the chase, ...."
Anonymous. Enquire Within upon Everything. 66th ed., 862nd thousand, Houlston and
Sons, London, 1883, HB. Section 135: Solitaire, p. 49. Mentions a 37 hole board but
shows a 33 hole board. This material presumably goes back some time before this
edition. It later shows Fox and Geese on the 33 hole board.
Hoffmann. 1893. Chap. X, no. 11: Solitaire problems, pp. 339-340 & 376-377
= Hoffmann-Hordern, pp. 232-233, with photo on p. 235. Three problems. Photo on
p. 235 shows a 33-hole board in a square frame, 1820-1840, and a 37-hole board with a
holding handle, 1840-1890.
Ernest Bergholt. Complete Handbook to the Game of Solitaire on the English Board of
Thirty-three Holes. Routledge, London, nd [Preface dated Nov 1920] -- facsimile
produced by Naoaki Takashima, 1993. This is the best general survey of the game prior
to Beasley.
King. Best 100. 1927. No. 68, pp. 28 & 55. = Foulsham's no. 24, pp. 9 & 13. 3 x 3 array of
men in the middle of a 5 x 5 board. Men can jump diagonally as well as orthogonally.
Object is to leave one man in the centre.
Rohrbough. Puzzle Craft. 1932. Note on Solitaire & French Solitaire, pp. 14-15 (= pp. 6-7
of 1940s?). 33 hole board, despite being called French.
B. M. Stewart. Solitaire on a checkerboard. AMM 48 (1941) 228-233. This surveys the
history and then considers the game on the 32 cell board comprising the squares of one
colour on a chessboard. He tilts this by 45o to get a board with 7 rows, having
2, 4, 6, 8, 6, 4, 2 cells in each row. He shows that each beginning-ending problem
SOURCES - page 131
which is permitted by the parity rules is actually solvable, but he gives examples to
show this need not happen on other boards.
Gardner. SA (Jun 1962). Much amended as: Unexpected, chap. 11, citing results of Beasley,
Conway, et al. Cites Leibniz and mentions Bastille story.
J. D. Beasley. Some notes on solitaire. Eureka 25 (Oct 1962) 13-18. No history of the game.
Jeanine Cabrera & René Houot. Traité Pratique du Solitaire. Librairie Saint-Germain, Paris,
1977. On p. 2, they give the story that it was invented by a prisoner in the Bastille, late
18C, and they even give the name of the reputed inventor: "Comte"(?) Pellisson. They
say that a Paul Pellisson-Fontanier was in the Bastille in 1661-1666 and was a man of
some note, but history records no connection between him and the game.
The Diagram Group. Baffle Puzzles -- 3: Practical Puzzles. Sphere, 1983. No. 12. On the
33-hole board place 16 markers: 1 in row 2; 3 in row 3; 5 in row 4; 7 in row 5;
making a triangle centred on the mid-line. Can you remove all the men, except for one
in the central square? Gives a solution in 15 jumps.
J. D. Beasley. The Ins and Outs of Peg Solitaire. OUP, 1985. History, pp. 3-7; Selected
Bibliography, pp. 253-261. PLUS Additional notes, from the author, 1p, Aug 1985. 57
references and 5 patents, including everything known before 1850.
Franco Agostini & Nicola Alberto De Carlo. Intelligence Games. (As: Giochi della
Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. This gives the
legend of the nobleman in the Bastille. Then says that "it would appear that a very
similar game" is mentioned by Ovid "and again, it was widely played in ancient China - hence its still frequent alternative name, "Chinese checkers."" I have included this as
an excellent example of how unreferenced statements are made in popular literature. I
have never seen either of these latter statements made elsewhere. The connection with
Ovid is pretty tenuous -- he mentions a game involving three in a row and otherwise is
pretty cryptic and I haven't seen anyone else claiming Ovid is referring to a solitaire
game -- cf 4.B.5. The connection with Chinese checkers is so far off that I wonder if
there is a translation problem -- i.e. does the Italian name refer to some game other than
what is known as Chinese checkers in English??
Nob Yoshigahara. Puzzlart. Tokyo, 1992. Coin solitaire, pp. 5 & 90. Four problems on a
4 x 4 board.
Marc Wellens, et al. Speelgoed Museum Vlaanderen -- Musée du Jouet Flandre -- Spielzeug
Museum Flandern -- Flanders Toy Museum. Speelgoedmuseum Mechelen, Belgium,
1996, p. 90 (in English), asserts 'It was invented by the French nobleman Palissen, who
had been imprisoned in the Bastille by Louis XIV' in the early 18C.
5.R.1.a.
TRIANGULAR VERSION
The triangular version of the game has only recently been investigated. The triangular
board is generally numbered as below.
1
2
4
3
6
9 10
11 12 13 14 15
7
5
8
Herbert M. Smith. US Patent 462,170 -- Puzzle. Filed: 13 Mar 1891; issued: 27 Oct 1891.
2pp + 1p diagrams. A board based on a triangular lattice.
Rohrbough. Puzzle Craft. 1932. Triangle Puzzle, p. 5 (= p. 6 in 1940s?). Remove peg 13
and leave last peg in hole 13.
Maxey Brooke. (Fun for the Money, Scribner's, 1963); reprinted as: Coin Games and
Puzzles; Dover, 1973. All the following are on the 15 hole board.
Prob. 1: Triangular jump, pp. 10-11 & 75. Remove one man and jump to leave one
man on the board. Says Wesley Edwards asserts there are just six solutions. He
removes the middle man of an edge and leaves the last man there.
Prob. 2: Triangular jump, Ltd., pp. 12-13 & 75. Removes some of the possible jumps.
Prob. 3: Headless triangle, pp. 14 & 75. Remove a corner man and leave last man
there.
M. Gardner. SA (Feb 1966) c= Carnival, 1975, chap. 2. Says a 15 hole version has been on
sale as Ke Puzzle Game by S. S. Adams for some years. Addendum cites Brooke and
SOURCES - page 132
Hentzel and says much unpublished work has been done.
Irvin Roy Hentzel. Triangular puzzle peg. JRM 6:4 (1973) 280-283. Gives basic theory for
the triangular version. Cites Gardner.
[Henry] Joseph & Lenore Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books
(Charter Communications), NY, 1975. Pennies for your thoughts, pp. 179-182.
Remove a coin and solve. Hint says to remove the coin at 13 and that you should be
able to have the last coin at 13. The solution has this property.
Alan G. Henney & Dagmar R. Henney. Computer oriented solutions. CM 4:8 (1978)
212-216. Considers the 'Canadian I. Q. Problem', which is the 15 hole board, but they
also permit such jumps as 1 to 13, removing 5. They find solutions from each initial
removal by random trial and error on a computer.
Putnam. Puzzle Fun. 1978. No. 15: Jumping coins, pp. 5 & 28. 15 hole version, remove peg
1 and leave last man there.
Benjamin L. Schwarz & Hayo Ahlburg. Triangular peg solitaire -- A new result. JRM 16:2
(1983-84) 97-101. General study of the 15 hole board showing that starting and ending
with 5 is impossible.
J. D. Beasley. The Ins and Outs of Peg Solitaire. Op. cit. above, 1985. Pp. 229-232 discusses
the triangular version, citing Smith, Gardner and Hentzel, saying that little has been
published on it.
Irvin Roy Hentzel & Robert Roy Hentzel. Triangular puzzle peg. JRM 18:4 (1985-86)
253-256. Develops theory.
John Duncan & Donald Hayes. Triangular solitaire. JRM 23:1 (1991) 26-37. Extended
analysis. Studies army advancement problem.
William A. Miller. Triangular peg solitaire on a microcomputer. JRM 23:2 (1991) 109-115
& 24:1 (1992) 11. Summarises and extends previous work. On the 10 hole triangular
board, the classic problem has essentially a unique solution -- the removed man must be
an edge man (e.g. 2) and the last man must be on the adjacent edge and a neighbour of
the starting hole (i.e. 3 if one starts with 2). On the 15 hole board, the removed man can
be anywhere and there are many solutions in each case.
Remove man from hole:
1
2
4
5
Number of solutions:
29760
14880
85258
1550
Considers the 'tree' formed by the first four rows and hole 13.
5.R.1.b.
OTHER SHAPES
New section. See also King and Stewart in 5.R.1 for some forms based on a square
board.
A
C D
B
F
E
G
H
I
J
Putnam. Puzzle Fun. 1978. No. 53: Checker star, pp. 10 & 34. Use the 10 points of a
pentagram, as above, and leave one of the inner points empty. Reduce to one man.
[Parity shows the one man must be at an outer point and any outer point can be
achieved. If one leaves an outer point empty, then the last man must be on an inner
point and any of these can be achieved.]
Hummerston. Fun, Mirth & Mystery. 1924.
Perplexity, pp. 22-23. Using the octagram board shown in 5.A, place 15 markers on it,
leaving cell 16 empty. It is possible to remove all but one man. [I can't see how
to apply parity to this board.]
Solplex, p. 25. In playing his Perplexity, specify where you will leave the last man?
Leap frog, Puzzle no. 22, pp. 64 & 175. Take a 4 x 3 board with the long edge
extended by one more cell at the upper left and lower right. Put white counters
on the 4 x 3 area, put a black counter in one of the extra cells and leave the other
extra cell empty. Remove all but the black man. Counting multiple jumps of the
same man as a single move, he does it in eight moves, getting the black man back
to its starting point.
SOURCES - page 133
5.R.2.FROGS AND TOADS: BBB_WWW TO WWW_BBB
In the simplest version, one has n black men at the left and n white men at the right of
a strip of 2n+1 cells, e.g. BBB_WWW. One can slide a piece forward (i.e. blacks go left
and whites go right) into an adjacent place or one can jump forward over one man of the other
colour into an empty place. The object is to reverse the colours, i.e. to get WWW_BBB.
S&B 121 & 125, shows versions.
One finds that the solution never has a man moving backward nor a man jumping
another man of the same colour. Some authors have considered relaxing these restrictions,
particularly if one has more blank spaces, when these unusual moves permit shorter solutions.
Perhaps the most general form of the one-dimensional problem would be the following.
Suppose we have m men at the left of the board, n men at the right and b blank spaces in
the middle. The usual case has b = 1, but when b > 1, the kinds of move permitted do
change the number of moves in a minimal solution. First, considering slides, can a piece slide
backward? Can a piece slide more than one space? If so, is there a maximum distance, s,
that it is allowed to slide? (The usual problem has s = 1.) Of course s  b. Second,
considering jumps, can a piece jump backward? Can a piece jump over a piece (or pieces) of
its own colour and/or a blank space (or spaces) and/or a mixture of these? If so, is there a
maximum number of pieces, p, that it can jump over? (The usual problem has p = 1.) It is
not hard to construct simple examples with s > 1 such that shorter solutions exist when
unusual moves are permitted. Are there situations where one can show that backward moves
are not needed?
The game is sometimes played on a 2-dimensional board, where one colour can move
down or right and the other can move up or left. See: Hyde ??; Lucas (1883); Ball;
Hoffmann and 5.R.3. Chinese checkers is a later variation of this same idea. On these more
complex boards, one is usually allowed to make multiple jumps and the object is usually to
minimize the number of moves to accomplish the interchange of pieces.
There is a trick version to convert full and empty glasses: FFFEEE to FEFEFE in one
move, which is done by pouring. I've just noted this in a 1992 book and I'll look for earlier
examples.
Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford,
1694. De Ludo dicto Ufuba wa Hulana, p. 233. This has a 5 x 5 board with each side
having 12 men, but the description is extremely brief. It seems to have two players,
but this may simply refer to the two types of piece. I'm not clear whether it's played like
solitaire (with the jumped pieces being removed) or like frogs & toads. I would be
grateful if someone could read the Latin carefully. The name of the puzzle is clearly
Arabic and Hyde cites an Arabic source, Hanzoanitas (not further identified on the
pages I have) -- I would be grateful to anyone who can track down and translate Arabic
sources.
American Agriculturist (Jun 1867). Spanish Puzzle. ??NYR -- copy sent by Will Shortz.
Anonymous. Every Little Boy's Book A Complete Cyclopædia of in and outdoor games with
and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and
fifty illustrations. Routledge, London, nd. HPL gives c1850, but the material is clearly
derived from Every Boy's Book, whose first edition was 1856. But the text considered
here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as unnamed editor),
nor in the 8th ed of 1868 (published for Christmas 1867), which was the first seriously
revised edition, with Edmund Routledge as editor, nor in the 13th ed. of 1878. So this
material is hard to date, though in 4.A.1, I've guessed this book may be c1868.
P. 12: Frogs and toads. "A new and fascinating game of skill for two players;
played on a leather board with twelve reptiles; the toads crawling, and the frogs
hopping, according to certain laws laid down in the rules. The game occupies but a few
minutes, but in playing it there is scarcely any limit to the skill that can be exhibited,
thus forming a lasting amusement. (Published by Jaques, Hatton Garden.)" This does
not sound like our puzzle, but perhaps it is related. Unfortunately Jaques' records were
destroyed in WW2, so it is unlikely they can shed any light on what the game was.
Does anyone know what it was?
Hanky Panky. 1872. Checker puzzle, p. 124. Three and three, with solution.
Mittenzwey. 1880. Prob. 239, pp. 44 & 94; 1895?: 267-268, pp. 48 & 96; 1917: 267-268,
SOURCES - page 134
pp. 44 & 91-92. Problem with 3 & 3 brown and white horses in stalls. 1895? adds a
version with 4 & 4.
Bazemore Bros. (Chattanooga, Tennessee). The Great "13" Puzzle! Copyright No. 1033 - O 1883. Hammond & Jones Printers. Advertising puzzle consisting of two 3 and 3
versions arranged in an X pattern.
Lucas. RM2. 1883.
Pp. 141-143. Finds number of moves for n and n.
Pp. 144-145. Considers game on 5 x 5, 7 x 7, ..., boards and gives number of moves.
Edward Hordern's collection has an example called Sphinxes and Pyramids from the 1880s.
Sophus Tromholt. Streichholzspiele. (1889; 5th ed, 1892.) Revised from 14th ed. of 1909
by R. Thiele; Zentralantiquariat der DDR, Leipzig, 1986. Prob. 11, 41, 81 are the game
for 4 & 4, 2 & 2, 3 & 3.
Ball. MRE, 1st ed., 1892, pp. 49-51. 3 & 3 case, citing Lucas, with generalization to n & n;
7 x 7 board, citing Lucas, with generalization to 2n+1 x 2n+1.
Berkeley & Rowland. Card Tricks and Puzzles. 1892. German counter puzzle, p. 112. 3 & 3
case.
Hoffmann. 1893. Chap. VI, pp. 269-270 & 282-284 = Hoffmann-Hordern, pp. 182-185, with
photo.
No. 17: The "Right and Left" puzzle. Three and three. Photo on p. 184 shows: a
cartoon from Punch (18 Dec 1880): The Irish Frog Puzzle -- with a Deal of
Croaking; and an example of a handsome carved board with square pieces with
black and white frogs on the tops, registered 1880. Hordern Collection, p. 77,
shows the latter board and two further versions: Combat Sino-Japonais
(1894-1895) and Anglais & Boers (1899-1902).
No. 18. Extends to a 7 x 5 board.
Puzzles with draughtsmen. The Boy's Own Paper 17 or 18?? (1894??) 751. 3 and 3.
Lucas. L'Arithmétique Amusante. 1895. Prob. XXXV: Le bal des crapauds et des
grenouilles, pp. 117-124. Does 2 and 2, 3 and 3, 4 and 4 and the general case of n
and n, showing it can be done in n(n+2) moves -- n2 jumps and 2n steps. The
general solution is attributed to M. Van den Berg. M. Schoute notes that each move
should make as little change as possible from the previous with respect to the two
aspects of changing type of piece and changing type of move.
Clark. Mental Nuts. 1904, no. 72; 1916, no. 62. A good study. 3 and 3.
Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY,
1909. Doola's Game, pp. 42-43 & 61-62. 3 and 3.
Anon. Prob. 47: The monkey's dilemma. Hobbies 30 (No. 762) (21 May 1910) 168 & 182
& (No. 765) (11 Jun 1910) 228. Basically 3 & 3, but there are eight posts for crossing
a river, with the monkeys on 1,2,3 and 6,7,8. The monkeys can jump onto the bank
and we want the monkeys to all get to the bank they are headed for, so this is not the
same as BBB..WWW to WWW..BBB. The solution doesn't spell out all the steps, so
it's not clear what the minimum number of moves is -- could we have a monkey
jumping another of the same colour?
Ahrens. MUS I. 1910. Pp. 17-19. Basically repeats some of Lucas's work from 1883 &
1895.
Williams. Home Entertainments. 1914. The cross-over puzzle, pp. 119-120. 3 and 3 with
red and white counters. Doesn't say how many moves are required.
Dudeney. AM. 1917. Prob. 216: The educated frogs, pp. 59-60 & 194. _WWWBBB to
BBBWWW_ with frogs able to jump either way over one or two men of either colour.
Solution in 10 jumps.
Ball. MRE, 9th ed., 1920, pp. 77-79, considers the m & n case, giving the number of steps in
the solution.
Blyth. Match-Stick Magic. 1921. Matchstick circle transfer, pp. 81-82. 3 and 3 in 15
moves.
Hummerston. Fun, Mirth & Mystery. 1924. The frolicsome frogs, Puzzle no. 2, pp. 17 &
172. Two 3 & 3 problems with the boards crossing at the centre cell. He notes that
the easiest solution is to solve the boards one at a a time. He says: "It is not good play
to jump a counter over another of the same colour."
Lynn Rohrbough, ed. Socializers. Handy Series, Kit G, Cooperative Recreation Service,
Delaware, Ohio, 1925. Six Frogs, p. 5. Dudeney's 1917 problem done in 11 moves.
Botermans et al. The World of Games. Op. cit. in 4.B.5. 1989. P. 235 describes this as The
Sphinx Puzzle, "very popular around the turn of the century, particularly in the United
SOURCES - page 135
States and France" and they show an example of the period labelled The Sphinx and
Pyramid Puzzle -- An Egyptian Novelty.
Haldeman-Julius. 1937. No. 162: Checker problem, pp. 18 & 29. 3 & 3.
See Harbin in 5.R.4 for a 1963 example.
Doubleday - 1. 1969.
Prob. 77: Square dance, pp. 93 & 171. = Doubleday - 5, pp. 103-104. Start with
_WWWBBB. He says they must change places, with a piece able to move into
the vacant space by sliding (either way) or by jumping one or two pieces of any
colour. Asks for a solution in 10 moves. His solution gets to BBBWWW_,
which does not seem to be 'changing places' to me.
Prob. 79: All change, pp. 95 & 171. = Doubleday - 5, pp. 105-106.
BB_WW
Start with the pattern at the right and change the whites and
BB_WW
blacks in 10 moves, where a piece can slide one place into an
adjacent vacant square or jump one or two pieces into a vacant square. However,
the solution simply does each row separately.
Katharina Zechlin. (Dekorative Spiele zum Selbermachen; Verlag Frech,
WWWWW
Stuttgart-Botnang, 1973.) Translated as: Making Games in Wood Games
BWWWW
you can build yourself. Sterling, 1975, pp. 24-27: The chess knight game.
BBOWW
5 x 5 board with 12 knights of each of two colours, arranged as at the right.
BBBBW
The object is to reverse them by knight's moves. Says it can be done within
BBBBB
50 moves and 'is almost impossible to do it in less than 45'.
Wickelgren. How to Solve Problems. Op. cit. in 5.O. 1974. Discrimination reversal
problem, pp. 78-81. _WWWBBB to BBBWWW with the extra place not specified in
the goal, with pieces allowed to move into the vacant space by sliding or by hopping
over one or two pieces. Gets to BBBWW_W in 9 moves. [I find it takes 10 moves to
get to BBBWWW_ .]
Joe Celko. Jumping frogs and the Dutch national flag. Abacus 4:1 (Fall 1986) 71-73. Same
as Wickelgren. Celko attributes this to Dudeney. Gives a solution to BBBWWW_ in
10 moves and asks for results for higher numbers.
Johnston Anderson. Seeing induction at work. MG 75 (No. 474) (Dec 1991) 406-414.
Example 2: Frogs, pp. 408-411. Careful proof that BB...BB_WW...WW to
WW...WW_BB...BB, with n counters of each colour, requires n2 + 2n moves.
5.R.3.FORE AND AFT -- 3 BY 3 SQUARES MEETING AT A CORNER
This is Frogs and Toads on part of the 5 x 5 board consisting of two 3 x 3 subarrays
at diagonally opposite corners. They overlap in the central square. One square has 8 black
men and the other has 8 white men, with the centre left vacant.
Ball. MRE, 1st ed., 1892, pp. 51-52. 51 move solution. In the third ed., 1896, pp. 69-70, he
says he believes he was the first to publish the puzzle but "that it has been since widely
distributed in connexion with an advertisement and probably now is well known". He
gives a 48 move solution.
Hoffmann. 1893. Chap. VI, no. 26: The "English Sixteen" puzzle, pp. 273-274 & 287
= Hoffmann-Hordern, pp. 188-189, with photo. Mentions that it is produced by Messrs
Heywood, as below. Solution in 52 moves, which he believes is minimal. Hordern
notes that the minimum is 46. Photo on p. 188 of the Heywood version, see next entry.
John Heywood, Manchester, produced a version called 'The English Sixteen Puzzle', undated,
but by 1893 as Hoffmann cites it. Photo in Hoffmann-Hordern, p. 188, dated
1880-1895.
Charles A. Emerson. US Patent 522,250 -- Puzzle. Applied: 3 Nov 1893; patented: 3 July
1894. 2pp + 1p diagrams. The Fore and Aft Puzzle. Says it can be done in 48, 49, 50,
51 or 52 moves.
Dudeney. Problem 66: The sixteen puzzle. Tit-Bits 33 (1 Jan & 5 Feb 1898) 257 & 355.
"It was produced, I believe, in America, many years ago, and has since been issued over
here in the form of an advertisement by a prominent commercial house." Solution in 46
moves. He says published solutions assert the minimum number of moves is 53, 52 or
50. The 46 move solution is given in Ball, MRE, 5th ed., 1911, 79-80.
Ball. MRE, 5th ed., 1911, pp. 79-80. Drops his historical claims and includes a 46 move
solution due to Dudeney.
Loyd. Fore and aft puzzle. Cyclopedia, 1914, pp. 108 & 353 (solution misprinted, but
SOURCES - page 136
claimed to be 47 moves in contrast to 52 move solutions 'in the puzzle books'.)
(c= MPSL1, prob. 4, pp. 3-4 & 121 (only referring to Dudeney's 46 move solution)).
Loyd Jr. SLAHP. 1928. A joke on granddad, pp. 29 & 93. Says 'our granddaddies, who
used to play this puzzle game 75 years ago, when it was universally popular. The
old-time books explain how the solution is accomplished in 52 moves, "the shortest
possible method."' He then asks for and gives a 46 move solution.
M. Adams. Puzzles That Everyone Can Do. 1931. Prob. 24, pp. 17 & 132: "General post".
Gives a solution which takes 46 moves, but gives no discussion of it.
Rohrbough. Puzzle Craft. 1932. Migration (or Fore and Aft), p. 12 (= p. 15 of 1940s?). Says
it was popular 75 years ago and it has recently been shown that it can be done in 46
moves, then gives a solution which stops at 42 moves!
M. Gardner. SA (Sep 1959) = 2nd Book, pp. 210-219. Discusses the puzzle. On
pp. 218-219, he gives Dudeney's 46 move solution and says 48 different solutions and
several proofs that 46 is minimal were sent to him.
Uwe Schult. Das Seemanns-Spiel: Mathematisch erledigt. Reported in Das Mathematische
Kabinett column, Bild der Wissenschaft 19:11 (Nov 1982) 181-184. (A version is
given in Neues aus dem Mathematischen Kabinett, ed. by Thiagar Devendran,
Hugendubel, Munich, 1985, pp. 102-103.) There are 218,790 possible patterns of the
pieces. Reversing black and white takes 46 moves and there are 1026 different
halfway positions that can occur in a 46 move solution. There are two patterns which
require 47 moves, namely, after reversing black and white, put one of the far corner
pieces in the centre.
Nob Yoshigahara, postcard to me on 18 Aug 1994, announces he has found the worst solution
-- in 58 moves.
5.R.4.REVERSING FROGS AND TOADS: _12...n TO _n...21 , ETC.
A piece can slide into the empty cell or jump another piece into the empty cell.
Dudeney. AM. 1917.
Prob. 214: The six frogs, pp. 59 & 193. Case of n = 6, solved in 21 moves, which he
says is minimal. In general, the minimal solution takes n(n+1)/2 moves,
including n steps, when n is even and (n2+3n-8)/2 moves, including 2n-4
steps, when n is odd. "This complete general solution is published here for the
first time."
Prob. 215: The grasshopper puzzle, pp. 59 & 193-194. Problem for a circular
arrangement. Example has n = 12. Says he invented it in 1900. Solvable in 44
moves. General solution is complex -- he says that for n > 4, it can be done in
(n2+4n-16)/4 moves when n is even and in (n2+6n-31)/4 moves when n is
odd.
Rohrbough. Puzzle Craft. 1932. The Reversible Frogs, p. 22 (= The Jumping Frogs,
pp. 20-21 of 1940s?). n = 8, citing Dudeney, AM.
Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. Hopover, p. 89. First gives 3 and 3
Frogs and Toads, then asks for complete reversal from 123_456 to 654_321.
[Henry] Joseph and Lenore Scott. Master Mind Pencil Puzzles. Tempo Books (Grosset &
Dunlap), NY, 1973 (& 1978?? -- both dates are given -- I'm presuming the 1978 is a
2nd ptg or a reissue under a different imprint??). Reverse the numbers, pp. 117-118.
Give the problem for n = 6 and a solution in 21 moves. For n even, the method
gives a solution in n(n+1)/2, it is not shown that this is optimal, nor is a general
method given for odd n.
[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. 1973. Op. cit. in 5.E. 13-hour
clock, pp. 43-44. Case n = 12 considered in a circle can be done in 44 moves.
Joe Celko. Jumping frogs and the Dutch national flag. Abacus 4:1 (Fall 1986) 71-73. Cites
Dudeney and gives the results.
Jim Howson. The Computer Weekly Book of Puzzlers. Computer Weekly Publications,
Sutton, Surrey, 1988, unpaginated. [The material comes from his column which started
in 1966, so an item may go back to then.] Prob. 54 -- same as the Scotts in Master
Mind Pencil Puzzles.
5.R.5.FOX AND GEESE, ETC.
SOURCES - page 137
There are a number of similar games on different boards -- too many to describe
completely here, so I will generally just cite extensive descriptions. See any of the main
books on games mentioned at the beginning of 4.B, such as Bell or Falkener. The key feature
is that one side has more, but weaker, pieces. These are sometimes called hunt games. The
standard Fox and Geese is played on a 33 hole Solitaire board, with diagonal moves allowed.
I have recently acquired but not yet read Murray's History of Board Games other than Chess
which should have lots of material.
Gretti's Saga, late 12C. Mention of Fox and Geese. Also in Edward IV's accounts. ??NYS -cited by Botermans et al, below.
Shackerley (or Schackerley or Shakerley) Marmion. A Fine Companion (a play). 1633. IN:
The Dramatic Works of Shackerley Marmion; William Paterson, Edinburgh & H.
Sotheran & Co., London, 1875. II, v, pp. 140-141. "..., let him sit in the shop ..., and
play at fox and geese with the foreman, ....." Earliest English occurrence of fox-andgeese. Quoted by OED and cited by Fiske, below.
Richard Lovelace. To His Honoured Friend On His Game of Chesse-Play or To Dr. F. B. on
his Book of Chesse. 1656?, published in his Posthume Poems, 1659. Lines 1-4. My
edition of Lovelace notes that F. B. was Francis Beale, author of 'Royall Game of
Chesse Play,' 1656. Lovelace died in 1658.
Sir, now unravell'd is the Golden Fleece,
Men that could onely fool at Fox and Geese,
Are new-made Polititians by the Book,
And can both judge and conquer with a look.
Henry Brooke. Fool of Quality. [A novel.] 1766-1768. Vol. I, p. 367. ??NYS -- quoted by
Fiske, below. "Can you play at no kind of game, Master Harry?" "A little at
fox-and-geese, madam."
Catel. Kunst-Cabinet. 1790.
Das Fuchs- und Hühnerspiel, pp. 51-52 & fig. 168 on plate VI. 11 chickens against one
fox on a 4 x 4 board with all diagonals drawn, giving 16 + 9 playing points.
Das Schaaf- und Wolfspiel, p. 52 & fig. 169 on plate VI, is the same game on the 33hole solitaire board with 11 sheep and one wolf, no diagonals
Bestelmeier. 1801.
Item 83: Das Schaaf- und Wolfspiel. Same diagram and game as Catel, p. 52.
Item 833: Ein Belagerungspiel. 33 hole board with a fortress on one arm, with
diagonals drawn.
Strutt. Op. cit. in 4.B.1. Fox and Geese. 1833: Book IV, chap. II, art. XIV, pp. 318-319.
= Strutt-Cox, p. 258 & plate opp. p. 246. Fig. 107 (= plate opp. p. 246) shows the 33
hole board with its diagonals drawn.
Gomme. Op. cit. in 4.B.1. I 141-142 refers to Strutt and Micklethwaite.
Illustrated Boy's Own Treasury. 1860. Fox and Geese, pp. 406-407. 33 hole Solitaire board
with diagonals drawn.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 320, p. 152: Fuchs und Gänse.
Shows 33 hole solitaire board with diagonals drawn.
Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S.
National Museum for 1893, pp. 489-537. Pp. 874-877 describes: the Japanese game of
Juroku Musashi (Sixteen Soldiers) with 16 men versus a general; the Chinese game
of Shap luk kon tséung kwan (The sixteen pursue the commander); another Chinese
game of Yeung luk sz' kon tséung kwan with 27 men against a commander (described
by Hyde -- ?? I didn't see this); the Malayan game of Dam Hariman (Tiger Game),
identical to the Hindu game of Mogol Putt'han (= Mogul Pathan (Mogul against
Pathan)), similar to a Peruvian game of Solitario and the Mexican game of Coyote; the
Siamese game of Sua ghin gnua (Tiger and Oxen) and the similar Burmese game of
Lay gwet kyah, with three big tigers versus 11 or 12 little tigers; the Samoan game of
Moo; the Hawaiian game of Konane; a similar Madagascarian game; the Hindu game
of Pulijudam (Tiger Game) with three tigers versus 15 lambs.
Fiske. Op. cit. in 4.B.1. 1905. Fox-and-Geese, pp. 146-156 & 359, discusses the history of
the game, especially as to whether it is identical to the old Norse game of Hnefatafl. On
p. 359, he says that John of Salisbury (c1150) used 'vulpes' as the name of a game, but
there is no indication of what it was. He says "the fox-and-geese board, in
comparatively modern times, has begun to be used for games more or less different in
their nature, especially for one called in England solitaire and in France "English
SOURCES - page 138
solitaire", and for another, known in Spain and Italy as asalto (assalto), in French as
assaut, in Danish as belejringsspel." He then surveys the various sources that he treated
under Mérelles -- see 4.B.1 and 4.B.5 for details. He is not sure that Brunet is really
describing the game in the Alfonso MS (op. cit. in 4.B.5 and below). He cites an 1855
Italian usage as Jeu de Renard or Giuoco della Volpe. In Come Posso Divertirmi?
(Milan, 1901, pp. 231-233), it is said that the game is usually played with 17 geese
rather than 13 -- Fiske notes that this assertion is of "some historical value, if it be true."
Moulidars calls it Marelle Quintuple, quotes Maison des Jeux Académiques (Paris,
1668) for a story that it was invented by the Lydians and gives the game with 13 or 17
geese. Asalto has 2 men against 24. Fiske quotes Shackley Marmion, above, for the
oldest English occurrence of fox-and-geese and then Henry Brooke, above. Fiske
follows with German, Swedish and Icelandic (with 13 geese) references.
H. Parker. Ancient Ceylon. Op. cit. in 4.B.1, 1909. Pp. 580-583 & 585 describe four forms
of The Leopards Game, with one tiger against seven leopards, three leopards against 15
dogs, two leopards against 24 cattle and one leopard against six cattle on a 12 x 12
board. The first two are played on a triangular board.
Robert Kanigel. The Man Who Knew Infinity. A Life of the Genius Ramanujan. (Scribner's,
NY, 1991); Abacus (Little, Brown & Co. (UK)), London, 1992. Pp. 18 & 377:
Ramanujan and his mother used to play the game with three tigers and fifteen goats on a
kind of triangular board.
The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year
1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as
the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete
reproduction in 194 Phototypic Plates. 2 vols., Karl W. Hirsemann, Leipzig, 1913.
See 4.B.5 for more details of this work. See below.
Botermans et al. The World of Games. Op. cit. in 4.B.5. 1989. P. 147 says De Cercar La
Liebre (Catch the Hare) occurs in the Alfonso MS and is the earliest example of a hunt
game in European literature, but undoubtedly derived from an Arabic game of the
Alquerque type -- I didn't see this when I briefly looked at the facsimile -- ??NYS.
They say Murray has noted that hunt games are popular in Asia, but not in Africa,
leading to the conjecture that they originated in Asia. They describe it on a 5 x 5 array
of points with verticals and horizontals and some diagonals drawn, with one hare
against 12 hunters.
Botermans et al. continue on pp. 148-155 to describe the following.
Shap Luk Kon Tseung Kwan (Sixteen Pursue the General) played on a 5 x 5 board
like Catch the Hare with an extra triangle on one side and capturing by
interception.
Yeung Luk Sz'Kon Tseung Kwan, seen in Nanking by Hyde and described by him in
1694, somewhat similar to the above, but with 26 rebels against a general.
(??NYS)
Fox and Geese, mentioned in Gretti's Saga of late 12C and in Edward IV's accounts.
They give a version called Lupo e Pecore from a 16C Venetian book, using a
Solitaire board extended by three points on each arm, giving 45 points. They
give a 1664 engraving showing Le Jeu du Roi which they say is a rather complex
form of fox and geese, but looks like a four-handed game on a cross-shaped
board with 7 x 5 arms on a 7 x 7 central square and 4 groups of 7 x 4 men.
Leopard games, from Southeast Asia, with a kind of triangular board. Len Choa, from
Thailand, has a tiger against six leopards. Hat Diviyan Keliya, from Sri Lanka,
has a tiger against seven leopards.
Tiger games, also from Southeast Asia, are similar to leopard games, but use an
extended Alquerque board (as in Catch the Hare). Rimau (Tiger), from Malaysia,
has 24 men versus a tiger and Rimau-Rimau (Tigers) is a version with two tigers
versus 22 men.
Murray. 1913.
P. 347 cites a 1901 Indian book for 2 lions against 32 goats on a chessboard.
P. 371 cites a Soyat (North Asia) example (19C?) of Bouge-Shodra (Boar's Chess) with
2 boars against 24 calves on a chessboard.
Pp. 569 & 616-617 cite the Alfonso MS of 1283 for 'De cercar la liebre', played on a
5 x 5 board with 10, 11 or 12 men against a hare.
P. 585 shows Cott. 6 (c1275) of 8 pawns against a king on a chessboard.
Pp. 587 & 590 give Cott. 11 = K6: Le Guy de Alfins with king and 4 bishops against a
SOURCES - page 139
king on a chessboard.
Pp. 589-590 shows K4 = CB249: Le Guy de Dames and No. 5 = K5: Le Guy de
Damoyselles, which have 16 pawns against a king on a chessboard.
P. 617 discusses Fox and Geese, with 13, 15 or 17 geese against a fox on the solitaire
board. Edward IV, c1470, bought "two foxis and 46 hounds". Murray says more
elaborate forms exist and refers to Hyde and Fiske (see 4.B.1 and 5.F.1 for more
on these), ??NYS.
Pp. 675 & 692 show CB258: Partitum regis Francorum with king and four pawns
against king on the chessboard. It says the first side wins.
P. 758 describes a 16C Venetian board (then) at South Kensington (V&A??) with the
Solitaire board for Fox and Geese and an enlarged board for Fox and Geese.
P. 857 mentions Fox and Geese in Iceland.
Family Friend 2 (1850) 59. Fox and geese. 4 geese against 1 fox on a chess board.
The Sociable. 1858. Fox and geese, p. 281. 17 geese against a fox on the solitaire board.
Four men versus a king on the draughts board, saying the first side wins even allowing
the king to be placed anywhere against the men who start on one side.
Stewart Culin. Korean Games, op. cit. in 4.B.5, 1895. Pp. 76-77 describes some games of
this type, in particular a Japanese game called Yasasukari Musashi with 16 soldiers
versus a general on a 5 x 5 board, taken from a 1714 (or 1712) Japanese book: Wa
Kan san sai dzu e "Japanese, Chinese, Three Powers picture collection", published in
Osaka.
Anonymous. Enquire Within upon Everything. 66th ed., 862nd thousand, Houlston and
Sons, London, 1883, HB. Section 2593: Fox and Geese, p. 364. 33 hole Solitaire board
with 17 geese against a fox. 4 geese against a fox on the chessboard. Says the geese
should win in both cases.
Slocum. Compendium. Shows Solitaire and Solitaire & Tactic Board from Gamage's 1913
catalogue. Like Bestelmeier's 833, but without diagonals.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Games involving
unequal forces, pp. 43-52. Discusses the following.
The Maharajah and the Sepoys. 1 against 16 on a chessboard.
Fox and Geese. Cites an Icelandic work of c1300 (probably Gretti's Saga?). 1 against
13 or 17 on a Solitaire board.
Lambs and Tigers, from India. 3 against 15.
Cows and Leopards, from SE Asia. 2 against 24.
Vultures and Crows, also called Kaooa, from India. 1 against 7 on a pentagram board.
The New Military Game of German Tactics, c1870. 2 against 24 on a Solitaire Board
with a fortress, as in Bestelmeier.
Yuri I. Averbakh. Board games and real events. IN: Alexander J. de Voogt, ed.; New
Approaches to Board Games Research: Asian Origins and Future Perspectives;
International Institute for Asian Studies, Leiden, 1995; pp. 17-23. Notes that Murray
believes hunt games evolved from war games, but he feels the opposite is true. He
describes a Nepalese game of Baghachal with four tigers versus 20 goats -- this is
Murray's 5.6.22. He corrects some of Murray's assertions about Boar Chess and
describes other Tuvinian hunt games: Bull's Chess and Calves' Chess, probably
borrowed from the Mongols. The latter has a three-in-a-row pattern and he wonders if
there is some connection with morris or noughts and crosses (which he says is "played
everywhere"). He mentions Cercar la Liebre from the Alfonso MS. Fox and Geese
type games are mentioned in the Icelandic sagas as 'the fox game'. He describes several
forms.
5.R.6.OCTAGRAM PUZZLE
One has an octagram and seven men. One has to place a man on a vacant point and
then slide him to an adjacent vacant point, then do the same with the next man, ..., so as to
cover seven of the points. The diagram is just an 8-cycle and is the same as the knight's
connections on the 3 x 3 board, so the octagram puzzle is equivalent to the 7 knights problem
mentioned in 5.F.1. Further, the 4 knights problem of 5.F.1 has the same 8-cycle, with men at
alternate points of it.
Versions with different numbers of points.
5 points: Rohrbough.
7 points: Mittenzwey; Meyer.
SOURCES - page 140
9 points: Dudeney.
10 points: Bell & Cornelius; Hoffmann; Cohen; Williams; Toymaker; Rohrbough; Putnam.
13 points: Berkeley & Rowland.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Pentalpha, p. 15.
Says that a pentagram board occurs at Kurna, Egypt, c-1400 and that the solitaire game
of Pentalpha is played in Crete. This has 9 men to be placed on the vertices and the
intersections of the pentagram. Each man must be placed on a vacant point, then slid
ahead two positions along one straight line. The intermediate point may be occupied,
but the ending point must be unoccupied. Unfortunately we don't know if the Egyptian
board was used for this game.
Pacioli. De Viribus. c1500.
Ff. 112r - 113v. .C(apitolo). LXVIII. D(e). cita ch' a .8. porti ch' cosa convi(e)ne
arepararli (Chap. 68. Of a city with 8 gates which admits of division ??). =
Peirani 158-160. Octagram puzzle with a complex story about a city with 8 gates
and 7 disputing factions to be placed at the gates.
F. IVv. = Peirani 8. The Index gives the above as Problem 83. Problem 82: De .8.
donne ch' sonno aun ballo et de .7. giovini quali con loro sa con pagnano (Of 8
ladies who are at a ball and of 7 youths who accompany them).
Schwenter. 1636. Part 2, exercise 36, pp. 149-150. Octagram.
Witgeest. Het Natuurlyk Tover-Boek. 1686. Prob. 4, pp. 224-225. Octagram, taken from
Schwenter.
Les Amusemens. 1749. P. xxxiii.
Catel. Kunst-Cabinet. 1790. Das Achteck, pp. 12-13 & fig. 36 on plate II. The rules are not
clearly described.
Bestelmeier. 1801. Item 290: Das Achteckspiel. Text copies part of Catel.
Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as
Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. Ff. 131-133 are an analysis
of the heptagram puzzle.
Rational Recreations. 1824. Feat 34, pp. 161-164. Octagram.
Endless Amusement II. 1826? Prob. 28, pp. 203-204. = New Sphinx, c1840, pp. 137-138.
Nuts to Crack IV (1835), no. 194 -- part of a long section called Tricks upon Travellers.
Family Friend (Dec 1858) 359. Practical puzzles -- 1. I don't have the answer.
The Boy's Own Magazine 3 (1857) 159 & 192. Puzzle of the points.
Illustrated Boy's Own Treasury. 1860. Practical Puzzles No. 6, pp. 396 & 436.
The Secret Out. 1859. To Place Seven Counters upon an Eight-Pointed Star, pp. 373-374.
J. J. Cohen, New York. Star puzzle. Advertising card for Star Soap, Schultz & Co.,
Zanesville, Ohio, Copyright May 1887. Reproduced in: Bert Hochberg; As advertised
Puzzles from the collection of Will Shortz; Games Magazine 17:1 (No. 113) 10-13, on
p. 11. Identical to pentalpha - see Bell & Cornelius above.
Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles No. IX: The reversi
puzzle, pp. 8-10. Version with 13 cards in a circle and one can move ahead by any
number of steps. If there are x cards and one moves ahead s steps, then x and s
must have no common factor.
Hoffmann. 1893.
Chap. VI, pp. 267-268 & 280-281 = Hoffmann-Hordern, pp. 180-181, with photos.
No. 13. No name. Basic octagram puzzle. Photo on p. 181 shows: The Seven Puzzles,
by W. & T. Darton, dated 1806-1811; a Tunbridge ware version dated 18251840; and Jeu de Zig-Zag, by M. D., Paris, 1891-1900.
No. 14. The "Okto" puzzle, pp. 268 & 281. Here the counters and points are coloured.
Photo on p. 181 of The "Okto" Puzzle by McGaw, Stevenson & Orr, Ltd. for
John Stewart, dated 1880-1895.
Chap. X, no. 8: Crossette, pp. 337 & 374-375 = Hoffmann-Hordern, pp. 229-230, with
photo. 10 counters in a circle. Start anywhere and move ahead three. Photo on p. 230
shows The Mystic Seven, a seven counter version, by the Lord Roberts Workshops,
1914-1920.
Mittenzwey. 1895? Prob. 329, pp. 58 & 106; 1917: 329, pp. 52 & 101. Heptagram.
Dudeney. Problem 58: A wreath puzzle. Tit-Bits 33 (6 & 27 Nov 1897) 99 & 153.
Complex nonagram puzzle involving moves in either direction and producing the
original word again.
Clark. Mental Nuts. 1897, no. 54; 1904, no. 80; 1916, no. 69. A little puzzle. Usual
SOURCES - page 141
octagram.
Benson. 1904. The eight points puzzle, pp. 250-251. c= Hoffmann, no. 13.
Slocum. Compendium. Shows the "Octo" Star Puzzle from Gamage's 1913 catalogue.
Williams. Home Entertainments. 1914.
Crossette, pp. 115-116. Ten points, advancing three places.
Eight points puzzle, pp. 120-121. Usual octagram.
"Toymaker". Top in Hole Puzzle. Work (23 Dec 1916) 200. 10 holes and one has to move to
the third position and reverse the top in that hole.
Blyth. Match-Stick Magic. 1921. Crossing the points, pp. 83-84.
Hummerston. Fun, Mirth & Mystery. 1924.
The sacred seven, Puzzle no. 5, pp. 26 & 173. Octagram puzzle on the outer points of
the diagram shown in 5.A.
The four rabbits, Puzzle no. 6, pp. 26 & 173. Using the octagram shown in 5.A, put
black counters on locations 1 and 2 and white counters on 7 and 8. The object
is to interchange the colours. This is like the 4 knights problem except the
corresponding 8-cycle has men at positions 1, 2, 5, 6. He counts a sequence of
steps by the same man as a move and hence solves it in 6 moves (comprising 16
steps).
Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Turning the tails, pp. 66-69. 8
coins in a circle, tails up. Count from a tail four ahead and reverse that coin. Get 7
heads up. Counting four ahead means that if you start at 1, you count 1, 2, 3, 4 and
reverse 4.
King. Best 100. 1927. No. 64, pp. 26-27 & 54.
Rohrbough. Puzzle Craft. 1932.
Count 4, p. 6. 10 points on a circle, moving ahead 3. (= Rohrbough; Brain Resters and
Testers; c1935, p. 21.)
Star Puzzle, p. 8 (= p. 10 of 1940s?). Consider the pentagram with its internal vertices.
First puzzle is Pentalpha. Second is to place a counter and move ahead three
positions. The object is to get four counters on the points, which is the same as
the pentagram puzzle, moving one position.
Jerome S. Meyer. Fun-to-do. Op. cit. in 5.C. 1948. Prob. 18: Odd man out, pp. 27 & 184.
Version with 7 positions in a circle and 6 men where one must place a man and then
move him three places ahead.
Putnam. Puzzle Fun. 1978. No. 63: Ten card turnover, pp. 11 & 35. Ten face down cards in
a circle. Mark a card, count ahead three and turnover.
5.R.7.PASSING OVER COUNTERS
The usual version is to have 8 counters in a row which must be converted to 4 piles of
two, but each move must pass a counter over two others. Martin Gardner pointed out to me
that the problem for 10, 12, 14, ... counters is easily reduced to that for 8. The problem is
impossible for 2, 4, 6. There are many later appearances of the problem than given here. In
describing solutions, 4/1 means move the 4th piece on top of the 1st piece.
There are trick solutions where a counter moves to a vacated space or even lands
between two spaces. See: Mittenzwey; Haldeman-Julius; Hemme.
Berkeley & Rowland give a problem where each move must pass a counter over two
piles. This makes the problem easier and it is solvable for any even number of counters  6,
but it gives more solutions. See: Berkeley & Rowland; Wood; Indoor Tricks & Games;
Putnam; Doubleday - 1.
One could also permit passing over one pile, which is solvable for any even number
 4.
Mittenzwey, Double Five Puzzle, Hummerston, and Singmaster & Abbott deal with the
problem in a circle and with piles to be left in specific locations.
Mittenzwey, Lucas and Putnam consider making piles of three by passing over 3, etc.
Kanchusen. Wakoku Chiekurabe. 1727. Pp. 38-39. Jukkoku-futatsu-koshi (Ten stones
jumping over two). Ten counters, one solution.
Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as
Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the
Essay of Games". F. 4v has "The question of the shillings passing at each time over two
or a certain number 8 is the least number Any number being given and any law of
SOURCES - page 142
transit Dr Roget" The layout suggests that Roget had posed the general version.
Adjacent is a diagram with a row of 10 counters and the first move 1 to 4 shown, but
with some unclear later moves.
Endless Amusement II. 1826? Prob. 10, p. 195. 10 halfpence. One solution: 4/1 7/3 5/9 2/6
8/10. = New Sphinx, c1840, pp. 135-135.
Nuts to Crack II (1833), no. 122. 10 counters, identical to Endless Amusement II.
Nuts to Crack V (1836), no. 68. Trick of the eight sovereigns. Usual form.
Young Man's Book. 1839. P. 234. Ingenious Problem. 10 halfpence. Identical to Endless
Amusement II.
Family Friend 2 (1850) 178 & 209. Practical Puzzle, No. VI. Usual form with eight counters
or coins. One solution.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868.
Mechanical puzzles, no. 2, p. 176 (1868: 187). Passing over coins. Gives two
symmetric solutions.
Magician's Own Book. 1857. Prob. 34: The counter puzzle, pp. 277 & 300. Identical to
Book of 500 Puzzles, prob. 34.
The Sociable. 1858. Prob. 16: Problem of money, pp. 291-292 & 308. Start with 10
half-dimes, says to pass over one, but solution has passing over two. One solution.
= Book of 500 Puzzles, 1859, prob. 16, pp. 9-10 & 26.
Book of 500 Puzzles. 1859.
Prob. 16: Problem of money, pp. 9-10 & 26. As in The Sociable.
Prob. 34: The counter puzzle, pp. 91 & 114. Eight counters, two solutions given.
Identical to Magician's Own Book.
The Secret Out. 1859. The Crowning Puzzle, p. 386. 'Crowning' is here derived from the
idea of crowning in draughts or checkers. One solution: 4/1 6/9 8/3 2/5 7/10.
Boy's Own Conjuring Book. 1860.
Prob. 33: The counter puzzle, pp. 240 & 264. Identical to Magician's Own Book,
prob. 34.
The puzzling halfpence, p. 342. Almost identical to The Sociable, prob. 16, with halfdimes replaced by halfpence.
Illustrated Boy's Own Treasury. 1860. Prob. 17, pp. 398 & 438. Same as prob. 34 in
Magician's Own Book but only gives one solution.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 593, part 6, pp. 299-411: Sechs
Knacknüsse. 10 counters, one solution.
Hanky Panky. 1872. Counter puzzle, p. 132. Gives two solutions for 8 counters and one for
10 counters.
Kamp. Op. cit. in 5.B. 1877. No. 12, p. 325.
Mittenzwey. 1880. Prob. 235-238, pp. 43-44 & 93-94; 1895?: 262-266, pp. 47-48 & 95-96;
1917: 262-266, pp. 43-44 & 91.
235 (262). Usual problem, with 10 counters. Two solutions.
--- (263). Added in 1895? Same with 8 counters. Two solutions.
236 (264). 12 numbered counters in a circle. Pass over two to leave six piles of two on
the first six positions. Solution is misprinted in all editions!
237 (265). 12 counters in a circle. Pass over three to leave six piles of two, except the
last move goes over six. The solution allows landing counters on vacated
locations!
238 (266). 15 counters in a row. Pass over 3 to leave five piles of three. The solution
allows landing a counter on a vacated location and landing a counter between two
locations!!
Lucas. RM2. 1883. Les huit pions, pp. 139-140. Solves for 8, 10, 12, ... counters. Says
Delannoy has generalized to the problem of mp counters to be formed into m piles
(m  4) of p by passing over p counters.
[More generally, using one of Berkeley & Rowland's variations (see below), one
can ask when the following problem is solvable: form a line of n = kp counters into k
piles of p by passing over q [counters or piles]. Does q have to be  p?]
Double Five Puzzle. c1890. ??NYS -- described by Slocum from his example. 10 counters
in a circle, but the final piles must alternate with gaps, e.g. the final piles are at the even
positions. This is also solvable for 4, 8, 12, 16, ..., and I conjectured it was only
solvable for 4n or 10 counters -- it is easy to see there is no solution for 2 or 6
counters and my computer gave no solutions for 14 or 18. For 4, 8, 10 or 12
counters, one can also leave the final piles in consecutive locations, but there is no such
SOURCES - page 143
solution for 6, 14, 16 or 18 counters. See Singmaster & Abbott, 1992/93, for the
resolution of these conjectures.
Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles.
No. VII: The halma puzzle, pp. 6-7. Arrange the first ten cards of a suit in a row so that
passing over two cards leaves five piles whose cards total 11 and are in the odd
places. Arrangement is 7,6,3,4,5,2,1,8,9,10. Move 2 to 9, 4 to 7, 8 to 3, 6 to
5, 10 to 1.
No. VIII: Another version, p. 7. With the cards in order and passing over two piles,
leave five piles of two. But this is so easy, he adds that one wants to leave as low
a total as possible on the tops of the piles. He moves 7 to 10, 6 to 3, 4 to 9,
1 to 5, 2 to 8, leaving a total of 20.
[However, this is not minimal -- there are six solutions leaving 18 exposed,
e.g. 1 to 4, 3 to 6, 7 to 10, 5 to 9, 2 to 8. For 6 cards, the minimum is 6,
achieved once; for 8 cards, the minimum is 11, achieved 3 times; for 12 cards,
the minimum is 27, achieved 10 times. For the more usual case of passing over
two cards, the minimum for 8 cards is 15, achieved twice; for 10 cards, the
minimum is 22, achieved 4 times, e.g. by 7 to 10, 5 to 2, 3 to 8, 1 to 4, 6 to 9;
for 12 cards, the minimum is 31, achieved 6 times; for 14 cards, the minimum is
42, achieved 8 times. For passing over one pile, the minimum for 4 cards is 3,
achieved once; the minimum for 6 cards is 7, achieved twice; the minimum for
8 cards is 13, achieved 3 times; for 10 cards, the minimum is 21, achieved 4
times; the minimum for 12 cards is 31, achieved 5 times. Maxima are obtained
by taking mirror images of the minimal solutions.]
Puzzles with draughtsmen. The Boy's Own Paper 17 or 18?? (1894??) 751. 8 men, passing
over two men each time. Notes that it can be extended to any even number of counters.
Clark. Mental Nuts. 1897, no. 63: Toothpicks; 1904, no. 83 & 1916, no. 70: Place 8
toothpicks in a row. One solution.
Parlour Games for Everybody. John Leng, Dundee & London, nd [1903 -- BLC], p. 32: The
five pairs. 10 counter version, one solution.
Wehman. New Book of 200 Puzzles. 1908. P. 15: The counter puzzle and Problem of
money. 8 and 10 counter versions, the latter using pennies. Two and one solutions.
Ahrens. MUS I. 1910. Pp. 15-17. Essentially repeats Lucas.
Manson. 1911. Decimal game, pp. 253-254. Ten rings on pegs. "Children are frequently
seen playing the game out of doors with pebbles or other convenient articles."
Blyth. Match-Stick Magic. 1921. Straights and crosses, pp. 85-87. 10 matchsticks, one must
pass over two of them. Two solutions, both starting with 4 to 1.
Hummerston. Fun, Mirth & Mystery. 1924.
The pairing puzzle, Puzzle no. 8, pp. 27 & 173. Essential 8 counters in a circle, with
four in a row being white, the other four being black. Moving only the whites,
and passing over two, form four piles of two.
Pairing the pennies, Puzzle no. 39, pp. 102 & 178. Ten pennies, one solution.
Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Marrying the coins,
pp. 113-115. Ten coins or eight coins, passing over 2. Gives two solutions for 10, not
noting that the case of 10 is immediately reduced to 8. Says there are several solutions
for 8 and gives two.
Wood. Oddities. 1927. Prob. 45: Fish in the basket, pp. 39-40. 12 fish in baskets in a circle.
Move a fish over two baskets, continuing moving in the same direction, to get get two
fish in each of six baskets, in the fewest number of circuits.
Rudin. 1936. No. 121, pp. 43 & 103. 10 matches. Two solutions.
J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?].
Straights and crosses, pp. 105-106. As in Blyth, 1926.
Indoor Tricks and Games. Success Publishing, London, nd [1930s??].
How to pair the pennies, p. 4. 8 pennies, one solution.
The ten rings. p. 4. 10 rings, passing over two piles, one solution.
Haldeman-Julius. 1937.
No. 91: Jumping pennies, pp. 11 & 25. Six pennies to be formed into two piles of three
by jumping over three pennies each time. Solution has a trick move. Jump 1 to
5, 6 to 3 and 2 to 1/5, which gives the position: 6/3 4 2/1/5. He then says: "No.
4 jumps over 5, 1 and 2 -- then jumps back over 5, 1 and 2 and lands upon 3 and
6, ...." Since the rules are not clear about where a jumping piece can land, the
trick move can be viewed as a legitimate jump to the vacant 6 position, then a
SOURCES - page 144
legitimate move over the 2/1/5 pile and the now vacant 4 position onto the 6/3
pile. If the pennies are considered as a cycle, this trick is not needed.
No. 148: Half dimes, pp. 16 & 143. 10 half dimes, passing over one dime (i.e. two
counters).
Sullivan. Unusual. 1947. Prob. 39: On the line. Ten pennies.
Doubleday - 1. 1969. Prob. 75: Money moves, pp. 91 & 170. Ten pennies. Jump over two
piles. Says there are several solutions and gives one, which sometimes jumps over three
or four pennies.
Putnam. Puzzle Fun. 1978.
No. 26: Pile up the coins, pp. 7 & 31. 12 in a row. Make four piles of three, passing
over three coins each time.
No. 27: Pile 'em up again, pp. 7 & 31. 16 in a row. Make four piles of four, passing
over four or fewer each time.
No. 60: Coin assembly, pp. 11 & 35. Ten in a row, passing over two each time.
No. 61: Alternative coin assembly, pp. 11 & 35. Ten in a row, passing over two piles
each time.
David Singmaster, proposer; H. L. Abbott, solver. Problem 1767. CM 18:7 (1992) 207 &
19:6 (1993). Solves the general version of the Double Five Puzzle, which the proposer
had not solved. One can leave the counters on even numbered locations if and only if
the number of counters is a multiple of 4 or a multiple of 10. One can leave the
counters in consecutive locations if and only if the number of counters is 4, 8, 10 or 12.
Heinrich Hemme. Email of 25 Feb 1999. Points out that the rules in the usual version should
say that the counter must land on a pile of a single coin. This would eliminate the trick
solutions given by Mittenzwey and Haldeman-Julius. Hemme says that without this
rule, the problem is easy and can be solved for 4 and 6 counters!
5.S. CHAIN CUTTING AND REJOINING
The basic problem is to minimise the cost or effort of reforming a chain from some
fragments.
Loyd. Problem 25: A brace of puzzles -- No. 25: The chain puzzle. Tit-Bits 31 (27 Mar
1897) & 32 (17 Apr 1897) 41. 13 lengths: 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 12. (Not in
the Cyclopedia.)
Loyd. Problem 42: The blacksmith puzzle. Tit-Bits 32 (10 & 31 Jul & 21 Aug 1897) 273,
327 & 385. Complex problem involving 10 pieces of lengths from 3 to 23 to be
joined.
Clark. Mental Nuts. 1897, no. 7 & 1904, no. 14: The chain question; 1916, no. 59: The
chain puzzle. 5 pieces of 3 links to make into a single length.
Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 9:4 (Feb 1903) 390-391 & 9:5 (Mar 1903)
490-491. The five chains. 5 pieces of 3 links to make into a single length.
Pearson. 1907. Part II, no. 67, pp. 128 & 205. 5 pieces of 3 links to make into a single
length.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. He attributes such puzzles to Loyd
(Tit-Bits prob. 25) and gives that problem.
Cecil H. Bullivant. Home Fun. T. C. & E. C. Jack, London, 1910. Part VI, Chap IV, No. 9:
The broken chain, pp. 518 & 522. 5 3-link pieces into an open chain.
Loyd. The missing link. Cyclopedia, 1914, pp. 222 (no solution) (c= MPSL2, prob. 25,
pp. 19 & 129). 6 5-link pieces into a loop.
Loyd. The necklace puzzle. Cyclopedia, 1914, pp. 48 & 345 (= MPSL1, prob. 47, pp. 45-46
& 138). 12 pieces, with large and small links which must alternate.
D. E. Smith. Number Stories. 1919. Pp. 119 & 143-144. 5 pieces of 3 links to make into
one length.
Hummerston. Fun, Mirth & Mystery. 1924. Q.E.D. -- The broken chain, Puzzle no. 38,
pp. 99 & 178. Pieces of lengths 2, 2, 3, 3, 4, 4, 6 to make into a closed loop.
Ackermann. 1925. Pp. 85-86. Identical to the Loyd example cited by Dudeney.
Dudeney. MP. 1926. Prob. 212: A chain puzzle, pp. 96 & 181 (= 536, prob. 513,
pp. 211-212 & 408). 13 pieces, with large and small links which must alternate.
King. Best 100. 1927. No. 7, pp. 9 & 40. 5 pieces of three links to make into one length.
William P. Keasby. The Big Trick and Puzzle Book. Whitman Publishing, Racine,
Wisconsin, 1929. A linking problem, pp. 161 & 207. 6 pieces comprising
SOURCES - page 145
2, 4, 4, 5, 5, 6 links to be made into one length.
5.S.1. USING CHAIN LINKS TO PAY FOR A ROOM
The landlord agrees to accept one link per day and the owner wants to minimise the
number of links he has to cut. The solution depends on whether the chain is closed in a cycle
or open at the ends. Some weighing problems in 7.L.2.c and 7.L.3 are phrased in terms of
making daily payment, but these are like having the chain already in pieces. See the Fibonacci
in 7.L.2.c.
New section. I recall that there are older versions.
Rupert T. Gould. The Stargazer Talks. Geoffrey Bles, London, 1944. A Few Puzzles -- write
up of a BBC talk on 10 Jan 1939, pp. 106-113. 63 link chain with three cuts. On
p. 106, he says he believes it is quite modern -- he first heard it in 1935. On p. 113, he
adds a postscript that he now believes it first appeared in John O'London's Weekly (16
Mar 1935) ??NYS.
Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14-16 & 30. No. 3. Man has
closed chain of 182 links and wants to stay 182 days. What is the minimum number of
links to be opened?
Birtwistle. Math. Puzzles & Perplexities. 1971. Pp. 13-16. Begins with seven link openended bracelet. Then how big a bracelet can be dealt with using only two cuts? Gets
23. Then does general case, getting n + (n+1)(2n+1 - 1).
Angela Fox Dunn. Second Book of Mathematical Bafflers. Dover, 1983. Selected from
Litton's Problematical Recreations, which appeared in 1959-1971. Prob. 26, pp. 28 &
176. 23 link case.
Howson. Op. cit. in 5.R.4. 1988. Prob. 30. Says a 23 link chain need only be cut twice,
giving lengths 1, 1, 3, 6, 12, which make all values up to 23. Asks for three cuts in a
63 link chain and the maximum length chain one can deal with in n cuts.
5.T. DIVIDING A CAKE FAIRLY
Mittenzwey. 1880. Prob. 200, pp. 37 & 89; 1895?: 225, pp. 41 & 91; 1917: 225, pp. 38 &
88. Family of 4 adults and 4 children. With three cuts, divide a cake so the adults and
the children get equal pieces. He makes two perpendicular diametrical cuts and then a
circular cut around the middle. He seems to mean the adults get equal pieces and the
children get equal pieces, not necessarily the same. But if the circular cut is at 2/2 of
the radius, then the areas are all equal. Not clear where this should go -- also entered in
5.Q.
B. Knaster. Sur le problème du partage pragmatique de H. Steinhaus. Annales de la Société
Polonaise de Mathématique 19 (1946) 228-230. Says Steinhaus proposed the problem
in a 1944 letter to Knaster. Outlines the Banach & Knaster method of one cutting 1/n
and each being allowed to diminish it -- last diminisher takes the piece. Also shows
that if the valuations are different, then everyone can get > 1/n in his measure. Gives
Banach's abstract formulations.
H. Steinhaus. Remarques sur le partage pragmatique. Ibid., 230-231. Says the problem isn't
solved for irrational people and that Banach & Knaster's method can form a game.
H. Steinhaus. The problem of fair division. Econometrica 16:1 (Jan 1948) 101-104. This is a
report of a paper given on 17 Sep. Gives Banach & Knaster's method.
H. Steinhaus. Sur la division pragmatique. (With English summary) Econometrica 17
(Supplement) (1949) 315-319. Gives Banach & Knaster's method.
Max Black. Critical Thinking. Prentice-Hall, Englewood Cliffs, (1946, ??NYS), 2nd ed.,
1952. Prob. 12, pp. 12 & 432. Raises the question but only suggests combining two
persons.
5.U. PIGEONHOLE RECREATIONS
van Etten. 1624. Prob. 89, part II, pp. 131-132 (not in English editions). Two men have
same number of hairs. Also: birds & feathers, fish & scales, trees & leaves, flowers
or fruit, pages & words -- if there are more pages than words on any page.
E. Fourrey. Op. cit. in 4.A.1, 1899, section 213: Le nombre de cheveux, p. 165. Two
Frenchmen have the same number of hairs. "Cette question fut posée et expliquée par
SOURCES - page 146
Nicole, un des auteurs de la Logique de Port-Royal, à la duchesse de Longueville."
[This would be c1660.]
The same story is given in a review by T. A. A. Broadbent in MG 25 (No. 264) (May 1941)
128. He refers to MG 11 (Dec 1922) 193, ??NYS. This might be the item reproduced
as MG 32 (No. 300) (Jul 1948) 159.
The question whether two trees in a large forest have the same number of leaves is said to
have been posed to Emmanuel Kant (1724-1804) when he was a boy. [W. Lietzmann;
Riesen und Zwerge im Zahlbereich; 4th ed., Teubner, Leipzig, 1951, pp. 23-24.]
Lietzmann says that an oak has about two million leaves and a pine has about ten
million needles.
Jackson. Rational Amusement. 1821. Arithmetical Puzzles, no. 9, pp. 2-3 & 53. Two
people in the world have the same number of hairs on their head.
Manuel des Sorciers. 1825. Pp. 84-85. ??NX Two men have the same number of hairs, etc.
Gustave Peter Lejeune Dirichlet. Recherches sur les formes quadratiques à coefficients et à
indéterminées complexes. (J. reine u. angew. Math. (24 (1842) 291-371) = Math.
Werke, (1889-1897), reprinted by Chelsea, 1969, vol. I, pp. 533-618. On pp. 579-580,
he uses the principle to find good rational approximations. He doesn't give it a name.
In later works he called it the "Schubfach Prinzip".
Illustrated Boy's Own Treasury. 1860. Arithmetical and Geometrical Problems, No. 34,
pp. 430 & 434. Hairs on head.
Pearson. 1907. Part II, no. 51, pp. 123 & 201. "If the population of Bristol exceeds by two
hundred and thirty-seven the number of hairs on the head of any one of its inhabitants,
how many of them at least, if none are bald, must have the same number of hairs on
their heads?" Solution says 474!
Dudeney. The Paradox Party. Strand Mag. 38 (No. 228) (Dec 1909) 670-676 (= AM,
pp. 137-141). Two people have same number of hairs.
Ahrens. A&N, 1918, p. 94. Two Berliners have same number of hairs.
Abraham. 1933. Prob. 43 -- The library, pp. 16 & 25 (12 & 113). All books have different
numbers of words and there are more books than words in the largest book. (My copy
of the 1933 ed. is a presentation copy inscribed 'For the Athenaeum Library No 43 p 16
R M Abraham Sept 19th 1933'.)
Perelman. FMP. c1935? Socks and gloves. Pp. 277 & 283-284. = FFF, 1957: prob. 25,
pp. 41 & 43; 1977, prob. 27, pp. 53-54 & 56. = MCBF, prob. 27, pp. 51 & 54. Picking
socks and gloves to get pairs from 10 pairs of brown and 10 pairs of black socks and
gloves.
P. Erdös & G. Szekeres. Op. cit. in 5.M. 1935. Any permutation of the first n2 - 1 integers
contains an increasing or a decreasing subsequence of length > n.
P. Erdös, proposer; M. Wachsberger & E. Weiszfeld, M. Charosh, solvers. Problem 3739.
AMM 42 (1935) 396 & 44 (1937) 120. n+1 integers from first 2n have one dividing
another.
H. Phillips. Question Time. Dent, London, 1937. Prob. 13: Marbles, pp. 7 & 179. 12 black,
8 red & 6 white marbles -- choose enough to get three of the same colour.
The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. Pp. 148-149, prob. 6.
Blind maid bringing stockings from a drawer of white and black stockings.
I am surprised that the context of picking items does not occur before Perelman, Phillips and
Home Book.
Sullivan. Unusual. 1943. Prob. 18: In a dark room. Picking shoes and socks to get pairs.
H. Phillips. News Chronicle "Quiz" No. 3: Natural History. News Chronicle, London, 1946.
Pp. 22 & 43. 12 blue, 9 red and 6 green marbles in a bag. Choose enough to have
three of one colour and two of another colour.
H. Phillips. News Chronicle "Quiz" No. 4: Current Affairs. News Chronicle, London, 1946.
Pp. 17 & 40. 6 yellow, 5 blue and 2 red marbles in a bag. Choose enough to have
three of the same colour.
L. Moser, proposer; D. J. Newman, solver. Problem 4300 -- The identity as a product of
successive elements. AMM 55 (1948) 369 & 57 (1950) 47. n elements from a group
of order n have a a subinterval with product = 1.
Doubleday - 2. 1971.
In the dark, pp. 145-146. How many socks do you have to pick from a drawer of white
and black socks to get two pairs (possibly different)?
Lucky dip, pp. 147-148. How many socks do you have to pick from a drawer of with
many white and black socks to get nine pairs (possibly different)? Gives the
SOURCES - page 147
general answer 2n+1 for n pairs. [Many means that the drawer contains more
than n pairs.]
Doubleday - 3. 1972. In the dark, pp. 35-36. Four sweaters and 5, 12, 4, 9 socks of the
same colours as the sweaters. Lights go out. He can only find two of the sweaters.
How many socks must he bring down into the light to be sure of having a pair matching
one of the sweaters?
5.V. THINK-A-DOT, ETC.
I managed to acquire one of these without instructions or packaging some years ago.
Michael Keller provided an example complete with instructions and packaging. I have
recently seen Dockhorn's article on variations of the idea. This is related to Binary
Recreations, 7.M.
The device was produced by E.S.R., Inc. The box or instructions give an address of
34 Label St., Montclair, New Jersey, 07042, USA, but the company has long been closed. In
Feb 2000, Jim McArdle wrote that he believed that this became the well known Edmund
Scientific Co. (101 East Gloucester Pike, Barrington, New Jersey, 08007, USA;
tel: 609-547 3488; email: scientifics@edsci.com; web: http://www.edmundscientific.com).
But he later wrote that investigation of the manuals of DifiComp, one of their other products,
reveals that there appears to be no connection. E.S.R. = Education Science Research. The
inventors of DigiComp, as listed in the patent for it, are: Irving J. Lieberman, William H,
Duerig and Charles D. Hogan, all of Montclair, and they were the founders of the company.
The DigiComp manuals say Think-A-Dot was later invented by John Weisbacker. There is a
website devoted to DigiComp which contains this material and/or pointers to related sites and
has a DigiComp emulator: http://members.aol.com/digicomp1/DigiComp.html .
www.yahoo.com has a Yahoo club called Friends of DigiComp. There is another website
with the DigiComp manual: http://galena.tj.edu.inter.net/digicomp/ .
E.S.R. Instructions, 8pp, nd -- but box says ©1965. No patent number anywhere but leaflet
says the name Think-A-Dot is trademarked.
E.S.R., Inc. Corporation. US trademark registration no. 822,770. Filed: 8 Dec 1965;
registered: 24 Jan 1967. First used 23 Aug 1965. Expired. The US Patent and
Trademark Office website entry says the owner is the company and gives no
information about the inventor(s). The name has been registered for a computer game
on 23 Jul 2002.
Benjamin L. Schwartz. Mathematical theory of Think-A-Dot. MM 40:4 (Sep 1967) 187-193.
Shows there are two classes of patterns and that one can transform any pattern into any
other pattern in the same class in at most 15 drops.
Ray Hemmings. Apparatus Review: Think-a-Dot. MTg 40 (1967) 45.
Sidney Kravitz. Additional mathematical theory of Think-A-Dot. JRM 1:4 (Oct 1968)
247-250. Considers problems of making ball emerge from one side and of viewing only
the back of the game.
Owen Storer. A think about Think-a-dot. MTg 45 (Winter 1968) 50-55. Gives an exercise to
show that any possible transformation can be achieved in at most 9 drops.
T. H. O'Beirne. Letter: Think-a-dot. MTg 48 (Autumn 1969) 13. Proves Steiner's (Storer??
- check) assertion about 9 drops and gives an optimal algorithm.
John A. Beidler. Think-A-Dot revisited. MM 46:3 (May 1973) 128-136. Answers a question
of Schwarz by use of automata theory. Characterizes all minimal sequences. Suggests
some generalized versions of the puzzle.
Hans Dockhorn. Bob's binary boxes. CFF 32 (Aug 1993) 4-6. Bob Kootstra makes boxes
with the same sort of T-shaped switch present in Think-A-Dot, but with just one
entrance. One switch with two exits is the simplest case. Kootstra makes a box with
three switches and four exits along the bottom, and the successive balls come out of the
exits in cyclic sequence. Using a reset connection between switches, he also makes a
two switch, three exit, box.
Boob Kootstra. Box seven. CFF 32 (Aug 1993) 7. Says he has managed to design and make
boxes with 5, 6, 7, 8 exits, again with successive balls coming out the exits in cyclic
order, but he cannot see any general method nor a way to obtain solutions with a
minimal number of movable parts (switches and reset levers). Further his design for 7
exits is awkward and the design of an optimal box for seven is posed as a contest
problem.
SOURCES - page 148
5.W. MAKING THREE PIECES OF TOAST
This involves an old-fashioned toaster which does one side of two pieces at a time. An
alternative version is frying steaks or hamburgers on a grill which holds two objects, assuming
each side has to be cooked the same length of time. The problem is probably older than these
examples.
Sullivan. Unusual. 1943. Prob. 7: For the busy housewife.
J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. P. 4 (26). Mentions
problem and solution.
Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 40:
Minute toast, pp. 18 & 93.
D. St. P. Barnard. 50 Daily Telegraph Brain-Twisters. 1985. Op. cit. in 4.A.4. Prob. 5: Well
done, pp. 16, 80, 103-104. Grilling three steaks on a grill which only holds two. He
complicates the problem in two ways: a) each side takes a minute to season before
cooking; b) the steaks want to be cooked 4, 3, 2 minutes per side.
Edward Sitarski. When do we eat? CM 27:2 (Mar 2001) 133-135. Hamburgers which
require time T per side. After showing that three hamburgers take 3T, he asks how
long it will take to cook H hamburgers. Easily shows that it can be done in HT,
except for H = 1, which takes 2T. Then remarks that this is an easy version of a
scheduling problem -- in reality, the hamburgers would have different numbers of sides,
there would be several grills and each hamburger would have different parts requiring
different grills, but in a particular order!
5.W.1.
BOILING EGGS
New section. These are essentially parodies of the Cistern Problem, 7.H.
McKay. Party Night. 1940. No. 28, p. 182. "An egg takes 3½ minutes to boil. How long
should 12 eggs take?"
Jonathan Always. Puzzles to Puzzle You. Op. cit. in 5.K.2. 1965. No. 88: A boiling
problem, pp. 29 & 82. "If it takes 3½ minutes to boil 2 eggs, how long will it take to
boil 4 eggs?"
John King, ed. John King 1795 Arithmetical Book. Published by the editor, who is the
great-great-grandson of the 1795 writer, Twickenham, 1995. P. 161, the editor
mentions "If a girl on a hilltop can see two miles, how far would two girls be able to
see?"
5.X. COUNTING FIGURES IN A PATTERN
New section -- there must be older examples. There are two forms of such problems
depending on whether one must use the lattice lines or just the lattice points.
For counting several shapes, see: Young World (c1960); Gooding (1994) in 5.X.1.
5.X.1.COUNTING TRIANGLES
Counting triangles in a pattern is always fraught with difficulties, so I have written a
program to do this, but I haven't checked all the examples here.
Pearson. 1907. Part II.
No. 74: A triangle of triangles, p. 74. Triangular array with four on a side, but with all
the altitudes also drawn. Gets 653 triangles of various shapes.
No. 75: Pharaoh's seal, pp. 75 & 174. Isosceles right triangles in a square pattern with
some diagonals.
Anon. Prob. 76. Hobbies 31 (No. 791) (10 Dec 1910) 256 & (No. 794) (31 Dec 1910) 318.
Make as many triangles as possible with six matches. From the solution, it seems that
the tetrahedron was expected with four triangles, but many submitted the figure of a
triangle with its altitudes drawn, but only one solver noted that this figure contains 16
triangles! However, if the altitudes are displaced to give an interior triangle, I find 17
triangles!!
SOURCES - page 149
Loyd. Cyclopedia. 1914. King Solomon's seal, pp. 284 & 378. = MPSL2, No. 142, pp. 100
& 165 c= SLAHP: Various triangles, pp. 25 & 91. How many triangles in the
triangular pattern with 4 on a side? Loyd Sr. has this embedded in a larger triangle.
Collins. Book of Puzzles. 1927. The swarm of triangles, pp. 97-98. Same as Pearson No.
74. He says there are 653 triangles and that starting with 5 on a side gives 1196 and
10,000 on a side gives 6,992,965,420,382. When I gave August's problem in the
Weekend Telegraph, F. R. Gill wrote that this puzzle with 5 on a side was given out as
a competition problem by a furniture shop in north Lancashire in the late 1930s, with a
three piece suite as a prize for the first correct solution.
Evelyn August. The Black-Out Book. Harrap, London, 1939. The eternal triangle, pp. 64 &
213. Take a triangle, ABC, with midpoints a, b, c, opposite A, B, C. Take a point d
between a and B. Draw Aa, ab, bc, ca, bd, cd. How many triangles? Answer is
given as 24, but I (and my program) find 27 and others have confirmed this.
Anon. Test your eyes. Mathematical Pie 7 (Oct 1952) 51. Reproduced in: Bernard Atkin,
ed.; Slices of Mathematical Pie; Math. Assoc., Leicester, 1991, pp. 15 & 71 (not
paginated - I count the TP as p. 1). Triangular pattern with 2 triangles on a side, with
the three altitudes drawn. Answer is 47 'obtained by systematic counting'. This is
correct. Cf Hancox, 1978.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. How many triangles,
pp. 43 & 130. Take a pentagon and draw the pentagram inside it. In the interior
pentagon, draw another pentagram. How many triangles are there? Answer is 85.
Young World. c1960. P. 57: One for Pythagoras. Consider a L-tromino. Draw all the
midlines to form 12 unit squares. Or take a 4 x 4 square array and remove a 2 x 2
array from a corner. Now draw the two main diagonals of the 4 x 4 square - except
half of one diagonal would be outside our figure. How many triangles and how many
squares are present? Gives correct answers of 26 & 17.
J. Halsall. An interesting series. MG 46 (No. 355) (Feb 1962) 55-56. Larsen (below) says he
seems to be the first to count the triangles in the triangular pattern with n on a side, but
he does not give any proof.
Although there are few references before this point, the puzzle idea was pretty well known
and occurs regularly. E.g. in the children's puzzle books of Norman Pulsford which
start c1965, he gives various irregular patterns and asks for the number of triangles or
squares.
J. E. Brider. A mathematical adventure. MTg (1966) 17-21. Correct derivation for the
number of triangles in a triangle. This seems to be the first paper after Halsall but is not
in Larsen.
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/12,
pp. 23 & 75. Consider an isosceles right triangle with legs along the axes from (0,0) to
(4,0) and (0,4). Draw the horizontals and verticals through the integer lattice points,
except that the lines through (1,1) only go from the legs to this point and stop. Draw
the diagonals through even-integral lattice points, e.g. from (2,0) to (0,2). How many
triangles. Says he found 27, but his secretary then found 29. I find 29.
Ripley's Puzzles and Games. 1966. Pp. 72-73 have several problems of counting triangles.
Item 3. Consider a Star of David with the diameters of its inner hexagon drawn. How
many triangles are in it? Answer: 20, which I agree with.
Item 4. Consider a 3 x 3 array of squares with their diagonals drawn. How many
triangles are there? Answer: 150, however, there are only 124.
Item 5. Consider five squares, with their midlines and diagonals drawn, formed into a
Greek cross. How many triangles are there? Answer: 104, but there are 120.
Doubleday - 2. 1971. Count down, pp. 127-128. How many triangles in the pentagram (i.e. a
pentagon with all its diagonals)? He says 35.
Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Triangular,
pp. 27 & 63. Count triangles in an irregular pattern.
[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. 1973. Op. cit. in 5.E. An
unusual star, pp. 49-50. Consider a pentagram and draw lines from each star point
through the centre to the opposite crossing point. How many triangles? They say 110.
[Henry] Joseph and Lenore Scott. Master Mind Pencil Puzzles. 1973. Op. cit. in 5.R.4.
Diamonds are forever, pp. 35-36. Hexagon with Star of David inside and another Star
of David in the centre of that one. How many triangles? Answer is 76.
Count the triangles, pp. 55-56. Ordinary Greek cross of five squares, with all the
diagonals and midlines of the five squares drawn. How many triangles> Answer
SOURCES - page 150
is 104.
C. P. Chalmers. Note 3353: More triangles. MG 58 (No. 403) (Mar 1974) 52-54. How
many triangles are determined by N points lying on M lines? (Not in Larsen.)
Nicola Davies. The 2nd Target Book of Fun and Games. Target (Universal-Tandem),
London, 1974. Squares and triangles, pp. 18 & 119. Consider a chessboard of 4 x 4
cells. Draw all the diagonals, except the two main ones. How many squares and how
many triangles?
Shakuntala Devi. Puzzles to Puzzle You. Op. cit. in 5.D.1. 1976. Prob. 136: The triangles,
pp. 85 & 133. How many triangles in a Star of David made of 12 equilateral triangles?
Michael Holt. Figure It Out -- Book Two. Granada, London, 1978. Prob. 67, unpaginated.
How many triangles in a Star of David made of 12 equilateral triangles?
Putnam. Puzzle Fun. 1978. No. 91: Counting triangles, pp. 12 & 37. Same as Doubleday 2.
D. J. Hancox, D. J. Number Puzzles For all The Family. Stanley Thornes, London, 1978.
Puzzle 8, pp. 2 & 47. Draw a line with five points on it, say A, B, C, D, E, making
four segments. Connect all these points to a point F on one side of the line and
to a point G on the other side of the line, with FCG collinear. How many
triangles are there? Answer is 24, which is correct.
Puzzle 53, pp. 24 & 54. Same as Anon.; Test Your Eyes, 1952. Answer is 36, but
there are 47.
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley,
Middlesex, 1984.
Problem 40, with Solution at the back of the book. Same as Doubleday - 2.
Problem 116, with Solution at the back of the book. Count the triangles in a 'butterfly'
pattern.
Sue Macy. Mad Math. The Best of DynaMath Puzzles. Scholastic, 1987. (Taken from
Scholastic's DynaMath magazine.) Shape Up, pp. 5 & 56.
Take a triangle, trisect one edge and join the points of trisection to the opposite
vertex. How many triangles? [More generally, if one has n points on a line and joins
them all to a vertex, there are 1 + 2 + ... + n-1 = n(n-1)/2 triangles.]
Take a triangle, join up the midpoints of the edges, giving four smaller triangles,
and draw one altitude of the original triangle. How many triangles?
1980 Celebration of Chinese New Year Contest Problem No. 5; solution by Leroy F. Meyers.
CM 17 (1991) 2 & 18 (1992) 272-273. n x n array of squares with all diagonals
drawn. Find the number of isosceles right triangles. [Has this also been done in half
the diagram? That is, how many isosceles right triangles are in the isosceles right
triangle with legs going from (0,0) to (n,0) and (0,n) with all verticals, horizontals
and diagonals through integral points drawn?]
Mogens Esrom Larsen. The eternal triangle -- a history of a counting problem. Preprint,
1988. Surveys the history from Halsall on. The problem was proposed at least five
times from 1962 and solved at least ten times. I have sent him the earlier references.
Marjorie Newman. The Christmas Puzzle Book. Hippo (Scholastic Publications), London,
1990. Star time, pp. 26 & 117. Consider a Star of David formed from 12 triangles,
but each of the six inner triangles is subdivided into 4 triangles. How many triangles
in this pattern? Answer is 'at least 50'. I find 58.
Erick Gooding. Polygon counting. Mathematical Pie No. 131 (Spring 1994) 1038 & Notes,
pp. 1-2. Consider the pentagram, i.e. the pentagon with its diagonals drawn. How
many triangles, quadrilaterals and pentagons are there? Gets 35, 25, 92, with some
uncertainty whether the last number is correct.
When F. R. Gill (See Pearson and Collins above) mentioned the problem of counting the
triangles in the figure with all the altitudes drawn, I decided to try to count them myself
for the figure with N intervals on each side. The theoretical counting soon gets really
messy and I adapted my program for counting triangles in a figure (developed to verify
the number found for August's problem). However, the number of points involved soon
got larger than my simple Basic could handle and I rewrote the program for this special
case, getting the answers of 653 and 1196 and continuing to N = 22. I expected the
answers to be like those for the simpler triangle counting problem so that there would
be separate polynomials for the odd and even cases, or perhaps for different cases (mod
3 or 4 or 6 or 12 or ??). However, no such pattern appeared for moduli 2, 3, 4 and I did
not get enough data to check modulus 6 or higher. I communicated this to Torsten
Sillke and Mogens Esrom Larsen. Sillke has replied with a detailed answer showing
SOURCES - page 151
that the relevant modulus is 60! I haven't checked through his work yet to see if this is
an empirical result or he has done the theoretical counting.
Heather Dickson, Heather, ed. Mind-Bending Challenging Optical Puzzles. Lagoon Books,
London, 1999, pp. 40 & 91. Gives the version m = n = 4 of the following. I have seen
other versions of this elsewhere, but I found the general solution on 4 Jul 2001 and am
submitting it as a problem to AMM.
Consider a triangle ABC. Subdivide the side AB into m parts by inserting
m-1 additional points. Connect these points to C. Subdivide the side AC into n
parts by inserting n-1 additional points and connect them to B. How many triangles
are in this pattern? The number is [m2n + mn2]/2. When m = n, we get n3, but I
cannot see any simple geometric interpretation for this.
5.X.2.COUNTING RECTANGLES OR SQUARES
I have just seen M. Adams. There are probably earlier examples of these types of
problems.
Anon. Prob. 63. Hobbies 30 (No. 778) (10 Sep 1910) 488 & 31 (No. 781) (1 Oct 1910) 2.
How many rectangles on a 4 x 4 chessboard? Solution says 100, which is correct, but
then says they are of 17 different types -- I can only get 16 types.
Blyth. Match-Stick Magic. 1921. Counting the squares, p. 47. Count the squares on a 4 x 4
chessboard made of matches with an extra unit square around the central point. The
extra unit square gives 5 additional squares beyond the usual 1 + 4 + 9 + 16.
King. Best 100. 1927. No. 9, pp. 10 & 40. = Foulsham's, no. 5, pp. 6 & 10. 4 x 4 board
with some diagonals yielding one extra square.
Loyd Jr. SLAHP. 1928. How many rectangles?, pp. 80 & 117. Asks for the number of
squares and rectangles on a 4 x 4 board (i.e. a 5 x 5 lattice of points). Says answers
are 1 + 4 + 9 + 16 and (1 + 2 + 3 + 4)2 and that these generalise to any size of board.
SOURCES - page 152
M. Adams. Puzzles That Everyone Can Do. 1931.
oo
Prob. 37, pp. 22 & 134: 20 counter problem. Given the pattern of
oo
20 counters at the tight, 'how many perfect squares are
oooooo
contained in the figure.' This means having their vertices
oooooo
at counters. There are surprisingly more than I expected.
oo
Taking the basic spacing as one, one can have squares of
oo
edge 1, 2, 5, 8, 13, giving 21 squares in all.
He then asks how many counters need to be removed in order to destroy all
the squares? He gives a solution deleting six counters.
Prob. 217, pp. 83 & 162: Match squares. He gives 10 matches making a row of three
equal squares and asks you to add 14 matches to form 14 squares. The answer is
to make a 3 x 3 array of squares and count all of the squares in it.
J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?].
Counting the squares, pp. 84-85. As in Blyth.
Indoor Tricks and Games. Success Publishing, London, nd [1930s??]. Square puzzle, p. 62.
Start with a square and draw its diagonals and midlines. Join the midpoints of the sides
to form a second level square inscribed in the first level original square. Repeat this
until the 9th level. How many squares are there? Given answer is 16, but in my copy
someone has crossed this out and written 45, which seems correct to me.
Meyer. Big Fun Book. 1940. No. 9, pp. 162 & 752. Draw four equidistant horizontal lines
and then four equidistant verticals. How many squares are formed? This gives a 3 x 3
array of squares, but he counts all sizes of squares, getting 9 + 4 + 1 = 14. (Also in
7.AU.)
Foulsham's New Party Book. Foulsham, London, nd [1950s?]. P. 103: How many squares?
4 x 4 board with some extra diagonals giving one extra square.
Although there are few references before this point, the puzzle idea was pretty well known
and occurs regularly in the children's puzzle books of Norman Pulsford which start
c1965. He gives various irregular patterns and asks for the number of triangles or
squares.
Jonathan Always. Puzzles to Puzzle You. Op. cit. in 5.K.2. 1965. No. 140: A surprising
answer, pp. 43 & 90. 4 x 4 chessboard with four corner cells deleted. How many
rectangles are there?
Anon. Puzzle page: Strictly for squares. MTg 30 (1965) 48 & 31 (1965) 39 & 32 (1965)
39. How many squares on a chessboard? First solution gets
S(8) = 1 + 4 + 9 + ... + 64 = 204. Second solution observes that there are skew squares
if one thinks of the board as a lattice of points and this gives
S(1) + S(2) + ... + S(8) = 540 squares.
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966.
Prob. 2/11, pp. 22 & 74. 4 x 4 array of squares bordered on two sides by bricks 1 x 2,
1 x 3, 2 x 1, 2 x 1. Count the squares and the rectangles. Gets 35 and 90.
Prob. 2/14, pp. 23 & 75. Pattern of squares making the shape of a person -- how many
squares in it?
Ripley's Puzzles and Games. 1966. Pp. 72-73 have several problems of counting squares.
Item 4. Consider a 3 x 3 array of squares with their diagonals drawn. The solution
says this has 30 squares. I get 31, but perhaps they weren't counting the whole
figure. I have computed the total number of squares for an n x n array and get
(2n3 + n2)/2 squares for n even and (2n3 + n2 -1)/2 squares for n odd.
Unnumbered item at lower right of p. 73. 4 x 4 array of squares with their diagonals
drawn, except that the four corner squares have only one diagonal -- the one not
pointing to an opposite corner -- and this reduces the number of squares by eight,
agreeing with the given answer of 64.
Doubleday - 2. 1971. Bed of nails, pp. 129-130. 20 points in the form of a Greek cross with
double-length arms (so that the axes are five times the width of the central square, or the
shape is a 9-omino). How many squares can be located on these points? He finds 21.
W. Antony Broomhead. Note 3315: Two unsolved problems. MG 55 (No. 394) (Dec 1971)
438. Find the number of squares on an n x n array of dots, i.e. the second problem in
MTg (1965) above, and another problem.
W. Antony Broomhead. Note 3328: Squares in a square lattice. MG 56 (No. 396) (May
1972) 129. Finds there are n2(n2 - 1)/12 squares and gives a proof due to John Dawes.
Editorial note says the problem appears in: M. T. L. Bizley; Probability: An
Intermediate Textbook; CUP, 1957, ??NYS. A. J. Finch asks the question for cubes.
SOURCES - page 153
Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Squares, pp. 26
& 63. Same as Briggs.
Nicola Davies. The 2nd Target Book of Fun and Games. 1974. See entry in 5.X.1.
Putnam. Puzzle Fun. 1978.
No. 107: Square the coins, pp. 17 & 40. 20 points in the form of a Greek cross made
from five 2 x 2 arrays of points. How many squares -- including skew ones?
Gets 21.
No. 108: Unsquaring the coins, pp. 17 & 40. How many points must be removed from
the previous pattern in order to leave no squares? Gets 6.
5.X.3.COUNTING HEXAGONS
M. Adams. Puzzle Book. 1939. Prob. C.157: Making hexagons, pp. 163 & 190. The
hexagon on the triangular lattice which is two units along each edge contains 8
hexagons. [It is known that the hexagon of side n contains n3 hexagons. I recently
discovered this but have found that it is known, though I don't know who discovered it
first.]
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley,
Middlesex, 1984. Problem 32, with Solution at the back of the book. Count the
hexagons in the hexagon of side three on the triangular lattice. They get 27.
5.X.4.COUNTING CIRCLES
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/13,
pp. 23 & 75. Pattern with hexagonal symmetry and lots of overlapping circles, some
incomplete.
5.Y. NUMBER OF ROUTES IN A LATTICE
The common earlier form was to have the route spell a word or phrase from the centre
to the boundaries of a diamond. I will call this a word diamond. Sometimes the phrase is a
palindrome and one reads to the centre and then back to the edge. See Dudeney, CP, for
analysis of the most common cases. I have seen such problems on the surface of a 3 x 3 x 3
cube. The problems of counting Euler or Hamiltonian paths are related questions, but dealt
with under 5.E and 5.F.
New section -- in view of the complexity of the examples below, there must be older,
easier, versions, but I have only found the few listed below. The first entry gives some ancient
lattices, but there is no indication that the number of paths was sought in ancient times.
Roger Millington. The Strange World of the Crossword. M. & J. Hobbs, Walton-on-Thames,
UK, 1974. (This seems to have been retitled: Crossword Puzzles: Their History and
Cult for a US ed from Nelson, NY.)
On pp. 38-39 & 162, he gives the cabalistic triangle shown below and says it is
thought to have been constructed from the opening letters of the Hebrew words Ab
(Father), Ben (Son), Ruach Acadash (Holy Spirit). He then asks how many ways one
can read ABRACADABRA in it, though there is no indication that the ancients did
this. His answer is 1024 which is correct.
A B R A C A D A B R A
A B R A C A D A B R
A B R A C A D A B
A B R A C A D A
A B R A C A D
A B R A C A
A B R A C
A B R A
A B R
A B
A
On pp. 39-40 he describes and illustrates an inscription on the Stele of Moschion
SOURCES - page 154
from Egypt, c300. This is a 39 x 39 square with a Greek text from the middle to the
corner, e.g. like the example in the following entry. The text reads:
ΟΥIΡIΔIΜΟΥΧIΩΝΥΓIΑΥΘΕIΥΤΟΝΠΟΔΑIΑΤΠΕIΑ
I Υ which means: Moschion to Osiris, for the treatment which cured his foot.
Millington does not ask for the number of ways to read the inscription, which is 4
BC(38,19) = 14 13810 55200.
Curiosities for the Ingenious selected from The most authentic Treasures of
EDCDE
Nature, Science and Art, Biography, History, and General Literature.
DCBCD
(1821); 2nd ed., Thomas Boys, London, 1822. Remarkable epitaph,
CBABC
p. 97. Word diamond extended to a square, based on 'Silo Princeps Fecit', D C B C D
with the ts at the corners. An example based on 'ABCDEF' is shown
EDCDE
at the right. Says this occurs on the tomb of a prince named Silo at the
entrance of the church of San Salvador in Oviedo, Spain. Says the epitaph can be read
in 270 ways. I find there are 4 BC(16, 8) = 51490 ways.
In the churchyard of St. Mary's, Monmouth, is the gravestone of John Rennie, died 31 May
1832, aged 33 years. This has the inscription shown below. Further down the stone it
gives his son's name as James Rennie. Apparently an N has been dropped to get a
message with an odd number of letters. I have good photos. Nothing asks for the
number of ways of reading the inscription. I get 4 BC(16,9) = 45760 ways.
eineRnhoJsJohnRenie
ineRnhoJsesJohnReni
neRnhoJseiesJohnRen
eRnhoJseiliesJohnRe
RnhoJseileliesJohnR
nhoJseilereliesJohn
hoJseilerereliesJoh
oJseilereHereliesJo
hoJseilerereliesJoh
nhoJseilereliesJohn
RnhoJseileliesJohnR
eRnhoJseiliesJohnRe
neRnhoJseiesJohnRen
ineRnhoJsesJohnReni
eineRnhoJsJohnRenie
Nuts to Crack I (1832), no. 200. The example from Curiosities for the Ingenious with
'SiloPrincepsFecit', but no indication of what is wanted -- perhaps it is just an amusing
picture.
W. Staniforth. Letter. Knowledge 16 (Apr 1893) 74-75. Considers
1 2 3 4 5 6
"figure squares" as at the right. "In how many different ways may
2 3 4 5 6 7
the figures in the square be read from 1 to 11 consecutively?"
3 4 5 6 7 8
He computes the answers for the n x n case for the first few
4 5 6 7 8 9
cases and finds a recurrence. "Has such a series of numbers any
5 6 7 8 9 10
mathematical designation?" The editor notes that he doesn't
6 7 8 9 10 11
know.
J. J. Alexander. Letter. Knowledge 16 (May 1893) 89. Says Staniforth's numbers are the
sums of the squares of the binomial coefficients BC(n, k), the formula for which is
BC(2n, n). Editor say he has received more than one note pointing this out and cites a
paper on such figure squares by T. B. Sprague in the Transactions of the Royal Society
of Edinburgh -- ??NYS, no more details provided.
Loyd. Problem 12: The temperance puzzle. Tit-Bits 31 (2 & 23 Jan 1897) 251 & 307. Red
rum & murder. = Cyclopedia, 1914, The little brown jug, pp. 122 & 355. c= MPSL2,
no. 61, pp. 44 & 141. Word diamond based on 'red rum & murder', i.e. the central line
is redrum&murder. He allows a diagonal move from an E back to an inner R and
this gives 372 paths from centre to edge, making 3722 = 138,384 in total.
Dudeney. Problem 57: The commercial traveller's puzzle. Tit-Bits 33 (30 Oct & 20 Nov
1897) 82 & 140. Number of routes down and right on a 10 x 12 board. Gives a
general solution for any board.
Dudeney. A batch of puzzles. The Royal Magazine 1:3 (Jan 1899) 269-274 & 1:4
SOURCES - page 155
(Feb 1899) 368-372. A "Reviver" puzzle. Complicated pattern based on 'reviver'.
544 solutions.
Dudeney. Puzzling times at Solvamhall Castle. London Magazine 7 (No. 42) (Jan 1902)
580-584 & 8 (No. 43) (Feb 1902) 53-56. The amulet. 'Abracadabra' in a triangle with
A at top, two B's below, three R's below that, etc. Answer: 1024. = CP, 1907,
No. 38, pp. 64-65 & 190. CF Millington at beginning of this section.
Dudeney. CP. 1907.
Prob. 30: The puzzle of the canon's yeoman, pp. 55-56 & 181-182. Word diamond
based on 'was it a rat I saw'. Answer is 63504 ways. Solution observes that for
a diamond of side n+1, with no diagonal moves, the number of routes from the
centre to an edge is 4(2n-1) and the number of ways to spell the phrase is this
number squared. Analyses four types with the following central lines:
A - 'yoboy'; B - 'level'; C - 'noonoon'; D - 'levelevel'.
In A, one wants to spell 'boy', so there are 4(2n-1) solutions.
In B, one wants to spell 'level' and there are [4(2n-1)]2 solutions.
In C, one wants to spell 'noon' and there are 8(2n-1) solutions.
In D, one wants to spell 'level' and there are complications as one can start
and finish at the edge. He obtains a general formula for the number of ways.
Cf Loyd, 1914.
Prob. 38: The amulet, pp. 64-65 & 190. See: Dudeney, 1902.
Pearson. 1907. Part II: A magic cocoon, p. 147. Word diamond based on 'cocoon', so the
central line is noocococoon. Because one can start at the non-central Cs, and can go
in as well as out, I get 948 paths. He says 756.
Loyd. Cyclopedia. 1914. Alice in Wonderland, pp. 164 & 360. = MPSL1, no. 109, pp. 107
& 161-162. Word diamond based on 'was it a cat I saw'. Cf Dudeney, 1907.
Dudeney. AM. 1917.
Prob. 256: The diamond puzzle, pp. 74 & 202. Word diamond based on
'dnomaidiamond'. This is type A of his discussion in CP and he states the
general formula. 252 solutions.
Prob. 257: The deified puzzle, pp. 74-75 & 202. Word diamond based on
'deifiedeified'. This is type D in CP and has 1992 solutions. He says
'madamadam' gives 400 and 'nunun' gives 64, while 'noonoon' gives 56.
Prob. 258: The voter's puzzle, pp. 75 & 202. Word diamond built on 'rise to vote sir'.
Cites CP, no. 30, for the result, 63504, and the general formula.
Prob. 259: Hannah's puzzle, pp. 75 & 202. 6 x 6 word square based on 'Hannah' with
Hs on the outside, As adjacent to the Hs and four Ns in the middle. Diagonal
moves allowed. 3468 ways.
Wood. Oddities. 1927. Prob. 44: The amulet problem, p. 39. Like the original
ABRACADABRA triangle, but with the letters in reverse order.
Collins. Book of Puzzles. 1927. The magic cocoon puzzle, pp. 169-170. As in Pearson.
Loyd Jr. SLAHP. 1928. A strolling pedagogue, pp. 38 & 97. Number of routes to opposite
corner of a 5 x 5 array of points.
D. F. Lawden. On the solution of linear difference equations. MG 36 (No. 317) (Sep 1952)
193-196. Develops use of integral transforms and applies it to find that the number of
king's paths going down or right or down-right from (0, 0) to (n, n) is Pn(3) where
Pn(x) is the Legendre polynomial.
Leo Moser. King paths on a chessboard. MG 39 (No. 327) (Feb 1955) 54. Cites Lawden and
gives a simpler proof of his result Pn(3).
Anon. Puzzle Page: Check this. MTg 36 (1964) 61 & 27 (1964) 65. Find the number of
king's routes from corner to corner when he can only move right, down or right-down.
Gets 48,639 routes on 8 x 8 board.
Ripley's Puzzles and Games. 1966. P. 32. Word diamond laid out differently so
AAA
one has to read from one side to the opposite side. Rotating by 45o, one gets
BB
the pattern at the right for edge three. One wants the number of ways to
CCC
read ABCDEF. In general, when the first line of As has n positions,
DD
the total number of ways to reach the first row is n. For each successive
EEE
row, the total number is alternately twice the number for the previous row less
FF
twice the end term of that row or just twice the the number for the previous
row. In our example with n = 3, the number of ways to reach the second row is
4 = 2x3 - 2x1. The number of ways to reach the third row is 8 = 2x4. The number of
ways to reach the fourth row is 12 = 2x8 - 2x2, then we get 24 = 2 x 12;
SOURCES - page 156
36 = 2x24 - 2x6. It happens that the first n end terms are the central binomial
coefficients BC(2k,k), so this is easy to calculate. I find the total number of routes, for
n = 2, 3, ..., 7, is 4, 18, 232, 1300, 6744, 33320, the last being the desired and
given answer for the given problem.
Pál Révész. Op. cit. in 5.I.1. 1969. On p. 27, he gives the number of routes for a king
moving forward on a chessboard and a man moving forward on a draughtsboard.
Putnam. Puzzle Fun. 1978. No. 8: Level - level, pp. 3 & 26. Form a wheel of 16 points
labelled LEVELEVELEVELEVE. Place 4 Es inside, joined to two consecutive Vs
and the intervening L. Then place a V in the middle, joined to these four Es. How
many ways to spell LEVEL? He gets 80, which seems right.
5.Z. CHESSBOARD PLACING PROBLEMS
See MUS I 285-318, some parts of the previous chapter and the Appendix in II 351360. See also 5.I.1, 6.T.
There are three kinds of domination problems.
In strong domination, a piece dominates the square it is on.
In weak domination, it does not, hence more pieces may be needed to dominate
the board.
Non-attacking domination is strong domination with no piece attacking another.
Graph theorists say the pieces are independent. This also may require more pieces than strong
domination, but it may require more or fewer pieces than weak domination.
The words 'guarded' or 'protected' are used for weak domination, but 'unguarded' or
'unprotected' may mean either strong or non-attacking domination.
Though these results seem like they must be old, the ideas seem to have originated with
the eight queens problem, c1850, (cf 5.I.1) and to have been first really been attacked in the
late 19C. There are many variations on these problems, e.g. see Ball, and I will not attempt to
be complete on the later variations. In recent years, this has become a popular subject in
graph theory, where the domination number, γ(G), is the size of the smallest strongly
dominating set on the graph G and the independent domination number, i(G), is the size of
the smallest non-attacking (= independent) dominating set.
Mario Velucchi has a web site devoted to the non-dominating queens problem and
related sites for similar problems. See: http://www.bigfoot.com/~velucchi/papers.html and
http://www.bigfoot.com/~velucchi/biblio.html.
Ball. MRE, 3rd ed., 1896, pp. 109-110: Other problems with queens. Says: "Captain Turton
has called my attention to two other problems of a somewhat analogous character,
neither of which, as far as I know, has been hitherto published, ...." These ask for ways
to place queens so as to attack as few or as many cells as possible -- see 5.Z.2.
Ball. MRE, 4th ed., 1905, pp. 119-120: Other problems with queens; Extension to other
chess pieces. Repeats above quote, but replaces 'hitherto published' by 'published
elsewhere', extends the previous text and adds the new section.
Ball. MRE, 5th ed., 1911. Maximum pieces problem; Minimum pieces problem,
pp. 119-122. [6th ed., 1914 adds that Dudeney has written on these problems in The
Weekly Dispatch, but this is dropped in the 11th ed. of 1939.] Considerably generalizes
the problems. On the 8 x 8 board, the maximum number of non-attacking kings is 16,
queens is 8, bishops is 14 [6th ed., 1914, adds there are 256 solutions], knights is 32
with 2 solutions and rooks is 8 with 88 solutions [sic, but changed to 8! in the 6th
ed.]. The minimum number of pieces to strongly dominate the board is 9 kings, 5
queens with 91 inequivalent solutions [the 91 is omitted in the 6th ed., since it is stated
later], 8 bishops, 12 knights, 8 rooks. The minimum number of pieces to weakly
dominate the board is 5 queens, 10 bishops, 14 knights, 8 rooks.
Dudeney. AM. 1917. The guarded chessboard, pp. 95-96. Discusses different ways pieces
can weakly or non-attackingly dominate n x n boards.
G. P. Jelliss. Multiple unguard arrangements. Chessics 13 (Jan/Jun 1982) 8-9. One can have
16 kings, 8 queens, 14 bishops, 32 knights or 8 rooks non-attackingly placed on a
8 x 8 board. He considers mixtures of pieces -- e.g. one can have 10 kings and 4
queens non-attacking. He tries to maximize the product of the numbers of each type in
a mixture -- e.g. scoring 40 for the example.
5.Z.1. KINGS
SOURCES - page 157
Ball. MRE, 4th ed., 1905. Other problems with queens; Extension to other chess pieces,
pp. 119-120. Says problems have been proposed for k kings on an n x n, citing
L'Inter. des math. 8 (1901) 140, ??NYS.
Gilbert Obermair. Denkspiele auf dem Schachbrett. Hugendubel, Munich, 1984. Prob. 27,
pp. 29 & 58. 9 kings strongly, and 12 kings weakly, dominate an 8 x 8 board.
5.Z.2. QUEENS
Here the graph is denoted Qn, but I will denote γ(Qn) by γ(n) and i(Qn) by i(n).
Murray. Pp. 674 & 691. CB249 (c1475) shows 16 queens weakly dominating an 8 x 8
board, but the context is unclear to me.
de Jaenisch. Op. cit. in 5.F.1. Vol. 3, 1863. Appendice, pp. 244-271. Most of this is due to
"un de nos anciens amis, Mr de R***". Finds and describes the 91 ways of placing 5
queens so as to non-attackingly dominate the 8 x 8 board. Then considers the n x n
board for n = 2, ..., 7 with strong and non-attacking domination. Up through 5, he
gives the number of pieces being attacked in each solution which allows one to
determine the weak solutions. For n < 6, he gets the answers in the table below, but
for n = 6, he gets 21 non-attacking solutions instead of 17?.
Ball. MRE, 3rd ed., 1896. Other problems with queens, pp. 109-110. "Captain [W. H.]
Turton has called my attention to two other problems of a somewhat analogous
character, neither of which, as far as I know, has been hitherto published, or solved
otherwise than empirically." The first is to place 8 queens so as to strongly dominate
the fewest squares. The minimum he can find is 53. (Cf Gardner, 1999.) The second is
to place m queens, m  5, so as to strongly dominate as many cells as possible. With
4 queens, the most he can find is 62.
Dudeney. Problem 54: The hat-peg puzzle. Tit-Bits 33 (9 & 30 Oct 1897) 21 & 82.
Problem involves several examples of strong domination by 5 queens on an 8 x 8
board leading to a non-attacking domination. He says there are just 728 such. This = 8
x 91. = Anon. & Dudeney; A chat with the Puzzle King; The Captain 2 (Dec? 1899)
314-320; 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89. = AM; 1917; pp. 93-94 &
221.
Ball. MRE, 4th ed., 1905, loc. cit. in 5.Z.1. Extends 3rd ed. by asking for the minimum
number of queens to strongly dominate a whole n x n board. Says there seem to be 91
ways of having 5 non-attacking queens on the 8 x 8, citing L'Inter. des Math. 8 (1901)
88, ??NYS.
Ball. MRE, 5th ed., 1911, loc. cit. in 5.Z. On pp. 120-122, he considers queens and states the
minimum numbers of queens required to strongly dominate the board and the numbers
of inequivalent solutions for 2 x 2, 3 x 3, ..., 7 x 7, citing the article cited in the 4th
ed. and Jaenisch, 1862, without a volume number. For n = 7, he gives the same unique
solution for strongly dominating as for non-attacking dominating. [In the 6th ed., this is
corrected and he says it is a solution.] He says Jaenisch also posed the question of the
minimum number of non-attacking queens to dominate the board and gives the numbers
and the number of inequivalent ways for the 4 x 4, .., 8 x 8, except that he follows
Jaenisch in stating that there are 21 solutions on the 6 x 6. [This is changed to 17 in the
6th ed.]
Dudeney. AM. 1917. Loc. cit. in 5.Z. He uses 'protected' for 'weakly', but he seems to copy
the values for 'strongly' from Jaenisch or Ball. His 'not protected' seems to mean 'nonattacking'. However, some values are different and I consequently am very uncertain as
to the correct values??
Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 24-25, he shows 5 queens are sufficient to
strongly dominate the board and says this is minimal.
Below, min. denotes the minimum number of queens to dominate and no. is the number of
inequivalent ways to do so.
SOURCES - page 158
STRONG WEAK NON-ATTACKING
min. no. min. no. min. no.
n
1
2
3
4
5
6
7
8
1
1
1
1
1
1
2
3
3 37
3
1
4
5 150
0
2
2
2
3
4
4
5
0
2
5
3
15
2
5?
41
1
1
1
1
1
1
3
2
3
2
4 17?
4
1
5 91
Rodolfo Marcelo Kurchan, proposer; Henry Ibstedt & proposer, solver. Prob. 1738 -- Queens
in space. JRM 21:3 (1989) 220 & 22:3 (1989) 237. How many queens are needed to
strongly dominate an n x n x n cubical board? For n = 3, 4, ..., 9, the best known
numbers are: 1, 4, 6, 8, 14, 20, 24. The solution is not clear if these are minimal, but it
seems to imply this.
Martin Gardner. Chess queens and maximum unattacked cells. Math Horizons (Nov 1999).
Reprinted in Workout, chap. 34. Considers the problem of Turton described in Ball,
3rd ed, above: place 8 queens on an 8 x 8 board so as to strongly dominate the fewest
squares. That is, leave the maximum number of unattacked squares. More generally,
place k queens on an n x n board to leave the maximum number of unattacked
squares. He describes a simple problem by Dudeney (AM, prob. 316) and recent work
on the general problem. He cites Velucchi, cf below, who provides the following table
of maximum numbers of unattacked cells and number of solutions for the maximum.
I'm not sure if some of these are still only conjectured.
n
Max
Sols
1
0
0
2
0
0
3
0
0
4
1
25
5
3
1
6
5
3
7
8
7 11
38 7
9
18
1
10
22
1
11
30
2
12
36
7
13
47
1
14
56
4
15
72
3
16
82
1
17
97
Mario Velucchi has a web site devoted to the non-dominating queens problem and related
sites for similar problems. See: http://www.bigfoot.com/~velucchi/papers.html and
http://www.bigfoot.com/~velucchi/biblio.html.
A. P. Burger & C. M. Mynhardt. Symmetry and domination in queens graphs. Bull. Inst.
Combinatorics Appl. 29 (May 2000) 11-24. Extends results to n = 30, 45, 69, 77.
Summarizes the field, with 14 references, several being earlier surveys. The table
below gives all known values. It will be seen that the case n = 4k + 1 seems easiest to
deal with. The values separated by strokes, /, indicate cases where the value is one of
the two given values, but it is not known which.
n
γ(n)
i(n)
4
2
3
5
3
3
6
3
4
7
4
4
8
5
5
9
5
5
10
5
5
11
5
5
12
6
7
13
7
7
14
7/8
8
15
8/9
9
16
8/9
9
17
9
9
n
γ(n)
i(n)
18
9
19
10
21
11
11
25
13
13
29
15
30
15
31
16
33
17
17
37
19
41
21
45
23
23
49
25
53
27
57
29
n
γ(n)
61
31
69
35
77
39
5.Z.3.BISHOPS
Dudeney. AM. 1917. Prob. 299: Bishops in convocation, pp. 89 & 215. There are 2n ways
to place 2n-2 bishops non-attackingly on an n x n board. At loc. cit. in 5.Z, he says
that for n = 2, ..., 8, there are 1, 2, 3, 6, 10, 20, 36 inequivalent placings.
Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 25-26, he shows the maximum number of
non-attacking bishops on one colour is 7 and there are 16 ways to place them.
Obermair. Op. cit. in 5.Z.1. 1984. Prob. 17, pp. 23 & 50. 8 bishops strongly, and 10 bishops
SOURCES - page 159
weakly, dominate the 8 x 8 board.
5.Z.4.KNIGHTS
Ball. MRE, 4th ed., 1905. Loc. cit. in 5.Z.1. Says questions as to the maximum number of
non-attacking knights and minimum number to strongly dominate have been
considered, citing L'Inter. des math. 3 (1896) 58, 4 (1897) 15-17 & 254, 5 (1898) 87
[5th ed. adds 230-231], ??NYS.
Dudeney. AM. 1917. Loc. cit. in 5.Z. Notes that if n is odd, one can have (n2+1)/2
non-attacking knights in one way, while if n is even, one can have n2/2 in two
equivalent ways.
Irving Newman, proposer; Robert Patenaude, Ralph Greenberg and Irving Newman, solvers.
Problem E1585 -- Nonattacking knights on a chessboard. AMM 70 (1963) 438 & 71
(1964) 210-211. Three easy proofs that the maximum number of non-attacking knights
is 32. Editorial note cites Dudeney, AM, and Ball, MRE, 1926, p. 171 -- but the
material is on p. 171 only in the 11th ed., 1939.
Gardner. SA (Oct 1967, Nov 1967 & Jan 1968) c= Magic Show, chap. 14. Gives
Dudeney's results for the 8 x 8. Golomb has noted that Greenberg's solution of E1585
via a knight's tour proves that there are only two solutions. For the k x k board,
k = 3, 4, ..., 10, the minimal number of knights to strongly dominate is:
4, 4, 5, 8, 10, 12, 14, 16. He says the table may continue: 21, 24, 28, 32, 37. Gives
numerous examples.
Obermair. Op. cit. in 5.Z.1. 1984. Prob. 16, pp. 21 & 47. 14 knights are necessary for weak
domination of the 8x8 board.
E. O. Hare & S. T. Hedetniemi. A linear algorithm for computing the knight's domination
number of a k x n chessboard. Technical report 87-May-1, Dept. of Computer
Science, Clemson University. 1987?? Pp. 1-2 gives the history from 1896 and Table 2
on p. 13 gives their optimal results for strong domination on k x n boards, 4  k  9,
k  n  12 and also for k = n = 10. For the k x k board, k = 3, ..., 10, they confirm
the results in Gardner.
Anderson H. Jackson & Roy P. Pargas. Solutions to the N x N knight's cover problem. JRM
23:4 (1991) 255-267. Finds number of knights to strongly dominate by a heuristic
method, which finds all solutions up through N = 10. Improves the value given by
Gardner for N = 15 to 36 and finds solutions for N = 16, ..., 20 with
42, 48, 54, 60, 65 knights.
5.Z.5.ROOKS
É. Lucas. Théorie des Nombres. Gauthier-Villars, Paris, 1891; reprinted by Blanchard, Paris,
1958. Section 128, pp. 220-223. Determines the number of inequivalent placings of n
nonattacking rooks on an n x n board in general and gives values for n  12. For
n = 1, ..., 8, there are 1, 1, 2, 7, 23, 115, 694, 5282 inequivalent ways.
Dudeney. AM. 1917. Loc. cit. at 5.Z. Notes there are n! ways to place n non-attacking
rooks and asks how many of these are inequivalent. Gives values for n = 1, ..., 5. AM
prob. 296, pp. 88 & 214, is the case n = 4.
D. F. Holt. Rooks inviolate. MG 58 (No. 404) (Jun 1974) 131-134. Uses Burnside's lemma
to determine the number of inequivalent solutions in general, getting Lucas' result in a
more modern form.
5.Z.6.MIXTURES
Ball. MRE, 5th ed., 1911. Loc. cit. in 5.Z. P. 122: "There are endless similar questions in
which combinations of pieces are involved." 4 queens and king or queen or bishop or
knight or rook or pawn can strongly dominate 8 x 8.
King. Best 100. 1927.
No. 77, pp. 30 & 57. 4 queens and a rook strongly dominate 8 x 8.
No. 78, pp. 30 & 57. 4 queens and a bishop strongly dominate 8 x 8.
SOURCES - page 160
5.AA.
CARD SHUFFLING
New section. I have been meaning to add this sometime, but I have just come across an
expository article, so I am now starting. The mathematics of this gets quite formidable. See
5.AD for a somewhat related topic.
A faro, weave, dovetail or perfect (riffle) shuffle starts by cutting the deck in half and
then interleaving the two halves. When the deck has an even number of cards, there are two
ways this can happen -- the original top card can remain on top (an out shuffle) or it can
become the second card of the shuffled deck (an in shuffle). E.g. if our deck is 123456, then
the out shuffle yields 142536 and the in shuffle yields 415263. Note that removing the first
and last cards converts an out shuffle on 2n cards to an in shuffle on 2n-2 cards. When the
deck has an odd number of cards, say 2n+1, we cut above or below the middle card and
shuffle so the top of the larger pile is on top, i.e. the larger pile straddles the smaller. If the cut
is below the middle card, we have piles of n+1 and n and the top card remains on top, while
cutting above the middle card leaves the bottom card on bottom. Removing the top or bottom
card leaves an in shuffle on 2n cards.
Monge's shuffle takes the first card and then alternates the next cards over and under the
resulting pile, so 12345678 becomes 86421357.
At G4G2, 1996, Max Maven gave a talk on some magic tricks based on card shuffling
and gave a short outline of the history. The following is an attempt to summarise his material.
The faro shuffle, done by inserting part of the deck endwise into the other part, but not done
perfectly, began to be used in the early 18C and a case of cheating using this is recorded in
1726. The riffle shuffle, which is the common American shuffle, depends on mass produced
cards of good quality and began to be used in the mid 19C. However, magicians did not
become aware of the possibilities of the perfect shuffle until the mid 20C, despite the early
work of Stanyans C. O. Williams and Charles T. Jordan in the 1910s.
Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Vol. 1, pp. 78-85: Of the
combinations of the cards. This describes a shuffle, where one takes the top two cards,
then puts the next two cards on top, then the next three cards underneath, then the next
two on top, then the next three underneath. For ten cards 1234567890, it produces
8934125670, a permutation of order 7. Tables of the first few repetitions are given for
10, 24, 27 and 32 cards, having orders 7, 30, 30, 156.
The Secret Out. 1859. Permutation table, pp. 394-395 (UK: 128-129). Describes Hooper's
shuffle for ten cards.
Bachet-Labosne. Problemes. 3rd ed., 1874. Supp. prob. XV, 1884: 214-222. Discusses
Monge's shuffle and its period.
John Nevil Maskelyne. Sharps and Flats. 1894. ??NYS -- cited by Gardner in the Addendum
of Carnival. "One of the earliest mentions". Called the "faro dealer's shuffle".
Ahrens. MUS I. 1910. Ein Kartenkunststück Monges, pp. 152-145. Expresses the general
form of Monge's shuffle and finds its order for n = 1, 2, ..., 10. Mentions the general
question of finding the order of a shuffle.
Charles T. Jordan. Thirty Card Mysteries. The author, Penngrove, California, 1919 (??NYS),
2nd ed., 1920 (?? I have copy of part of this). Cited by Gardner in the Addendum to
Carnival. First magician to apply the shuffle, but it was not until late 1950s that
magicians began to seriously use and study it. The part I have (pp. 7-10) just describes
the idea, without showing how to perform it. The text clearly continues to some
applications of the idea. This material was reprinted in The Bat (1948-1949).
Frederick Charles Boon. Shuffling a pack of cards and the theory of numbers. MG 15 (1930)
17-20. Considers the Out shuffle and sees that it relates to the order of 2 (mod 2n+1)
and gives some number theoretic observations on this. Also considers odd decks.
J. V. Uspensky & M. A. Heaslet. Elementary Number Theory. McGraw-Hill, NY, 1939.
Chap. VIII: Appendix: On card shuffling, pp. 244-248. Shows that an In shuffle of a
deck of 2n cards takes the card in position i to position 2i (mod 2n+1), so the order
of the permutation is the exponent or order of 2 (mod 2n+1), which is 52 when n =
26. [Though not discussed, this shows that the order of the Out shuffle is the order of
2 (mod 2n-1), which is only 8 for n = 26. And the order of a shuffle of 2n+1 cards is
the order of 2 (mod 2n+1).] Monge's shuffle is more complex, but leads to
congruences (mod 4n+1) and has order equal to the smallest exponent e such that
2e  ±1 (mod 4n+1), which is 12 for n = 26.
SOURCES - page 161
T. H. R. Skyrme. A shuffling problem. Eureka 7 (Mar 1942) 17-18. Describes Monge's
shuffle with the second card going under or over the first. Observes that in the under
shuffle for an even number of cards, the last card remains fixed, while the over shuffle
for an odd number of cards also leaves the last card fixed. By appropriate choice, one
always has the n-th card becoming the first. Finds the order of the shuffle essentially as
in Uspensky & Heaslet. Makes some further observations.
N. S. Mendelsohn, proposer and solver. Problem E792 -- Shuffling cards. AMM 54 (1947)
545 ??NYS & 55 (1948) 430-431. Shows the period of the out shuffle is at most 2n2. Editorial notes cite Uspensky & Heaslet and MG 15 (1930) 17-20 ??NYS.
Charles T. Jordan. Trailing the dovetail shuffle to its lair. The Bat (Nov, Dec 1948; Jan, Feb,
Mar, 1949). ??NYS -- cited by Gardner. I have No. 59 (Nov 1948) cover & 431-432,
which reprints some of the material from his book.
Paul B. Johnson. Congruences and card shuffling. AMM 63 (1956) 718-719. ??NYS -- cited
by Gardner.
Alexander Elmsley. Work in Progress. Ibidem 11 (Sep 1957) 222. He had previously coined
the terms 'in' and 'out' and represented them by I and O. He discovers and shows that
to put the top card into the k-th position, one writes k-1 in binary and reads off the
sequence of 1s and 0s, from the most significant bit, as I and O shuffles. He asks
but does not solve the question of how to move the k-th card to the top -- see Bonfeld
and Morris.
Alexander Elmsley. The mathematics of the weave shuffle, The Pentagram 11:9 (Jun 1957)
70-71; 11:10 (Jul 1957) 77-79; 11:11 (Aug 1957) 85; 12 (May 1958) 62. ??NYR -cited by Gardner in the bibliography of Carnival, but he doesn't give the Ibidem
reference in the bibliography, so there may be some confusion here?? Morris only cites
Pentagram.
Solomon W. Golomb. Permutations by cutting and shuffling. SIAM Review 3 (1961)
293-297. ??NYS -- cited by Gardner. Shows that cuts and the two shuffles generate all
permutations of an even deck. However, for an odd deck of n cards, the two kinds of
shuffles can be intermixed and this only changes the cyclic order of the result. Since
cutting also only changes the cyclic order, the number of possible permutations is n
times the order of the shuffle.
Gardner. SA (Oct 1966) = Carnival, chap. 10. Defines the in and out shuffles as above and
gives the relation to the order of 2. Notes that it is easier to do the inverse operations,
which consist of extracting every other card. Describes Elmsley's method. Addendum
says no easy method is known to determine shuffles to bring the k-th card to the top.
Murray Bonfeld. A solution to Elmsley's problem. Genii 37 (May 1973) 195-196. Solves
Elmsley's 1957 problem by use of an asymmetric in-shuffle where the top part of the
deck has 25 cards, so the first top card becomes second and the last two cards remain in
place. (If one ignores the bottom two cards this is an in-shuffle of a 50 card deck.)
S. Brent Morris. The basic mathematics of the faro shuffle. Pi Mu Epsilon Journal 6 (1975)
86-92. Obtains basic results, getting up to Elmsley's work. His reference to Gardner
gives the wrong year.
Israel N. Herstein & Irving Kaplansky. Matters Mathematical. 1974; slightly revised 2nd
ed., Chelsea, NY, 1978. Chap. 3, section 4: The interlacing shuffle, pp. 118-121.
Studies the permutation of the in shuffle, getting same results as Uspensky & Heaslet.
S. Brent Morris. Faro shuffling and card placement. JRM 8:1 (1975) 1-7. Shows how to do
the faro shuffle. Gives Elmsley's and Bonfeld's results.
Persi Diaconis, Ronald L. Graham & William M. Kantor. The mathematics of perfect
shuffles. Adv. Appl. Math. 4 (1983) 175-196. ??NYS.
Steve Medvedoff & Kent Morrison. Groups of perfect shuffles. MM 60:1 (1987) 3-14.
Several further references to check.
Walter Scott. Mathematics of card sharping. M500 125 (Dec 1991) 1-7. Sketches Elmsley's
results. States a peculiar method for computing the order of 2 (mod 2n+1) based on
adding translates of the binary expansion of 2n+1 until one obtains a binary number of
all 1s. The number of ones is the order a and the method is thus producing the
smallest a such that 2a-1 is a multiple of 2n+1.
John H. Conway & Richard K. Guy. The Book of Numbers. Copernicus (Springer-Verlag),
NY, 1996. Pp. 163-165 gives a brief discussion of perfect shuffles and Monge's shuffle.
5.AB.
FOLDING A STRIP OF STAMPS
SOURCES - page 162
É. Lucas. Théorie des Nombres. Gauthier-Villars, Paris, 1891; reprinted by Blanchard, Paris,
1958. P. 120.
Exemple II -- La bande de timbres-poste. -- De combien de manières peut-on replier,
sur un seul, une bande de p timbres-poste?
Exemple III -- La feuille de timbres-poste. -- De combien de manières peut-on replir,
sur un seul, une feuille rectangulaire de pq timbres-poste?
"Nous ne connaissons aucune solution de ces deux problèmes difficiles proposés par M.
Em. Lemoine."
M. A. Sainte-Laguë. Les Réseaux (ou Graphes). Mémorial des Sciences Mathématiques,
fasc. XVIII. Gauthier-Villars, Paris, 1926. Section 62: Problème des timbres-poste,
pp. 39-41. Gets some basic results and finds the numbers for a strip of n,
n = 1, 2, ..., 10 as: 1, 2, 6, 16, 50, 144, 448, 7472, 17676, 41600.
Jacques Devisme. Contribution a l'étude du problème des timbres-poste. Comptes-Rendus du
Deuxième Congrès International de Récréation Mathématique, Paris, 1937. Librairie du
"Sphinx", Bruxelles, 1937, pp. 55-56. Cites Lucas (but in the wrong book!) and SainteLaguë. Studies the number of different forms of the result, getting numbers:
1, 2, 3, 8, 18, 44, 115, 294, 783.
5.AC.
PROPERTIES OF THE SEVEN BAR DIGITAL DISPLAY
┌─┐
2
The seven bar display, in the form of a figure 8, as at the right, is
│ │ 1
3
now the standard form for displaying digits on calculators, clocks, etc.
├─┤
4
This lends itself to numerous problems of a combinatorial/numerical
│ │ 5
7
New Section.
└─┘
6
For reference, we number the seven bars in the reverse-S pattern
shown. We can then refer to a pattern by its binary 7-tuple or its decimal equivalent. E.g. the
number one is displayed by having bars 3 and 7 on, which gives a binary pattern 1000100
corresponding to decimal 68. NOTE that there is some ambiguity with the 6 / 9. Most
versions use the upper / lower bar for these, i.e. 1101111 / 1111011, but the bar is sometimes
omitted, giving 1001111 / 1111001. I will assume the first case unless specified.
I have been interested in these for some time for several reasons. First, my wife has
such a clock on her side of the bed and she often has a glass of water in front of it, causing
patterns to be reversed. At other times the clock has been on the floor upside down, causing a
different reversal of patterns. Second, segments often fail or get stuck on and I have tried to
analyse which would be the worst segment to fail or get stuck. As an example, the clock in
my previous car went from 16:59 to 15:00. Third, I have analysed which segment(s) in a
clock are used most/least often.
Birtwistle. Calculator Puzzle Book. 1978. Prob. 35: New numbers, pp. 26-27 & 83. Asks
for the number of new digits one can make, subject to their being connected and full
height. Says it is difficult to determine when these are distinct -- e.g. calculators differ
as to the form of their 6s and 9s -- so he is not sure how to count, but he gives 22
examples. I find there are 55 connected, full-height patterns.
Gordon Alabaster, proposer & Robert Hill, solver. Problem 134.3 -- Clock watching. M500
134 (Aug 1993) 17 & 135 (Oct 1993) 14-15. Proposer notes that one segment of the
units digit of the seconds on his station clock was stuck on, but that the sequence of
symbols produced were all proper digits. Which segment was stuck? Asks if there are
answers for 2, ..., 6 segments stuck on. Solver gives systematic tables and discusses
problems of how to determine which segment(s) are stuck and whether one can deduce
the correct time when the stuck segments are known.
Martin Watson. Email to NOBNET, 17 Apr 2000 08:17:32 PDT [NOBNET 2334]. Observes
that the 10 digits have a total of 49 segments and asks if they can be placed on a 4 x 5
square grid. He calls these forms 'digigrams'. He had been unable to find a solution but
Leonard Campbell has found 5 distinct solutions, though they do no differ greatly. He
has the pieces and some discussion on his website:
http://martnal.tripod.com/puzzles.html . Dario Uri [22 Apr 2000 14:44:35 +0200]
found two extra solutions, but Rick Eason [22 Apr 2000 09:37: -0400] also found these,
but points out that these have an error due to misreading the lattice which gives the two
bars of the 1 being parallel instead of end to end. Eason's program also found the 5
solutions.
SOURCES - page 163
5.AD.
STACKING A DECK TO PRODUCE A SPECIAL EFFECT
New section. This refers to the process of arranging a deck of cards or a stack of coins
so that dealing it by some rule produces a special effect. In many cases, this is just inverting
the permutation given by the rule and the Josephus problem (7.B) is a special case. Other
cases involve spelling out the names of cards, etc.
Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Alternate heads, pp. 61-63.
Stack of eight coins. Place one on the table and the next on the bottom of the stack.
The sequence of placed coins is to alternate heads and tails. How do you arrange the
stack? Answer is HHTHHTTT. This is the same process as counting out by 2s -- see
7.B.
Doubleday - 2. 1971. Heads and tails, pp. 105-106. Same as Blyth, but with six coins and
solution HTTTHH.
5.AE.
REVERSING CUPS
New section. There are several versions of this and they usually involve parity. The
basic move is to reverse two of the cups. The classic problem seems to be to start with UDU
and produce DDD in three moves. A trick version is to demonstrate this several times to
someone and then leave him to start from DUD. Another easy problem is to leave three cups
as they were after three moves. This is equivalent to a 3 x 3 array with an even number in
each row and column -- see 6.AO.2. These problems must be much older than I have, but the
following are the only examples I have yet noted.
Anonymous. Social Entertainer and Tricks (thus on spine, but running title inside is New
Book of Tricks). Apparently a compilation with advertisements for Johnson Smith
(Detroit, Michigan) products, c1890?. P. 38a: Bottoms up. Given UDU, produce
DDD in three moves.
Young World. c1960. P. 39: Water switch. Full and empty glasses: FFFEEE. Make them
alternately full and empty in one move.
Putnam. Puzzle Fun. 1978.
No. 3: Tea for three, pp. 1 & 25. Cups given as UDU. Produce DDD in three moves.
No. 16: Glass alignment, pp. 5 & 28. Six cups arranged UUUDDD. Produce an
alternating row. He gets UDUDUD in three moves. I can get DUDUDU in
four moves.
5.AF. SPOTTING DICE
New section. In the early 1980s, I asked Richard Guy what was the 'standard'
configuration for a die and later asked Ray Bathke if he used a standard pattern. Assuming
opposite sides add to seven there are two handednesses. But also the spot pattern of the two,
three and six has two orientations, giving 16 different patterns of die. Ray said that when he
furnished dice with games, some customers had sent them back because they weren't the
same. Within about three years, I had obtained examples of all sixteen patterns! Indeed, I
often found several patterns in a single batch from one manufacturer. Ray Bathke also
pointed out that the small dice that come from the oriental games have the two arranged either
horizontally or vertically rather than diagonally, giving another 16 patterns. I have only
obtained five of these, but with both handednesses included. I used this idea in one of my
Brain Twisters, cf below.
Since the 2, 3 and 6 faces all meet at a corner, one has just to describe this corner. The
2, 3, 6 can be clockwise around the corner or anti-clockwise. Note that 236 is clockwise if
and only if 132 is clockwise. The position of the 2 and 3 can be described by saying whether
the pattern points toward or away from the corner. If we place the 2 upward, then 6 will be a
vertical face and we can describe it by saying whether the lines of three spots are vertical or
horizontal. Guy told me a system for describing a die, but it's not in Winning Ways and I've
forgotten it, so I'll invent my own.
We write the sequence 236 if 236 is arranged clockwise at the 236 corner and we
write 263 otherwise. When looked at cornerwise, with the 2 on top, the pattern of the 2
may appear vertical or horizontal. We write 2 when it is vertical and 2 when it is
horizontal. (For oriental dice, the 2 will appear on a diagonal and can be indicated by 2 or
SOURCES - page 164
2. If we now rotate the cube to bring the 3 on top, its pattern will appear either vertical or
horizontal and we write 3 or 3. Putting the 2 back on top, the 6 face will be upright and
the lines of three spots will be either vertical or horizontal, which we denote by 6 or 6.
David Singmaster. Dicing around. Weekend Telegraph (16 Dec 1989). = Games & Puzzles
No. 15 (Jun 1995) 22-23 & 16 (Jul 1995) 43-44. How many dice are there? Describes
the normal 16 and mentions the other 16.
Ian Stewart. The lore and lure of dice. SA (Nov 1997) ??. He asserts that the standard
pattern has 132 going clockwise at a corner, except that the Japanese use the mirrorimage version in playing mah-jongg. His picture has both 2 and 3 toward the 236
corner and the 6 being vertical, i.e. in pattern 236. He discusses crooked dice of various
sorts and that the only way to make all values from 1 to 12 equally likely is to have
123456 on one die and 000666 on the other.
Ricky Jay. The story of dice. The New Yorker (11 Dec 2000) 90-95.
5.AG.
RUBIK'S CUBE AND SIMILAR PUZZLES
I have previously avoided this as being too recent to be covered in a historical work, but
it is now old enough that it needs to be covered, and there are some older references.
Much of the history is given in my Notes on Rubik's Cube and my Cubic Circular. Jaap
Scherphuis has sent me a file of puzzle patents and several dozen of them could be
entered here, but I will only enter older or novel items. Scherphuis's file has about a
dozen patents for the 4 x 4 x 4 and 5 x 5 x 5 cubes! See Section 5.A for predecessors
of the idea. However, this Section will mostly deal with puzzles where pieces are
permuted without having any empty places, so these are generally permutation puzzle.
5.AG.1.
RUBIK'S CUBE
New section. Much to be added.
Richard E. Korf. Finding optimal solutions to Rubik's Cube using pattern databases. Proc.
Nat. Conf. on Artificial Intelligence (AAAI-97), Providence, Rhode Island, Jul 1997,
pp. 700-705. Studies heuristic methods of finding optimal solutions of the Cube.
Claims to be the first to find optimal solutions for random positions of the Cube -- but I
think others such as Kociemba and Reid were doing it up to a decade earlier. For ten
random examples, he found optimal solutions took 16 moves in one case, 17 moves in
three cases, 18 moves in six cases, from which he asserts the median optimal solution
length seems to be 18. He uses the idea of axial moves and obtains the lower bound of
18 for God's Algorithm, as done in my Notes in 1980. Cites various earlier work in the
field, but only one reference to the Cube literature.
Richard E. Korf & Ariel Felner. Disjoint pattern database heuristics. Artificial Intelligence
134 (2002) 9-22. Discusses heuristic methods of solving the Fifteen Puzzle, Rubik's
Cube, etc. Asserts the median optimal solution length for the Cube is only 18. Seems
to say one of the problems in the earlier paper took a couple of weeks running time, but
improved methods of Kociemba and Reid can find optimal solutions in about an hour.
5.AG.2.
HUNGARIAN RINGS, ETC.
New section. Much to be added.
William Churchill. US Patent 507,215 -- Puzzle. Applied: 28 May 1891; patented: 24 Oct
1893. 1p + 1p diagrams. Two rings of 22 balls, intersecting six spaces apart.
Hiester Azarus Bowers. US Patent 636,109 -- Puzzle. Filed: 16 Aug 1899; patented 31 Oct
1899. 2pp + 1p diagrams. 4 rotating discs which overlap in simple lenses.
Ivan Moscovich. US Patent 4,509,756 -- Puzzle with Elements Transferable Between
Closed-loop Paths. Filed: 18 Dec 1981; patented: 9 Apr 1985. Cover page + 3pp +
2pp diagrams. Two rings of 18 balls, each stretched to have two straight sections with
semicircular ends. The rings cross in four places, at the ends of the straight sections, so
adjacent crossing points are separated by two balls. I'm not sure this was ever produced.
Mentions three circular rings version, but there each pair of rings only overlaps in two
places so this is a direct generalization of the Hungarian Rings.
David Singmaster. Hungarian Rings groups. Bull. Inst. Math. Appl. 20:9/10 (Sep/Oct 1984)
SOURCES - page 165
137-139. [The results were stated in Cubic Circular 5 & 6 (Autumn & Winter 1982)
9-10.] An article by Philippe Paclet [Des anneaux et des groupes; Jeux et Stratégie 16
(Aug/Sep 1982) 30-32] claimed that all puzzles of two rings have groups either the
symmetric or the alternating group on the number of balls. This article shows this is
false and determines the group in all cases. If we have rings of size m, n and the
intersections are distances a, b apart on the two rings. Then the group, G(m, n, a, b)
is the symmetric group on m+n-2 if mn is even and is the alternating group if mn is
odd; except that G(4, 4, 1, 1) is the exceptional group described in R. M. Wilson's
1974 paper: Graph puzzles, homotopy and the alternating group -- cited in Section 5.A
under The Fifteen Puzzle -- and is also the group generated by two adjacent faces on the
Rubik Cube acting on the six corners on those faces; and except that G(2a, 2b, a, b)
keeps antipodal pairs at antipodes and hence is a subgroup of the wreath product
Z2 wr Sa+b-1, with three cases depending on the parities of a and b.
Bala Ravikumar. The Missing Link and the Top-Spin. Report TR94-228, Department of
Computer Science and Statistics, University of Rhode Island, Jan 1994. Top-Spin has a
cycle of 20 pieces and a small turntable which permits inverting a section of four pieces.
After developing the group theory and doing the Fifteen Puzzle and the Missing Link,
he shows the state space of Top-Spin is S20.
SOURCES - page 166
6.
GEOMETRIC RECREATIONS
6.A. PI
This is too big a topic to cover completely. The first items should be consulted for
older material and the general history. Then I include material of particular interest. See also
6.BL which has some formulae which are used to compute π. I have compiled a separate file
on the history of π.
Augustus De Morgan. A Budget of Paradoxes. (1872); 2nd ed., edited by D. E. Smith,
(1915), Books for Libraries Press, Freeport, NY, 1967.
J. W. Wrench Jr. The evolution of extended decimal approximations to π. MTr 53
(Dec 1960) 644-650. Good survey with 55 references, including original sources.
Petr Beckmann. A History of π. The Golem Press, Boulder, Colorado, (1970); 2nd ed.,
1971.
Lam Lay-Yong & Ang Tian-Se. Circle measurements in ancient China. HM 13 (1986)
325-340. Good survey of the calculation of π in China.
Dario Castellanos. The ubiquitous π. MM 61 (1988) 67-98 & 148-163. Good survey of
methods of computing π.
Joel Chan. As easy as pi. Math Horizons 1 (Winter 1993) 18-19. Outlines some recent work
on calculating π and gives several of the formulae used.
David Singmaster. A history of π. M500 168 (Jun 1999) 1-16. A chronology. (Thanks to
Tony Forbes and Eddie Kent for carefully proofreading and amending my file.)
Aristophanes. The Birds. -414. Lines 1001-1005. In: SIHGM I 308-309. Refers to 'circlesquarers', possibly referring to the geometer/astronomer Meton.
E. J. Goodwin. Quadrature of the circle. AMM 1 (1894) 246-247.
House Bill No. 246, Indiana Legislature, 1897. "A bill for an act introducing a new
mathematical truth ..." In Edington's paper (below), p. 207, and in several of the
newspaper reports.
(Indianapolis) Journal (19 Jan 1897) 3. Mentions the Bill in the list of bills introduced.
Die Quadratur des Zirkels. Täglicher Telegraph (Indianapolis) (20 Jan 1897) ??.
Surveys attempts since -2000 and notes that Lindemann and Weierstrass have
shown that the problem is impossible, like perpetual motion.
A man of 'genius'. (Indianapolis) Sun (6 Feb 1897) ??. An interview with Goodwin,
who says: "The astronomers have all been wrong. There's about 40,000,000
square miles on the surface of this earth that isn't here." He says his results are
revelations and gives several rules for the circle and the sphere.
Mathematical Bill passed. (Indianapolis) Journal (6 Feb 1897) 5. "This is the strangest
bill that has ever passed an Indiana Assembly." Gives whole text of the Bill.
Dr. Goodwin's theaorem (sic) Resolution adopted by the House of Representatives.
(Indianapolis) News (6 Feb 1897) 4. Gives whole text of the Bill.
The Mathematical Bill Fun-making in the Senate yesterday afternoon -- other action.
(Indianapolis) News (13 Feb 1897) 11. "The Senators made bad puns about it,
...." The Bill was indefinitely postponed.
House Bills in the Senate. (Indianapolis) Sentinel (13 Feb 1897) 2. Reports the Bill
was killed.
(No heading??) (Indianapolis) Journal (13 Feb 1897) 3, col. 4. "... indefinitely
postponed, as not being a subject fit for legislation."
Squaring the circle. (Indianapolis) Sunday Journal (21 Feb 1897) 9. Says Goodwin has
solved all three classical impossible problems. Says π = 3.2, using the fact that
2 = 10/7, giving diagrams and a number of rules.
My thanks to Underwood Dudley for locating and copying the above newspaper items.
C. A. Waldo. What might have been. Proc. Indiana Acad. Science 26 (1916) 445-446.
W. E. Edington. House Bill No. 246, Indiana State Legislature, 1897. Ibid. 45 (1935)
206-210.
A. T. Hallerberg. House Bill No. 246 revisited. Ibid. 84 (1975) 374-399.
Manuel H. Greenblatt. The 'legal' value of pi, and some related mathematical anomalies.
American Scientist 53 (Dec 1965) 427A-434A. On p. 427A he tries to interpret the bill
SOURCES - page 167
and obtains three different values for π.
David Singmaster. The legal values of pi. Math. Intell. 7:2 (1985) 69-72. Analyses
Goodwin's article, Bill and other assertions to find 23 interpretable statements giving 9
different values of π !
Underwood Dudley. Mathematical Cranks. MAA, 1992. Legislating pi, pp. 192-197.
C. T. Heisel. The Circle Squared Beyond Refutation. Published by the author, 657 Bolivar
Rd., Cleveland, Ohio, 1st ed., 1931, printed by S. J. Monck, Cleveland; 2nd ed., 1934,
printed by Lezius-Hiles Co., Cleveland, ??NX + Supplement: "Fundamental Truth",
1936, ??NX, distributed by the author from 2142 Euclid Ave., Cleveland. This is
probably the most ambitious publication of a circle-squarer -- Heisel distributed copies
all around the world.
Underwood Dudley. πt: 1832-1879. MM 35 (1962) 153-154. He plots 45 values of π as a
function of time over the period 1832-1879 and finds the least-squares straight line
which fits the data, finding that πt = 3.14281 + .0000056060 t, for t measured in years
AD. Deduces that the Biblical value of 3 was a good approximation for the time and
that Creation must have occurred when πt = 0, which was in -560,615.
Underwood Dudley. πt. JRM 9 (1976-77) 178 & 180. Extends his previous work to 50
values of π over 1826-1885, obtaining πt = 4.59183 - .000773 t. The fact that πt is
decreasing is worrying -- when πt = 1, all circles will collapse into straight lines and
this will certainly be the end of the world, which is expected in 4646 on 9 Aug at
20:55:33 -- though this is only the expected time and there is considerable variation in
this prediction. [Actually, I get that this should be on 11 Aug. However, it seems to me
that circles will collapse once πt = 2, as then the circumference corresponds to going
back and forth along the diameter. This will occur when t = 3352.949547, i.e. in 3352,
on 13 Dec at 14:01:54 -- much earlier than Dudley's prediction, so start getting ready
now!]
6.B. STRAIGHT LINE LINKAGES
See Yates for a good survey of the field.
James Watt. UK Patent 1432 -- Certain New Improvements upon Fire and Steam Engines,
and upon Machines worked or moved by the same. Granted: 28 Apr 1784; complete
specification: 24 Aug 1784. 14pp + 1 plate. Pp. 4-6 & Figures 7-12 describe Watt's
parallel motion. Yates, below, p. 170 quotes one of Watt's letters: "... though I am not
over anxious after fame, yet I am more proud of the parallel motion than of any other
invention I have ever made."
P. F. Sarrus. Note sur la transformation des mouvements rectilignes alternatifs, en
mouvements circulaires; et reciproquement. C. R. Acad. Sci. Paris 36 (1853)
1036-1038. 6 plate linkage. The name should be Sarrus, but it is printed Sarrut on this
and the following paper.
Poncelet. Rapport sur une transformation nouvelle des mouvements rectilignes alternatifs en
mouvements circulaires et reciproquement, par Sarrut. Ibid., 36 (1853) 1125-1127.
A. Peaucellier. Lettre au rédacteur. Nouvelles Annales de Math. (2) 3 (1864) 414-415. Poses
the problem.
A. Mannheim. Proces-Verbaux des sceances des 20 et 27 Juillet 1867. Bull. Soc.
Philomathique de Paris (1867) 124-126. ??NYS. Reports Peaucellier's invention.
Lippman Lipkin. Fortschritte der Physik (1871) 40-?? ??NYS
L. Lipkin. Über eine genaue Gelenk-Geradführung. Bull. Acad. St. Pétersbourg [=? Akad.
Nauk, St. Petersburg, Bull.] 16 (1871) 57-60. ??NYS
L. Lipkin. Dispositif articulé pour la transformation rigoureuse du mouvement circulaire en
mouvement rectiligne. Revue Univers. des Mines et de la Métallurgie de Liége 30:4
(1871) 149-150. ??NYS. (Now spelled Liège.)
A. Peaucellier. Note sur un balancier articulé a mouvement rectiligne. Journal de Physique 2
(1873) 388-390. (Partial English translation in Smith, Source Book, vol. 2,
pp. 324-325.) Says he communicated it to Soc. Philomath. in 1867 and that Lipkin has
since also found it. There is also an article in Nouv. Annales de Math. (2) 12 (1873)
71-78 (or 73?), ??NYS.
E. Lemoine. Note sur le losange articulé du Commandant du Génie Peaucellier, destiné a
SOURCES - page 168
remplacer le parallélogramme de Watt. J. de Physique 2 (1873) 130-134. Confirms
that Mannheim presented Peaucellier's cell to Soc. Philomath. on 20 Jul 1867.
Develops the inversive geometry of the cell.
[J. J. Sylvester.] Report of the Annual General Meeting of the London Math. Soc. on 13 Nov
1873. Proc. London Math. Soc. 5 (1873) 4 & 141. On p. 4 is: "Mr. Sylvester then gave
a description of a new instrument for converting circular into general rectilinear motion,
and into motion in conics and higher plane curves, and was warmly applauded at the
close of his address." On p. 141 is an appendix saying that Sylvester spoke "On recent
discoveries in mechanical conversion of motion" to a Friday Evening's Discourse at the
Royal Institution on 23 Jan 1874. It refers to a paper 20 pages long but is not clear if or
where it was published.
H. Hart. On certain conversions of motion. Cambridge Messenger of Mathematics 4 (1874)
82-88 and 116-120 & Plate I. Hart's 5 bar linkage. Obtains some higher curves.
A. B. Kempe. On some new linkages. Messenger of Mathematics 4 (1875) 121-124 & Plate
I. Kempe's linkages for reciprocating linear motion.
H. Hart. On two models of parallel motions. Proc. Camb. Phil. Soc. 3 (1876-1880) 315-318.
Hart's parallelogram (a 5 bar linkage) and a 6 bar one.
V. Liguine. Liste des travaux sur les systèms articulés. Bull. d. Sci. Math. 18 (or (2) 7)
(1883) 145-160. ??NYS - cited by Kanayama. Archibald; Outline of the History of
Mathematics, p. 99, says Linguine is entirely included in Kanayama.
Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and Novelties of
Construction. A second volume to accompany his previous Mechanical Movements,
Powers and Devices. Norman W. Henley Publishing Co, NY, (1904), 2nd ed., 1910.
This is filled with many types of mechanisms. Pp. 245-247 show five straight-line
linkages and some related mechanisms.
(R. Kanayama). (Bibliography on linkages. Text in Japanese, but references in roman type.)
Tôhoku Math. J. 37 (1933) 294-319.
R. C. Archibald. Bibliography of the theory of linkages. SM 2 (1933-34) 293-294.
Supplement to Kanayama.
Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed.,
Educational Publishers, St. Louis, 1949. Pp. 82-101 & 168-191. Gets up to outlining
Kempe's proof that any algebraic curve can be drawn by a linkage.
R. H. Macmillan. The freedom of linkages. MG 34 (No. 307) (Feb 1960) 26-37. Good
survey of the general theory of linkages.
Michael Goldberg. Classroom Note 312: A six-plate linkage in three dimensions. MG 58
(No. 406) (Dec 1974) 287-289.
6.C. CURVES OF CONSTANT WIDTH
Such curves play an essential role in some ways to drill a square hole, etc.
L. Euler. Introductio in Analysin Infinitorum. Bousquet, Lausanne, 1748. Vol. 2, chap. XV,
esp. § 355, p. 190 & Tab. XVII, fig. 71. = Introduction to the Analysis of the Infinite;
trans. by John D. Blanton; Springer, NY, 1988-1990; Book II, chap. XV: Concerning
curves with one or several diameters, pp. 212-225, esp. § 355, p. 221 & fig. 71, p. 481.
This doesn't refer to constant width, but fig. 71 looks very like a Reuleaux triangle.
L. Euler. De curvis triangularibus. (Acta Acad. Petropol. 2 (1778(1781)) 3-30) = Opera
Omnia (1) 28 (1955) 298-321. Discusses triangular versions.
M. E. Barbier. Note sur le problème de l'aiguille et jeu du joint couvert. J. Math. pures appl.
(2) 5 (1860) 273-286. Mentions that perimeter = π * width.
F. Reuleaux. Theoretische Kinematik; Vieweg, Braunschweig, 1875. Translated: The
Kinematics of Machinery. Macmillan, 1876; Dover, 1964. Pp. 129-147.
Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and Novelties of
Construction. A second volume to accompany his previous Mechanical Movements,
Powers and Devices. Norman W. Henley Publishing Co, NY, (1904), 2nd ed., 1910.
Item 642: Turning a square by circular motion, p. 247. Plain face, with four pins
forming a centred square, is turned by the lathe. A triangular follower is against
the face, so it is moved in and out as a pin moves against it. This motion is
conveyed by levers to the tool which moves in and out against the work which is
driven by the same lathe.
Item 681: Geometrical boring and routing chuck, pp. 257-258. Shows it can make
SOURCES - page 169
rectangles, triangles, stars, etc. No explanation of how it works.
Item 903A: Auger for boring square holes, pp. 353-354. Uses two parallel rotating
cutting wheels.
H. J. Watts. US Patents 1,241,175-7 -- Floating tool-chuck; Drill or boring member;
Floating tool-chuck. Applied: 30 Nov 1915; 1 Nov 1916; 22 Nov 1916; all patented:
25 Sep 1917. 2 + 1, 2 + 1, 4 + 1 pp + pp diagrams. Devices for drilling square holes
based on the Reuleaux triangle.
T. Bonnesen & W. Fenchel. Theorie der konvexen Körper. Berlin, 1934; reprinted by
Chelsea, 1971. Chap. 15: Körper konstanter Breite, pp. 127-141. Surveys such curves
with references to the source material.
G. D. Chakerian & H. Groemer. Convex bodies of constant width. In: Convexity and Its
Applications; ed. by Peter M. Gruber & Jörg M. Wills; Birkhäuser, Boston, 1983.
Pp. 49-96. (??NYS -- cited in MM 60:3 (1987) 139.) Bibliography of some 250 items
since 1930.
6.D. FLEXAGONS
These were discovered by Arthur H. Stone, an English graduate student at Princeton in
1939. American paper was a bit wider than English and would not fit into his notebooks, so
he trimmed the edge off and had a pile of long paper strips which he played with and
discovered the basic flexagon. Fellow graduate students Richard P. Feynman, Bryant
Tuckerman and John W. Tukey joined in the investigation and developed a considerable
theory. One of their fathers was a patent attorney and they planned to patent the idea and
began to draw up an application, but the exigencies of the 1940s led to its being put aside,
though knowledge of it spread as mathematical folklore. E.g. Tuckerman's father, Louis B.
Tuckerman, lectured on it at the Westinghouse Science Talent Search in the mid 1950s.
S&B, pp. 148-149, show several versions. Most square versions (tetraflexagons or
magic books) don't fold very far and are really just extended versions of the Jacob's Ladder -see 11.L
Martin Gardner. Cherchez la Femme [magic trick]. Montandon Magic Co., Tulsa, Okla.,
1946. Reproduced in: Martin Gardner Presents; Richard Kaufman and Alan
Greenberg, 1993, pp. 361-363. [In: Martin Gardner Presents, p. 404, this is attributed
to Gardner, but Gardner told me that Roger Montandon had the copyright -- ?? I have
learned a little more about Gardner's early life -- he supported himself by inventing and
selling magic tricks about this time, so it may be that Gardner devised the idea and sold
it to Montandon.]. A hexatetraflexagon.
"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and Sciences, Croydon,
1947. A trick book, pp. 42-43. Same hexatetraflexagon.
Sidney Melmore. A single-sided doubly collapsible tessellation. MG 31 (No. 294) (1947)
106. Forms a Möbius strip of three triangles and three rhombi. He sees it has two
distinct forms, but doesn't see the flexing property!!
Margaret Joseph. Hexahexaflexagrams. MTr 44 (Apr 1951) 247-248. No history.
William R. Ransom. A six-sided hexagon. SSM 52 (1952) 94. Shows how to number the 6
faces. No history.
F. G. Maunsell. Note 2449: The flexagon and the hexahexaflexagram. MG 38 (No. 325)
(Sep 1954) 213-214. States that Joseph is first article in the field and that this is first
description of the flexagon. Gives inventors' names, but with Tulsey for Tukey.
R. E. Rogers & Leonard L. D'Andrea. US Patent 2,883,195 -- Changeable Amusement
Devices and the Like. Applied: 11 Feb 1955; patented: 21 Apr 1959. 2pp + 1p
correction + 2pp diagrams. Clearly shows the 9 and 18 triangle cases and notes that one
can trim the triangles into hexagons so the resulting object looks like six small
hexagons in a ring.
M. Gardner. Hexa-hexa-flexagon and Cherchez la femme. Hugard's MAGIC Monthly 13:9
(Feb 1956) 391. Reproduced in his: Encyclopedia of Impromptu Magic; Magic Inc.,
Chicago, 1978, pp. 439-442. Describes hexahexa and the hexatetra of
Gardner/Montandon & Willane.
M. Gardner. SA (Dec 1956) = 1st Book, chap. 1. His first article in SA!!
Joan Crampin. Note 2672: On note 2449. MG 41 (No. 335) (Feb 1957) 55-56. Extends to a
general case having 9n triangles of 3n colours.
C. O. Oakley & R. J. Wisner. Flexagons. AMM 64:3 (Mar 1957) 143-154.
SOURCES - page 170
Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, section 61,
pp. 24-25: Hexaflexagons. Describes the simplest case, citing Joseph.
Roger F. Wheeler. The flexagon family. MG 42 (No. 339) (Feb 1958) 1-6. Improved
methods of folding and colouring.
M. Gardner. SA (May 1958) = 2nd Book, chap. 2. Tetraflexagons and flexatube.
P. B. Chapman. Square flexagons. MG 45 (1961) 192-194. Tetraflexagons.
Anthony S. Conrad & Daniel K. Hartline. Flexagons. TR 62-11, RIAS, (7212 Bellona
Avenue, Baltimore 12, Maryland,) 1962, 376pp. This began as a Science Fair project in
1956 and was then expanded into a long report. The authors were students of Harold V.
McIntosh who kindly sent me one of the remaining copies in 1996. They discover how
to make any chain of polygons into a flexagon, provided certain relations among angles
are satisfied. The bibliography includes almost all the preceding items and adds the
references to the Rogers & D'Andrea patent, some other patents (??NYS) and a number
of ephemeral items: Conrad produced an earlier RIAS report, TR 60-24, in 1960;
Allan Phillips wrote a mimeographed paper on hexaflexagons; McIntosh wrote an
unpublished paper on flexagons; Mike Schlesinger wrote an unpublished paper on
Tuckerman tree theory.
Sidney H. Scott. How to construct hexaflexagons. RMM 12 (Dec 1962) 43-49.
William R. Ransom. Protean shapes with flexagons. RMM 13 (Feb 1963) 35-37. Describes
3-D shapes that can be formed. c= Madachy, below.
Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. The magic book, pp. 124-125. As in
Gardner's Cherchez la Femme and Willane.
Pamela Liebeck. The construction of flexagons. MG 48 (No. 366) (Dec 1964) 397-402.
Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Other flexagon
diversions, pp. 76-81. Describes 3-D shapes that one can form. Based on Ransom,
RMM 13.
Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 66-75.
Describes various tetra- and hexa-flexagons.
Douglas A. Engel. Hexaflexagon + HFG = slipagon! JRM 25:3 (1993) 161-166. Describes
his slipagons, which are linked flexagons.
Robert E. Neale (154 Prospect Parkway, Burlington, Vermont, 05401, USA). Self-designing
tetraflexagons. 12pp document received in 1996 describing several ways of making
tetraflexagons without having to tape or paste. He starts with a creased square sheet,
then makes some internal tears or cuts and then folds things through to miraculously
obtain a flexagon! A slightly rearranged version appeared in: Elwyn R. Berlekamp &
Tom Rodgers, eds.; The Mathemagician and Pied Puzzler A Collection in Tribute to
Martin Gardner; A. K. Peters, Natick, Massachusetts, 1999, pp. 117-126.
Jose R. Matos. US Patent 5,735,520 -- Fold-Through Picture Puzzle. Applied: 7 Feb 1997;
patented: 7 Apr 1998. Front page + 6pp diagrams + 13pp text. Robert Byrnes sent an
example of the puzzle. This is a square in thin plastic, 100mm on an edge. Imagine a
2 x 2 array of squares with their diagonals drawn. Fold along all the diagonals and
between the squares. This gives an array of 16 isosceles right triangles. Now cut from
the centres of the four squares to the centre of the whole array. This produces an X cut
in the middle. This object can now be folded through itself in various ways to produce
a double thickness square of half the area with various logos. The example is 100mm
along the edge of the large square and has four logos advertising Beanoland (at
Chessington, 3 versions) and Strip Cheese. The patent is assigned to Lulirama
International, but Byrnes says it has not been a commercial success as it is too
complicated. The patent cites 19 earlier patents, back to 1881, and discusses the history
of such puzzles. It also says the puzzle can form three dimensional objects.
6.E. FLEXATUBE
This is the square cylindrical tube that can be inverted by folding. It was also invented
by Arthur H. Stone, c1939, cf 6.D.
J. Leech. A deformation puzzle. MG 39 (No. 330) (Dec 1955) 307. Doesn't know source.
Says there are three solutions.
M. Gardner. Flexa-tube puzzle. Ibidem 7 (Sep 1956) 129. Cites the inventors of the
flexagons and the articles of Maunsell and Leech (but he doesn't have its details). (I
have a note that this came with attached sample, but the copy I have doesn't indicate
SOURCES - page 171
such.)
T. S. Ransom. Flexa-tube solution. Ibidem 9 (Mar 1957) 174.
M. Gardner. SA (May 1958) = 2nd Book, chap. 2. Says Stone invented it and shows
Ransom's solution.
H. Steinhaus. Mathematical Snapshots. Not in the Stechert, NY, 1938, ed. nor the OUP, NY,
1950 ed. OUP, NY: 1960: pp. 189-193 & 326; 1969 (1983): pp. 177-181 & 303.
Erroneous attribution to the Dowkers. Shows a different solution than Ransom's.
John Fisher. John Fisher's Magic Book. Muller, London, 1968. Homage to Houdini,
pp. 152-155. Detailed diagrams of the solution, but no history.
Highland Games (2 Harpers Court, Dingwall, Ross-Shire, IV15 9HT) makes a version called
Table Teaser, made in a strip with end pieces magnetic. Pieces are coloured so to
produce several folding and inverting problems other than the usual one. Bought in
1995.
6.F. POLYOMINOES, ETC.
See S&B, pp. 15-18. See 6.F.1, 6.F.3, 6.F.4 & 6.F.5 for early occurrences of
polyominoes. See Lammertink, 1996 & 1997 for many examples in two and three
dimensions.
NOTATION. Each of the types of puzzle considered has a basic unit and pieces are
formed from a number of these units joined edge to edge. The notation N: n1, n2, .... denotes
a puzzle with N pieces, of which ni pieces consist of i basic units. If ni are single digit
numbers, the intervening commas and spaces will be omitted, but the digits will be grouped
by fives, e.g. 15: 00382 11.
Polyiamonds: Scrutchin; John Bull; Daily Sketch; Daily Mirror; B. T.s Zig-Zag;
Daily Mail; Miller (1960); Guy (1960); Reeve & Tyrrell; O'Beirne (2 & 9 Nov 1961);
Gardner (Dec 1964 & Jul 1965); Torbijn; Meeus; Gardner (Aug 1975);
Guy (1996, 1999); Knuth,
Polycubes: Rawlings (1939); Editor (1948); Niemann (1948); French (1948); Editor
(1948); Niemann (1948); Gardner (1958); Besley (1962); Gardner (1972)
Solid Pentominoes: Nixon (1948); Niemann (1948); Gardner (1958); Miller (1960);
Bouwkamp (1967, 1969, 1978); Nelson (2002);
Cylindrical Pentominoes: Yoshigahara (1992);
Polyaboloes: Hooper (1774); Book of 500 Puzzles (1859); J. M. Lester (1919); O'Beirne
(21 Dec 61 & 18 Jan 62)
Polyhexes: Gardner (1967); Te Riele & Winter
Polysticks: Benjamin; Barwell; General Symmetrics; Wiezorke & Haubrich; Knuth;
Jelliss;
Polyrhombs or Rhombiominoes: Lancaster (1918); Jones (1992).
Polylambdas: Roothart.
Polyspheres -- see Section 6.AZ.
GENERAL REFERENCES
G. P. Jelliss. Special Issue on Chessboard Dissections. Chessics 28 (Winter 1986) 137-152.
Discusses many problems and early work in Fairy Chess Review.
Branko Grünbaum & Geoffrey C. Shephard. Tilings and Patterns. Freeman, 1987. Section
9.4: Polyiamonds, polyominoes and polyhexes, pp. 497-511. Good outline of the field
with a number of references otherwise unknown.
Michael Keller. A polyform timeline. World Game Review 9 (Dec 1989) 4-5. This outlines
the history of polyominoes and other polyshapes. Keller and others refer to polyaboloes
as polytans.
Rodolfo Marcelo Kurchan (Parana 960 5 "A", 1017 Buenos Aires, Argentina). Puzzle Fun,
starting with No. 1 (Oct 1994). This is a magazine entirely devoted to polyomino and
other polyform puzzles. Many of the classic problems are extended in many ways here.
In No. 6 (Aug 1995) he presents a labelling of the 12 hexiamonds by the letters A, C,
H, I, J, M, O, P, S, V, X, Y, which he obtained from Anton Hanegraaf. I have never
seen this before.
Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Vol. 1, recreation 23, pp. 64-66.
SOURCES - page 172
Considers figures formed of isosceles right triangles. He has eight of these, coloured
with eight colours, and uses some of them to form "chequers or regular four-sided
figures, different either in form or colour".
Book of 500 Puzzles. 1859. Triangular problem, pp. 74-75. Identical to Hooper, dropping
the last sentence.
Dudeney. CP. 1907. Prob. 74: The broken chessboard, pp. 119-121 & 220-221. The 12
pentominoes and a 2 x 2.
A. Aubry. Prob. 3224. Interméd. Math 14 (1907) 122-124. ??NYS -- cited by Grünbaum &
Shephard who say Aubry has something of the idea or the term polyominoes.
G. Quijano. Prob. 3430. Interméd. Math 15 (1908) 195. ??NYS -- cited by Grünbaum &
Shephard, who say he first asked for the number of n-ominoes.
Thomas Scrutchin. US Patent 895,114 -- Puzzle. Applied: 20 Feb 1908; patented: 4 Aug
1908. 2pp + 1p diagrams. Mentioned in S&B, p. 18. A polyiamond puzzle -- triangle
of side 8, hence with 64 triangles, apparently cut into 10 pieces (my copy is rather faint
-- replace??).
Thomas W. Lancaster. US Patent 1,264,944 -- Puzzle. Filed: 7 May 1917; patented: 7 May
1918. 2pp + 1p diagrams. For a general polyrhomb puzzle making a rhombus. His
diagram shows an 11 x 11 rhombus filled with 19 pieces formed from 4 to 10
rhombuses.
John Milner Lester. US Patent 1,290,761 -- Game Apparatus. Filed: 6 Feb 1918; patented:
7 Jan 1919. 2pp + 3pp diagrams. Fairly general assembly puzzle claims. He
specifically illustrates a polyomino puzzle and a polyabolo puzzle. The first has a
Greek cross of edge 3 (hence containing 45 unit cells) to be filled with polyominoes - 11: 01154. The second has an 8-pointed star formed by superimposing two 4 x 4
squares. This has area 20 and hence contains 40 isosceles right triangles of edge 1,
which is the basic unit of this type of puzzle. There are 11: 0128 pieces.
Blyth. Match-Stick Magic. 1921. Spots and squares, pp. 68-73. He uses matchsticks broken
in thirds, so it is easier to describe with units of one-third. 6 units, 4 doubles and
2 triples. Some of the pieces have black bands or spots. Object is to form polyomino
shapes without pieces crossing, but every intersection must have a black spot. 19
polyomino shapes are given to construct, including 7 of the pentominoes, though some
of the shapes are only connected at corners.
"John Bull" Star of Fortune Prize Puzzle. 1922. This is a puzzle with 20 pieces, coloured
red on one side, containing 6 through 13 triangles to be assembled into a star of
David with 4 triangles along each edge (hence 12 x 16 = 192 triangles). Made by
Chad Valley. Prize of £250 for a red star matching the key solution deposited at a bank;
£150 for solution closest to the key; £100 for a solution with 10 red and 10 grey
pieces, or as nearly as possible. Closing date of competition is 27 Dec 1922. Puzzle
made by Chad Valley Co. as a promotional item for John Bull magazine, published by
Odhams Press. A copy is in the toy shop of the Buckleys Shop Museum, Battle, East
Sussex, to whom I am indebted for the chance to examine the puzzle and a photocopy
of the puzzle, box and solution.
Daily Sketch Jig-Saw Puzzle. By Chad Valley. Card polyiamonds. 39: 0,0,1,5,6, 12,9,6,
with a path printed on one side, to assemble into a shape of 16 rows of 15 with four
corners removed and so the printed sides form a continuous circuit. In box with shaped
bottom. Instructions on inside cover and loose sheet to submit solution. No dates
given, but appears to be 1920s, though it is somewhat similar to the Daily Mail Crown
Puzzle of 1953 -- cf below -- so it might be much later.
Daily Mirror Zig-Zag £500 Prize Puzzle. By Chad Valley. Card polyiamonds.
29: 0,0,1,1,4, 5,6,3,1,4, 1,2,1. One-sided pieces to fit into frame in card box. Three
pieces are duplicated and one is triplicated. Solution and claim instructions appeared in
Daily Mirror (17 Jan 1930) 1-2. See: Tom Tyler & Felicity Whiteley; Chad Valley
Promotional Jig-Saw Puzzles; Magic Fairy Publishing, Petersfield, Hampshire, 1990,
p. 55.
B. T.s Zig-Zag. B.T. is a Copenhagen newspaper. Polyiamond puzzle.
33: 0,0,1,2,5, 6,7,2,2,4, 1,1,1, Some repetitions, so I only see 20 different shapes. To
be fit into an irregular frame. Solution given on 23 Nov 1931, pp. 1-2. (I have a
photocopy of the form to fill in; an undated set of rules, apparently from the paper,
saying the solutions must be received by 21 Nov; and the pages giving the solution;
provided by Jan de Geus.)
Herbert D. Benjamin. Problem 1597: A big cutting-out design -- and a prize offer.
SOURCES - page 173
Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:9 (Dec 1934)
92. Finds the 35 hexominoes and asks if they form a 14 x 15 rectangle. Cites
Dudeney (Tribune (20 Dec 1906)); Loyd (OPM (Apr-Jul 1908)) (see 6.F.1); Dudeney
(CP, no. 74) (see above) and some other chessboard dissections. Jelliss says this is the
first dissection problem in this journal.
F. Kadner. Solution 1597. Problemist Fairy Chess Supplement (later called Fairy Chess
Review) 2:10 (Feb 1935) 104-105. Shows the 35 hexominoes cannot tile a rectangle
by two arguments, both essentially based on two colouring. Gives some other results
and some problems are given as 1679-1681 -- ??NYS.
William E. Lester. Correction to 1597. Problemist Fairy Chess Supplement (later called Fairy
Chess Review) 2:11 (Apr 1935) 121. Corrects an error in Kadner. Finds a number of
near-solutions. Editor says Kadner insists the editor should take credit for the twocolouring form of the previous proof.
Frans Hansson, proposer & solver?. Problem 1844. Problemist Fairy Chess Supplement
(later called Fairy Chess Review) 2:12 (Jun 1935) 128 & 2:13 (Aug 1935) 135. Finds
both 3 x 20 pentomino rectangles.
W. E. Lester & B. Zastrow, proposers. Problem 1923. Problemist Fairy Chess Supplement
(later called Fairy Chess Review) 2:13 (Aug 1935) 138. Take an 8 x 8 board and
remove its corners. Fill this with the 12 pentominoes.
H. D. Benjamin, proposer. Problem 1924. Problemist Fairy Chess Supplement (later called
Fairy Chess Review) 2:13 (Aug 1935) 138. Dissect an 8 x 8 into the 12 pentominoes
and the I-tetromino. Need solution -- ??NYS.
Thomas Rayner Dawson & William E. Lester. A notation for dissection problems. Fairy
Chess Review 3:5 (Apr 1937) 46-47. Gives all n-ominoes up to n = 6. Describes the
row at a time notation. Shows the pentominoes and a 2 x 2 cover the chessboard with
the 2 x 2 in any position. Asserts there are 108 7-ominoes and 368 8-ominoes -citing F. Douglas & W. E. L[ester] for the hexominoes and J. Niemann for the
heptominoes.
H. D. Benjamin, proposer. Problem 3228. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect
a 5 x 5 into the five tetrominoes and a pentomino so that the pentomino touches all the
tetrominoes along an edge. Asserts the solution is unique. Refers to problems
3026-3030 -- ??NYS.
H. D. Benjamin, proposer. Problem 3229. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect
an 8 x 8 into the 12 pentominoes and a tetromino so that all pieces touch the edge of
the board. Asserts only one tetromino works.
T. R. D[awson], proposer. Problems 3230-1. Fairy Chess Review 3:12 (Jun 1938) 129.
Extends prob. 3229 to ask for solutions with 12 pieces on the edge, using two other
tetrominoes. Thinks it cannot be done with the remaining two tetrominoes.
Editorial note: The colossal count. Fairy Chess Review 3:12 (Jun 1938) 131. Describes
progress on enumerating 8-ominoes (four people get 368 but Niemann gets 369) and
9-ominoes (numbers vary from 1237 to 1285). All workers are classifying them by
the size of the smallest containing rectangle.
W. H. Rawlings, proposer. Problem 3930. Fairy Chess Review 4:3 (Nov 1939) 28. How
many pentacubes are there? Ibid. 4:4 (Feb 1940) 75, reports that both 25 or 26 are
claimed, but the editor has only seen 24. Ibid. 4:5 (Apr 1940) 85, reports that
R. J. F[rench] has clearly shown there are 23 -- but this considers reflections as equal - cf the 1948 editorial note.
R. J. French, proposer and solver. Problem 4149. Fairy Chess Review 4:3 (Nov 1939) 43 &
4:6 (Jun 1940) 93. Asks for arrangement of the pentominoes with the largest hole and
gives one with 127 squares in the hole. (See: G. P. Jelliss; Comment on Problem
1277; JRM 22:1 (1990) 69. This reviews various earlier solutions and comments on
Problem 1277.)
J. Niemann. Item 4154: "The colossal count". Fairy Chess Review 4:3 (Nov 1939) 44-45.
Announces that there are 369 8-ominoes, 1285 9-ominoes and 4654 10-ominoes,
but Keller and Jelliss note that he missed a 10-omino which was not corrected until
1966.
H. D. Benjamin. Unpublished notes. ??NYS -- cited and briefly described in G. P. Jelliss;
Prob. 48 -- Aztec tetrasticks; G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss says Benjamin
studied polysticks, which he called 'lattice dissections' around 1946-1948 and that some
results by him and T. R. Dawson were entered in W. Stead's notebooks but nothing is
known to have been published. For orders 1, 2, 3, 4, there are 1, 2, 5, 16 polysticks.
SOURCES - page 174
Benjamin formed these into a 6 x 6 lattice square. Jelliss then mentions Barwell's
rediscovery of them and goes on to a new problem -- see Knuth, 1999.
D. Nixon, proposer and solver. Problem 7560. Fairy Chess Review 6:16 (Feb 1948) 12 &
6:17 (Apr 1948) 131. Constructs 3 x 4 x 5 from solid pentominoes.
Editorial discussion: Space dissection. Fairy Chess Review 6:18 (Jun 1948) 141-142. Says
that several people have verified the 23 pentacubes but that 6 of them have mirror
images, making 29 if these are considered distinct. Says F. Hansson has found 77
6-cubes (these exclude mirror images and the 35 solid 6-ominoes). Gives many
problems using n-cubes and/or solid polyominoes, which he calls flat n-cubes -- some
are corrected in 7:2 (Oct 1948) 16 (erroneously printed as 108).
J. Niemann. The dissection count. Item 7803. Fairy Chess Review 7:1 (Aug 1948) 8
(erroneously printed as 100). Reports on counting n-cubes. Gets the following.
n=
4
5
6
7
flat pieces
5
12
35
108
non-flats
2
11
77
499
TOTAL
7
23
112
607
mirror images
1
6
55
416
GRAND TOTAL
8
29
167
1023
R. J. French. Space dissections. Fairy Chess Review 7:2 (Oct 1948) 16 (erroneously printed
as 108). French writes that he and A. W. Baillie have corrected the number of 6-cubes
to 35 + 77 + 54 = 166. Baillie notes that every 6-cube lies in two layers -- i.e. has
some width  2 -- and asks for the result for n-cubes as prob. 7879. [I suspect the
answer is that n  3k implies that an n-cube has some width  k.] Editor adds some
corrections to the discussion in 6:18.
Editorial note. Fairy Chess Review 7:3 (Dec 1948) 23. Niemann and Hansson confirm the
number 166 given in 7:2.
Daily Mail Crown Puzzle. Made by Chad Valley Co. 1953. 26 pieces, coloured on one side,
to be fit into a crown shape. 11 are border pieces and easily placed. The other 15 are
polyiamonds: 15: 00112 24012 11. Prize of £100 for solution plus best slogan, entries
due on 8 Jun 1953.
S. W. Golomb. Checkerboards and polyominoes. AMM 61 (1954) 675-682. Mostly
concerned with covering the 8 x 8 board with copies of polyominoes. Shows one
covering with the 12 pentominoes and the square tetromino. Mentions that the idea
can be extended to hexagons. S&B, p. 18, and Gardner (Dec 1964) say he mentions
triangles, but he doesn't.
Walter S. Stead. Dissection. Fairy Chess Review 9:1 (Dec 1954) 2-4. Gives many
pentomino and hexomino patterns -- e.g. one of each pattern of 8 x 8 with a 2 x 2
square deleted. "The possibilities of the 12 fives are not infinite but they will provide
years of amusement." Includes 3 x 20, 4 x 15, 5 x 12 and 6 x 10 rectangles. No
reference to Golomb. In 1955, Stead uses the 108 heptominoes to make a 28 x 28
square with a symmetric hole of size 28 in the centre -- first printed as cover of
Chessics 28 (1986).
Jules Pestieau. US Patent 2,900,190 -- Scientific Puzzle. Filed: 2 Jul 1956; patented: 18 Aug
1959. 2pp + 1p diagrams. For the 12 pentominoes! Diagram shows the 6 x 10
solution with two 5 x 6 rectangles and shows the two-piece non-symmetric
equivalence of the N and F pieces. Pieces have markings on one side which may be
used -- i.e. pieces may not be turned over. Mentions possibility of using n-ominoes.
Gardner. SA (Dec 1957) = 1st Book, chap. 13. Exposits Golomb and Stead. Gives number
of n-ominoes for n = 1, ..., 7. 1st Book describes Scott's work. Says a pentomino set
called 'Hexed' was marketed in 1957. (John Brillhart gave me and my housemates an
example in 1960 -- it took us two weeks to find our first solution.)
Dana Scott. Programming a Combinatorial Puzzle. Technical Report No. 1, Dept. of Elec.
Eng., Princeton Univ., 1958, 20pp. Uses MANIAC to find 65 solutions for
pentominoes on an 8 x 8 board with square 2 x 2 in the centre. Notes that the 3 x 20
pentomino rectangle has just two solutions. In 1999, Knuth notes that the total number
of solutions with the 2 x 2 being anywhere does not seem to have ever been published
and he finds 16146.
M. Gardner. SA (Sep 1958) c= 2nd Book, chap. 6. First general mention of solid
pentominoes, pentacubes, tetracubes. In the Addendum in 2nd Book, he says Theodore
Katsanis of Seattle suggested the eight tetracubes and the 29 pentacubes in a letter to
Gardner on 23 Sep 1957. He also says that Julia Robinson and Charles W. Stephenson
SOURCES - page 175
both suggested the solid pentominoes.
C. Dudley Langford. Note 2793: A conundrum for form VI. MG 42 (No. 342) (Dec 1958)
287. 4 each of the L, N, and T (= Y) tetrominoes make a 7 x 7 square with the
centre missing. Also nine pieces make a 6 x 6 square but this requires an even number
of Ts.
M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 8. Use the pentominoes to make two 5 x 5 squares at the same time.
Solution just says there are several ways to do so.
J. C. P. Miller. Pentominoes. Eureka 23 (Oct 1960) 13-16. Gives the Haselgroves' number
of 2339 solutions for the 6 x 10 and says there are 2 solutions for the 3 x 20. Says
Lehmer suggests assembling 12 solid pentominoes into a 3 x 4 x 5 and van der Poel
suggests assembling the 12 hexiamonds into a rhombus.
C. B. & Jenifer Haselgrove. A computer program for pentominoes. Ibid., 16-18. Outlines
program which found the 2339 solutions for the 6 x 10. It is usually said that they
also found all solutions of the 3 x 20, 4 x 15 and 5 x 12, but I don't see it mentioned
here and in JRM 7:3 (1974) 257, it is reported that Jenifer (Haselgrove) Leech stated
that only the 6 x 10 and 3 x 20 were done in 1960, but that she did the 5 x 12 and
4 x 15 with a new program in c1966. See Fairbairn, c1962, and Meeus, 1973.
Richard K. Guy. Some mathematical recreations I & II. Nabla [= Bull. Malayan Math. Soc.]
7 (Oct & Dec 1960) 97-106 & 144-153. Considers handed polyominoes, i.e.
polyominoes when reflections are not considered equivalent. Notes that neither the 5
plain nor the 7 handed tetrominoes can form a rectangle. The 10 chequered handed
tetrominoes form 4 x 10 and 5 x 8 rectangles and he has several solutions of each.
There is no 2 x 20 rectangle. Discusses MacMahon pieces -- cf 5.H.2 -- and
polyiamonds. He uses the word 'hexiamond', but not 'polyiamond' -- in an email of 8
Apt 2000, Guy says that O'Beirne invented all the terms. He considers making a
'hexagon' from the 19 hexiamonds. Part II considers solid problems and uses the term
'solid pentominoes'.
Solomon W. Golomb. The general theory of polyominoes: part 2 -- Patterns and
polyominoes. RMM 5 (Oct 1961) 3-14. ??NYR.
J. E. Reeve & J. A. Tyrrell. Maestro puzzles. MG 45 (No. 353) (Oct 1961) 97-99. Discusses
hexiamond puzzles, using the 12 reversible pieces. [The puzzle was marketed under
the name 'Maestro' in the UK.]
T. H. O'Beirne. Pell's equation in two popular problems. New Scientist 12 (No. 258)
(26 Oct 1961) 260-261.
T. H. O'Beirne. Pentominoes and hexiamonds. New Scientist 12 (No. 259) (2 Nov 1961)
316-317. This is the first use of the word 'polyiamond'. He considers the 19 one-sided
pieces. He says he devised the pieces and R. K. Guy has already published many
solutions in Nabla. He asks for the number of ways the 18 one-sided pentominoes can
fill a 9 x 10. In 1999, Knuth found this would take several months.
T. H. O'Beirne. Some hexiamond solutions: and an introduction to a set of 25 remarkable
points. New Scientist 12 (No. 260) (9 Nov 1961) 378-379.
Maurice J. Povah. Letter. MG 45 (No. 354) (Dec 1961) 342. States Scott's result of 65 and
the Haselgroves' result of 2339 (computed at Manchester). Says he has over 7000
solutions for the 8 x 8 board using a 2 x 2.
T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961)
751-752.
T. H. O'Beirne. Some tetrabolic difficulties. New Scientist 13 (No. 270) (18 Jan 1962)
158-159. These two columns are the first mention of tetraboloes, so named by
S. J. Collins.
R. A. Fairbairn. Pentomino Problems: The 6 x 10, 5 x 12, 4 x 15, and 3 x 20 Rectangles -The Complete Drawings. Unpublished MS, undated, but c1962, based on the
Haselgroves' work of 1960. ??NYS -- cited by various authors, e.g. Madachy (1969),
Torbijn (1969), Meeus (1973). Madachy says Fairbairn is from Willowdale, Ontario,
and takes some examples from his drawings. However, the dating is at variance with
Jenifer Haselgrove's 1973 statement - cf Haselgrove, 1960. Perhaps this MS is
somewhat later?? Does anyone know where this MS is now? Cf Meeus, 1973.
Serena Sutton Besley. US Patent 3,065,970 -- Three Dimensional Puzzle. Filed: 6 Jul 1960;
issued: 27 Nov 1962. 2pp + 4pp diagrams. For the 29 pentacubes, with one piece
duplicated giving a set of 30. Klarner had already considered omitting the 1 x 1 x 5
and found that he could make two separate 2 x 5 x 7s. Besley says the following can be
SOURCES - page 176
made: 5 x 5 x 6, 3 x 5 x 10, 2 x 5 x 15, 2 x 3 x 25; 3 x 5 x 6, 3 x 3 x 10, 2 x 5 x 9,
2 x 3 x 15; 3 x 4 x 5, 2 x 5 x 6, 2 x 3 x 10 (where the latter three are made with the
12 solid pentominoes and the previous four are made with the 18 non-planar
pentacubes) but detailed solutions are only given for the 5 x 5 x 6, 3 x 5 x 6, 3 x 4 x 5.
Mentions possibility of n-cubes.
M. Gardner. Polyiamonds. SA (Dec 1964) = 6th Book, chap. 18. Exposits basic ideas and
results for the 12 double sided hexiamonds. Poses several problems which are
answered by readers. The six-pointed star using 8 pieces has a unique solution. John
G. Fletcher and Jenifer (Haselgrove) Leech both showed the 3 x 12 rhombus is
impossible. Fletcher found the 3 x 11 rhombus has 24 solutions, all omitting the 'bat'.
Leech found 155 solutions for the 6 x 6 rhombus and 74 solutions for the 4 x 9.
Mentions there are 160 9-iamonds, one with a hole.
John G. Fletcher. A program to solve the pentomino problem by the recursive use of macros.
Comm. ACM 8 (1965) 621-623. ??NYS -- described by Knuth in 1999 who says that
Fletcher found the 2339 solutions for the 6 x 10 in 10 minutes on an IBM 7094 and
that the program remains the fastest known method for problems of placing the 12
pentominoes.
M. Gardner. Op art. SA (Jul 1965) = 6th Book, chap. 24. Shows the 24 heptiamonds and
discusses which will tile the plane.
Solomon W. Golomb. Tiling with polyominoes. J. Combinatorial Theory 1 (1966) 280-296.
??NYS. Extended by his 1970 paper.
T. R. Parkin. 1966. ??NYS -- cited by Keller. Finds 4655 10-ominoes.
M. Gardner. SA (Jun 1967) = Magic Show, chap. 11. First mention of polyhexes.
C. J. Bouwkamp. Catalogue of Solutions of the Rectangular 3 x 4 x 5 Solid Pentomino
Problem. Dept. of Math., Technische Hogeschool Eindhoven, July 1967, reprinted
1981, 310pp.
C. J. Bouwkamp. Packing a rectangular box with the twelve solid pentominoes.
J. Combinatorial Thy. 7 (1969) 278-280. He gives the numbers of solutions for
rectangles as 'known'.
2 x 3 x 10 can be packed in
12 ways, which are given.
2 x 5 x 6 can be packed in
264 ways.
3 x 4 x 5 can be packed in 3940 ways. (See his 1967 report.)
T. R. Parkin, L. J. Lander & D. R. Parkin. Polyomino enumeration results. Paper presented
at the SIAM Fall Meeting, Santa Barbara, 1 Dec 1967. ??NYS -- described by
Madachy, 1969. Gives numbers of n-ominoes, with and without holes, up to n = 15,
done two independent ways.
Joseph S. Madachy. Pentominoes -- Some solved and unsolved problems. JRM 2:3 (Jul
1969) 181-188. Gives the numbers of Parkin, Lander & Parkin. Shows various
examples where a rectangle splits into two congruent halves. Discusses various other
problems, including Bouwkamp's 3 x 4 x 5 solid pentomino problem. Bouwkamp
reports that the final total of 3940 was completed on 16 Mar 1967 after about three
years work using three different computers, but that a colleague's program would now
do the whole search in about three hours.
P. J. Torbijn. Polyiamonds. JRM 2:4 (Oct 1969) 216-227. Uses the double sided
hexiamonds and heptiamonds. A few years before, he found, by hand, that there are
156 ways to cover the 6 x 6 rhombus with the 12 hexiamonds and 74 ways for the 4
x 9, but could find no way to cover the 3 x 12. The previous year, John G. Fletcher
confirmed these results with a computer and he displays all of these -- but this
contradicts Gardner (Dec 64) -- ?? He gives several other problems and results,
including using the 24 heptiamonds to form 7 x 12, 6 x 14, 4 x 21 and 3 x 28
rhombuses.
Solomon W. Golomb. Tlling with sets of polyominoes. J. Combinatorial Theory 9 (1970)
60-71. ??NYS. Extends his 1966 paper. Asks which heptominoes tile rectangles and
says there are two undecided cases -- cf Marlow, 1985. Gardner (Aug 75) says Golomb
shows that the problem of determining whether a given finite set of polyominoes will
tile the plane is undecidable.
C. J. Boukamp & D. A. Klarner. Packing a box with Y-pentacubes. JRM 3:1 (1970) 10-26.
Substantial discussion of packings with Y-pentominoes and Y-pentacubes. Smallest
boxes are 5 x 10 and 2 x 5 x 6 and 3 x 4 x 5.
Fred Lunnon. Counting polyominoes. IN: Computers in Number Theory, ed. by
A. O. L. Atkin & B. J. Birch; Academic Press, 1971, pp. 347-372. He gets up through
SOURCES - page 177
18-ominoes, but the larger ones can have included holes. The numbers for n = 1, 2, ...,
are as follows: 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591,
901971, 3426576, 13079255, 50107911, 192622052. These values have been quoted
numerous times.
Fred Lunnon. Counting hexagonal and triangular polyominoes. IN: Graph Theory and
Computing, ed. by R. C. Read; Academic Press, 1972, pp. 87-100. ??NYS -- cited by
Grünbaum & Shephard.
M. Gardner. SA (Sep 1972). c= Knotted, chap. 3. Says the 8 tetracubes were made by
E. S. Lowe Co. in Hong Kong and marketed as "Wit's End". Says an MIT group found
1390 solutions for the 2 x 4 x 4 box packed with tetracubes. He reports that several
people found that there are 1023 heptacubes -- but see Niemann, 1948, above. Klarner
reports that the heptacubes fill a 2 x 6 x 83.
Jean Meeus. Some polyomino and polyamond [sic] problems. JRM 6:3 (1973) 215-220.
(Corrections in 7:3 (1974) 257.) Considers ways to pack a 5 x n rectangle with some
n pentominoes. A. Mank found the number of ways for n = 2, 3, ..., 11 as follows,
and the number for n = 12 was already known:
0, 7, 50, 107, 541, 1387, 3377, 5865, 6814, 4103, 1010.
Says he drew out all the solutions for the area 60 rectangles in 1972 (cf Fairbairn,
c1962). Finds that 520 of the 6 x 10 rectangles can be divided into two congruent
halves, sometimes in two different ways. For 5 x 12, there are 380; for 4 x 15, there
are 94. Gives some hexomino rectangles by either deleting a piece or duplicating one,
and an 'almost 11 x 19'. Says there are 46 solutions to the 3 x 30 with the 18 onesided pentominoes and attributes this to Mrs (Haselgrove) Leech, but the correction
indicates this was found by A. Mank.
Jenifer Haselgrove. Packing a square with Y-pentominoes. JRM 7:3 (1974) 229. She finds
and shows a way to pack 45 Y-pentominoes into a 15 x 15, but is unsure if there are
more solutions. In 1999, Knuth found 212 solutions. She also reports the
impossibility of using the Y-pentominoes to fill various other rectangles.
S. W. Golomb. Trademark for 'PENTOMINOES'. US trademark 1,008,964 issued
15 Apr 1975; published 21 Jan 1975 as SN 435,448. (First use: November 1953.)
[These appear in the Official Gazette of the United States Patent Office (later Patent and
Trademark Office) in the Trademarks section.]
M. Gardner. Tiling with polyominoes, polyiamonds and polyhexes. SA (Aug 75) (with
slightly different title) = Time Travel, chap. 14. Gives a tiling criterion of Conway.
Describes Golomb's 1966 & 1970 results.
C. J. Bouwkamp. Catalogue of solutions of the rectangular 2 x 5 x 6 solid pentomino
problem. Proc. Koninklijke Nederlandse Akad. van Wetenschappen A81:2 (1978)
177-186. Presents the 264 solutions which were first found in Sep 1967.
H. Redelmeier. Discrete Math. 36 (1981) 191-203. ??NYS -- described by Jelliss. Obtains
number of n-ominoes for n  24.
Karl Scherer. Problem 1045: Heptomino tessellations. JRM 14:1 (1981-82) 64.
XX
Says he has found that the heptomino at the right fills a 26 x 42 rectangle.
XXXXX
See Dahlke below.
David Ellard. Poly-iamond enumeration. MG 66 (No. 438) (Dec 1982) 310-314. For
n = 1, ..., 12, he gets 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, 1186, 3342 n-iamonds. One
of the 8-iamonds has a hole and there are many later cases with holes.
Anon. 31: Polyominoes. QARCH 1:8 (June 1984) 11-13. [This is an occasional publication
of The Archimedeans, the student maths society at Cambridge.] Good survey of
counting and asymptotics for the numbers of polyominoes, up to n = 24, polycubes,
etc. 10 references.
T. W. Marlow. Grid dissections. Chessics 23 (Autumn, 1985) 78-79.
X
XX
Shows XXXXX fills a 23 x 24 and XXXXX fills a 19 x 28.
Herman J. J. te Riele & D. T. Winter. The tetrahexes puzzle. CWI Newsletter [Amsterdam]
10 (Mar 1986) 33-39. Says there are: 7 tetrahexes, 22 pentahexes, 82 hexahexes,
333 heptahexes, 1448 octahexes. Studies patterns of 28 hexagons. Shows the
triangle cannot be constructed from the 7 tetrahexes and gives 48 symmetric patterns
that can be made.
Karl A. Dahlke. Science News 132:20 (14 Nov 1987) 310. (??NYS -- cited in JRM 21:3 and
XX
22:1 and by Marlow below.) Shows XXXXX fills a 21 x 26 rectangle.
SOURCES - page 178
The results of Scherer and Dahlke are printed in JRM 21:3 (1989) 221-223 and
Dahlke's solution is given by Marlow below.
Karl A. Dahlke. J. Combinatorial Theory A51 (1989) 127-128. ??NYS -- cited in JRM 22:1.
Announces a 19 x 28 solution for the above heptomino problem, but the earlier
21 x 26 solution is printed by error. The 19 x 28 solution is printed in JRM 22:1
(1990) 68-69.
Tom Marlow. Grid dissections. G&PJ 12 (Sep/Dec 1989) 185. Prints Dahlke's result.
Brian R. Barwell. Polysticks. JRM 22:3 (1990) 165-175. Polysticks are formed of unit
lengths on the square lattice. There are: 1, 2, 5, 16, 55 polysticks formed with
1, 2, 3, 4, 5 unit lengths. He forms 5 x 5 squares with one 4-stick omitted, but he
permits pieces to cross. He doesn't consider the triangular or hexagonal cases. See also
Blyth, 1921, for a related puzzle. Cf Benjamin, above, and Wiezorke & Haubrich,
below.
General Symmetrics (Douglas Engel) produced a version of polysticks, ©1991, with 4
3-sticks and 3 4-sticks to make a 3 x 3 square array with no crossing of pieces.
Nob Yoshigahara. Puzzlart. Tokyo, 1992. Ip-pineapple (pineapple delight), pp. 78-81.
Imagine a cylindrical solution of the 6 x 10 pentomino rectangle and wrap it around a
cylinder, giving each cell a depth of one along the radius. Hence each cell is part of an
annulus. He reduces the dimensions along the short side to make the cells look like
tenths of a slice of pineapple. Nob constructed and example for Toyo Glass's puzzle
series and it was later found to have a unique solution.
Kate Jones, proposer; P. J. Torbijn, Jacques Haubrich, solvers. Problem 1961 -Rhombiominoes. JRM 24:2 (1992) 144-146 & 25:3 (1993) 223-225. A rhombiomino
or polyrhomb is a polyomino formed using rhombi instead of squares. There are 20
pentarhombs. Fit them into a 10 x 10 rhombus. Various other questions. Haubrich
found many solutions. See Lancaster, 1918.
Bernard Wiezorke & Jacques Haubrich. Dr. Dragon's polycons. CFF 33 (Feb 1994) 6-7.
Polycons (for connections) are the same as the polysticks described by Barwell in 1990,
above. Authors describe a Taiwanese version on sale in late 1993, using 10 of the
4-sticks suitably shortened so they fit into the grooves of a 4 x 4 board -- so crossings
are not permitted. (An n x n board has n+1 lines of n edges in each direction.) They
fit 15 of the 4-sticks onto a 5 x 5 board and determine all solutions.
CFF 35 (Dec 1994) 4 gives a number of responses to the article. Brain Barwell
wrote that he devised them as a student at Oxford, c1970, but did not publish until
1990. He expected someone to say it had been done before, but no one has done so. He
also considered using the triangular and hexagonal lattice. He had just completed a
program to consider fitting 15 of the 4-sticks onto a 5 x 5 board and found over
180,000 solutions, with slightly under half having no crossings, confirming the results
of Wiezorke & Haubrich.
Dario Uri also wrote that he had invented the idea in 1984 and called them
polilati (polyedges). Giovanni Ravesi wrote about them in Contromossa (Nov 1984) 23
-- a defunct magazine.
Chris Roothart. Polylambdas. CFF 34 (Oct 1994) 26-28. A lambda is a 30o-60o-90o
triangle. These may be joined along corresponding legs, but not along hypotenuses.
For n = 1, 2, 3, 4, 5, there are 1, 4, 4, 11, 12 n-lambdas. He gives some problems
using various sets of these pieces.
Richard Guy. Letters of 29 May and 13 Jun 1996. He is interested in using the 19 one-sided
hexiamonds. Hexagonal rings of hexagons contain 1, 6, 12 hexagons, so the hexagon
with three hexagons on a side has 19 hexagons. If these hexagons are considered to
comprise six equilateral triangles, we have a board with 19 x 6 triangles. O'Beirne
asked for the number of ways to fill this board with the one-sided hexiamonds. Guy has
collected over 4200 solutions. A program by Marc Paulhus found 907 solutions in
eight hours, from which it initially estimated that there are about 30,000 solutions.
The second letter gives the final results -- there are 124,518 solutions. This is modulo
the 12 symmetries of the hexagon. In 1999, Knuth found 124,519 and Paulhus has
rerun his program and found this number.
Ferdinand Lammertink. Polyshapes. Parts 1 and 2. The author, Hengelo, Netherlands, 1996
& 1997. Part 1 deals with two dimensional puzzles. Good survey of the standard
polyform shapes and many others.
Hilarie Korman. Pentominoes: A first player win. IN: Games of No Chance; ed. by Richard
Nowakowski; CUP, 1997??, ??NYS - described in William Hartston; What
SOURCES - page 179
mathematicians get up to; The Independent Long Weekend (29 Mar 1997) 2. This
studies the game proposed by Golomb -- players alternately place one of the
pentominoes on the chess board, aligned with the squares and not overlapping the
previous pieces, with the last one able to play being the winner. She used a Sun IPC
Sparcstation for five days, examining about 22 x 109 positions to show the game is a
first player win.
Nob Yoshigahara found in 1994 that the smallest box which can be packed with Wpentacubes is 5 x 6 x 6. In 1997, Yoshya (Wolf) Shindo found that one can pack the 6
x 10 x 10 with Z-pentacubes, but it is not known if this is the smallest such box. These
were the last unsolved problems as to whether a box could be packed with a planar
pentacube (= solid pentomino).
Marcel Gillen & Georges Philippe. Twinform 462 Puzzles in one. Solutions for Gillen's
puzzle exchange at IPP17, 1997, 32pp + covers. Take 6 of the pentominoes and place
them in a 7 x 5 rectangle, then place the other six to make the same shape on top of the
first shape. There are 462 (= BC(12,6)/2) possible puzzles and all of them have
solutions. Taking F, T, U, W, X, Z for the first layer, there is just one solution; all
other cases have multiple solutions, totalling 22,873 solutions, but only one solution
for each case is given here.)
Richard K. Guy. O'Beirne's hexiamond. In: The Mathemagican and Pied Puzzler; ed. by
Elwyn Berlekamp & Tom Rodgers, A. K. Peters, Natick, Massachusetts, 1999, pp.
85-96. He relates that O'Beirne discovered the 19 one-sided hexiamonds in c1959 and
found they would fill a hexagonal shape in Nov 1959 and in Jan 1960 he found a
solution with the hexagonal piece in the centre. He gives Paulhus's results (see Guy's
letters of 1996), broken down in various ways. He gives the number of double-sided
(i.e. one can turn them over) and single-sided n-iamonds for n = 1, ..., 7. Cf Ellard,
1982, for many more values for the double-sided case.
n
1 2 3 4 5 6 7
double
1 1 1 3 4 12 24
single
1 1 1 4 6 19 44
In 1963, Conway and Mike Guy considered looking for 'symmetric' solutions for filling
the hexagonal shape with the 19 one-sided hexiamonds. A number of these are
described.
Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by
the author. Available as: http://www-cs-faculty.stanford.edu/~knuth/preprints.html .
In this he introduces a new technique for backtrack programming which runs faster
(although it takes more storage) and is fairly easy to adapt to different problems. In this
approach, there is a symmetry between pieces and cells. He applies it to several
polyshape problems, obtaining new, or at least unknown, results. He extends Scott's
1958 results to get 16146 ways to pack the 8 x 8 with the 12 pentominoes and the
2 x 2. He describes Fletcher's 1965 work. He extends Haselgrove's 1974 work and
finds 212 ways to fit 15 Y-pentominoes in a 15 x 15. Describes Torbijn's 1969 work
and Paulhus' 1996 work on hexiamonds, correcting the latter's number to 124,519. He
then looks for the most symmetric solutions for filling the hexagonal shape with the 19
one-sided hexiamonds, in the sense discussed by Guy (1999). He then considers the 18
one-sided pentominoes (cf Meeus (1973)) and tries the 9 x 10, but finds it would take
a few months on his computer (a 500 MHz Pentium III), so he's abandoned it for now.
He then considers polysticks, citing an actual puzzle version that I've not seen. He
adapts his program to them. He considers the 'welded tetrasticks' which have internal
junction points. There are six of these and ten if they are taken as one-sided. The ten
can be placed in a 4 x 4 grid. There are 15 unwelded, one-sided, tetrasticks, but they
do not form a square, nor indeed any nice shape. He considers all 25 one-sided
tetrasticks and asks if they can be fit into what he calls an Aztec Diamond, which is the
shape looking like a square tilted 45o on the square lattice. The rows contain 1, 3, 5, 7,
9, 7, 5, 3, 1 cells. He thinks an exhaustive search is beyond present computing power.
G. P. Jelliss. Prob. 48 -- Aztec tetrasticks. G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss first
discusses Benjamin's work on polysticks (see at 1946-1948 above) and Barwell's
rediscovery of them (see above). He then describes Knuth's Dancing Links and gives
the Aztec Diamond problem. Jelliss has managed to get all but one of the polysticks
into the shape, but feels it is impossible to get them all in.
Harry L. Nelson. Solid pentomino storage, Question and answer. 1p HO at G4G5, 2002.
1: Can one put all the solid pentominoes into a cube of edge 4.5? What is the smallest
SOURCES - page 180
cube into which they can all be placed? He gives 2 solutions to 1 and a solution due to
Wei-Hwa Huang for a cube of edge 4.405889..., which is conjectured to be minimal. In
fact, one edge of the packing is actually 4, so the volume is less than (4.405889...)3.
This leads me to ask what is the smallest volume of a cuboid, with edges less than 5,
that contains all the solid pentominoes. In Summer 2002, Harry gave me a set of solid
pentominoes in a box with a list of various rectangles and boxes to fit them into: 3 x
22; 3 x 21; 3 x 20; 4 x 16; 4 x 15; 5 x 13; 5 x 12; 6 x 11; 6 x 10; 7 x 9; 8 x 8;
2 x 4 x 8; 2 x 5 x 7; 2 x 5 x 6; 2 x 6 x 6; 3 x 4 x 6; 3 x 4 x 5; 3 x 5 x 5; 4 x 4 x 5;
the given box: 4.4 x 4.4 x 4.9.)
6.F.1.
OTHER CHESSBOARD DISSECTIONS
See S&B, pp. 12-14. See also 6.F.5 for dissections of uncoloured boards.
Jerry Slocum. Compendium of Checkerboard Puzzles. Published by the author, 1983.
Outlines the history and shows all manufactured versions known then to him: 33 types
in 61 versions. The first number in Slocum's numbers is the number of pieces.
Jerry Slocum & Jacques Haubrich. Compendium of Checkerboard Puzzles. 2nd ed.,
published by Slocum, 1993. 90 types in 161 versions, with a table of which pieces are
in which puzzles, making it much easier to see if a given puzzle is in the list or not.
This gives many more pictures of the puzzle boxes and also gives the number of
solutions for each puzzle and sometimes prints all of them. The Slocum numbers are
revised in the 2nd ed. and I use the 2nd ed. numbers below. (There was a 3rd ed. in
1997, with new numbering of 217 types in 376 versions. NYR. Haubrich is working
on an extended version with Les Barton providing information.)
Henry Luers. US Patent 231,963 -- Game Apparatus or Sectional Checker Board. Applied:
7 Aug 1880; patented: 7 Sep 1880. 1p + 1p diagrams. 15: 01329. Slocum 15.5.1.
Manufactured as: Sectional Checker Board Puzzle, by Selchow & Righter. Colour
photo of the puzzle box cover is on the front cover of the 1st ed. of Slocum's booklet.
B&W photo is on p. 14 of S&B.
?? UK patent application 16,810. 1892. Not granted, so never published. I have spoken to
the UK Patent Office and they say the paperwork for ungranted applications is
destroyed after about three to five years. (Edward Hordern's collection has an example
with this number on it, by Feltham & Co. In the 2nd ed., the cover is reproduced and it
looks like the number may be 16,310, but that number is for a locomotive vehicle.)
14: 00149. Slocum 14.20.1. Manufactured as: The Chequers Puzzle, by Feltham & Co.
Hoffmann. 1893. Chap. III, no. 16: The chequers puzzle, pp. 97-98 & 129-130
= Hoffmann-Hordern, pp. 88-89, with photos. 14: 00149. Slocum 14.20.1. Says it is
made by Messrs. Feltham, who state it has over 50 solutions. He gives two solutions.
Photo on p. 89 of a example by Feltham & Co., dated 1880-1895.
At the end of the solution, he says Jacques & Son are producing a series of three
"Peel" puzzles, which have coloured squares which have to be arranged so the same
colour is not repeated in any row or column. Photo on p. 89 shows an example, 9: 023,
with the trominoes all being L-trominoes. This makes a 5 x 5 square, but the colours
have almost faded into indistinguishability.
Montgomery Ward & Co. Catalog No 57, Spring & Summer, 1895. Facsimile by Dover,
1969, ??NX. P. 237 describes item 25470: "The "Wonder" Puzzle. The object is to
place 18 pieces of 81 squares together, so as to form a square, with the colors running
alternately. It can be done in several different ways."
Dudeney. Problem 517 -- Make a chessboard. Weekly Dispatch (4 & 18 Oct 1903), both
p. 10. 8: 00010 12111 001. Slocum 8.3.1.
Benson. 1904. The chequers puzzle, pp. 202-203. As in Hoffmann, with only one solution.
Dudeney. The Tribune (20 & 24 Dec 1906) both p. 1. ??NX Dissecting a chessboard.
Dissect into maximum number of different pieces. Gets 18: 2,1,4,10,0, 0,0,1. Slocum
18.1, citing later(?) Loyd versions.
Loyd. Sam Loyd's Puzzle Magazine (Apr-Jul 1908) -- ??NYS, reproduced in: A. C. White;
Sam Loyd and His Chess Problems; 1913, op. cit. in 1; no. 58, p. 52. = Cyclopedia,
1914, pp. 221 & 368, 250 & 373. = MPSL2, prob. 71, pp. 51 & 145. = SLAHP:
Dissecting the chessboard, pp. 19 & 87. Cut into maximum number of different pieces
-- as in Dudeney, 1906.
SOURCES - page 181
Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY,
1909. The rug, pp. 7-13 & 65. 14: 00149. Not in Slocum.
Loyd. A battle royal. Cyclopedia, 1914, pp. 97 & 351 (= MPSL1, prob. 51, pp. 49 & 139).
Same as Dudeney's prob. 517 of 1903.
Dudeney. AM. 1917. Prob. 293: The Chinese chessboard, pp. 87 & 213-214. Same as Loyd,
p. 221.
Western Puzzle Works, 1926 Catalogue. No. 79: "Checker Board Puzzle, in 16 pieces", but
the picture only shows 14 pieces. 14: 00149. Picture doesn't show any colours, but
assuming the standard colouring of a chess board, this is the same as Slocum 14.15.
John Edward Fransen. US Patent 1,752,248 -- Educational Puzzle. Applied: 19 Apr 1929;
patented: 25 Mar 1930. 1p + 1p diagrams. 'Cut thy life.' 11: 10101 43001.
Slocum 11.3.1.
Emil Huber-Stockar. Patience de l'echiquier. Comptes-Rendus du Premier Congrès
International de Récréation Mathématique, Bruxelles, 1935. Sphinx, Bruxelles, 1935,
pp. 93-94. 15: 01329. Slocum 15.5. Says there must certainly be more than 1000
solutions.
Emil Huber-Stockar. L'echiquier du diable. Comptes-Rendus du Deuxième Congrès
International de Récréation Mathématique, Paris, 1937. Librairie du "Sphinx",
Bruxelles, 1937, pp. 64-68. Discusses how one solution can lead to many others by
partial symmetries. Shows several solutions containing about 40 altogether. Note at
end says he has now got 5275 solutions. This article is reproduced in Sphinx 8 (1938)
36-41, but without the extra pages of diagrams. At the end, a note says he has 5330
solutions. Ibid., pp. 75-76 says he has got 5362 solutions and ibid. 91-92 says he has
5365. By use of Bayes' theorem on the frequency of new solutions, he estimates c5500
solutions. Haubrich has found 6013. Huber-Stocker intended to produce a book of
solutions, but he died in May 1939 [Sphinx 9 (1939) 97].
F. Hansson. Sam Loyd's 18-piece dissection -- Art. 48 & probs. 4152-4153. Fairy Chess
Review 4:3 (Nov 1939) 44. Cites Loyd's Puzzles Magazine. Asserts there are many
millions of solutions! He determines the number of chequered handed n-ominoes for
n = 1, 2, ..., 8 is 2, 1, 4, 10, 36, 110, 392, 1371. The first 17 pieces total 56 squares.
Considers 8 ways to dissect the board into 18 different pieces. Problems ask for the
number of ways to choose the pieces in each of these ways and for symmetrical
solutions. Solution in 4:6 (Jun 1940) 93-94 (??NX of p. 94) says there are a total of
3,309,579 ways to make the choices.
C. Dudley Langford. Note 2864: A chess-board puzzle. MG 43 (No. 345) (Oct 1959) 200.
15: 01248. Not in Slocum. Two diagrams followed by the following text. "The pieces
shown in the diagrams can be arranged to form a square with either side uppermost. If
the squares of the underlying grid are coloured black and white alternately, with each
white square on the back of a black square, then there is at least one more way of
arranging them as a chess-board by turning some of the pieces over." I thought this
meant that the pieces were double-sided with the underside having the colours being the
reverse of the top and the two diagrams were two solutions for this set of pieces.
Jacques Haubrich has noted that the text is confusing and that the second diagram is
NOT using the set of double-sided pieces which are implied by the first diagram. We
are not sure if the phrasing is saying there are two different sets of pieces and hence two
problems or if we are misinterpreting the description of the colouring.
B. D. Josephson. EDSAC to the rescue. Eureka 24 (Oct 1961) 10-12 & 32. Uses the EDSAC
computer to find two solutions of a 12 piece chessboard dissection. 12: 00025 41.
Slocum 12.9.
Leonard J. Gordon. Broken chessboards with unique solutions. G&PJ 10 (1989) 152-153.
Shows Dudeney's problem has four solutions. Finds other colourings which give only
one solution. Notes some equivalences in Slocum.
6.F.2.
COVERING DELETED CHESSBOARD WITH DOMINOES
See also 6.U.2.
There is nothing on this in Murray.
Pál Révész. Op. cit. in 5.I.1. 1969. On p. 22, he says this problem comes from John [von]
Neumann, but gives no details.
Max Black. Critical Thinking, op. cit. in 5.T. 1946 ed., pp. 142 & 394, ??NYS. 2nd ed.,
SOURCES - page 182
1952, pp. 157 & 433. He simply gives it as a problem, with no indication that he
invented it.
H. D. Grossman. Fun with lattice points: 14 -- A chessboard puzzle. SM 14 (1948) 160.
(The problem is described with 'his clever solution' from M. Black, Critical Thinking,
pp. 142 & 394.)
S. Golomb. 1954. Op. cit. in 6.F.
M. Gardner. The mutilated chessboard. SA (Feb 1957) = 1st Book, pp. 24 & 28.
Gamow & Stern. 1958. Domino game. Pp. 87-90.
Robert S. Raven, proposer; Walter P. Targoff, solver. Problem 85 -- Deleted checkerboard.
In: L. A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959, pp. 52
& 227.
R. E. Gomory. (Solution for deletion of any two squares of opposite colour.) In: M. Gardner,
SA (Nov 1962) = Unexpected, pp. 186-187. Solution based on a rook's tour. (I don't
know if this was ever published elsewhere.)
Michael Holt. What is the New Maths? Anthony Blond, London, 1967. Pp. 68 & 97. Gives
the 4 x 4 case as a problem, but doesn't mention that it works on other boards. (I
include this as I haven't seen earlier examples in the educational literature.)
David Singmaster. Covering deleted chessboards with dominoes. MM 48 (1975) 59-66.
Optimum extension to n-dimensions. For an n-dimensional board, each dimension
must be  2. If the board has an even number of cells, then one can delete any n-1
white cells and any n-1 black cells and still cover the board with dominoes (i.e.
2 x 1 x 1 x ... x 1 blocks). If the board has an odd number of cells, then let the corner
cells be coloured black. One can then delete any n black cells and any n-1 white cells
and still cover the board with dominoes.
I-Ping Chu & Richard Johnsonbaugh. Tiling deficient boards with trominoes. MM 59:1
(1986) 34-40. (3,n) = 1 and n  5 imply that an n x n board with one cell deleted can
be covered with L trominoes. Some 5 x 5 boards with one cell deleted can be tiled,
but not all can.
6.F.3.
DISSECTING A CROSS INTO Zs AND Ls
The L pieces are not always drawn carefully, and in some cases the unit pieces are not
all square. I have enlarged and measured those which are not clear and approximated them as
n-ominoes.
Minguet. 1733. Pp. 119-121 (1755: 85-86; 1822: 138-139; 1864: 116-117). The problem has
two parts. The first is a cross into 5 pieces: L-tetromino, 2 Z-pentominoes,
L-hexomino, Z-hexomino. The two hexominoes are like the corresponding
pentominoes lengthened by one unit. Similar to Les Amusemens, but one Z is longer
and one L is shorter. The diagram shows 8 L and Z shaped pieces formed from
squares, but it is not clear what the second part of the problem is doing -- either a piece
or a label is erroneous or missing. Says one can make different figures with the pieces.
Les Amusemens. 1749. P. xxxi. Cross into 3 Z pentominoes and 2 L pieces. Like
Minguet, but the Ls are much lengthened and are approximately a L-heptomino and an
L-octomino.
Catel. Kunst-Cabinet. 1790. Das mathematische Kreuz, p. 10 & fig. 27 on plate I. As in Les
Amusemens, but the Ls are approximately a 9-omino and a 10-omino.
Bestelmeier. 1801. Item 274 -- Das mathematische Kreuz. Cross into 6 pieces, but the
picture has an erroneous extra line. It should be the reversal of the picture in Catel.
Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as
Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the
Essay of Games". F. 4v has the dissection of the cross into 3 Z pentominoes and two
L pieces. I don't have a copy of this, but my sketch looks like the Ls are a tetromino
and a pentomino, or possibly a pentomino and a hexomino.
Manuel des Sorciers. 1825. Pp. 204-205, art. 21. ??NX. Dissect a cross into three Zs and
two Ls. My notes don't indicate the size of the Ls.
Boy's Own Book. 1843 (Paris): 435 & 440, no. 3. As in Les Amusemens, but with the Ls
apparently intended to be a pentomino and a hexomino. = Boy's Treasury, 1844,
pp. 424-425 & 428. = de Savigny, 1846, pp. 353 & 357, no. 2, except the solution has
been redrawn with some slight changes and so the proportions are less clear.
Family Friend 3 (1850) 330 & 351. Practical puzzle, No. XXI. As in Les Amusemens.
SOURCES - page 183
Magician's Own Book. 1857. Prob. 31: Another cross puzzle, pp. 276 & 299. As in Les
Amusemens.
Landells. Boy's Own Toy-Maker. 1858. P. 152. As in Les Amusemens.
Book of 500 Puzzles. 1859. Prob. 31: Another cross puzzle, pp. 90 & 113. As in Les
Amusemens. = Magician's Own Book.
Indoor & Outdoor. c1859. Part II, p. 127, prob. 5: The puzzle of the cross. As in Les
Amusemens.
Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 24, pp. 399 & 439. Identical to
Magician's Own Book.
Boy's Own Conjuring book. 1860. Prob. 30: Another cross puzzle, pp. 239 & 263.
= Magician's Own Book, 1857.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob. 584-2, pp. 286 & 404. 4 Z pentominoes to make a (Greek) cross. (Also entered
in 6.F.5.)
Prob. 584-8, pp. 287 & 405. 3 Z pentominoes, L tetromino and L pentomino to
make a Greek cross. Despite specifically asking for a Greek cross, the answer is
a standard Latin cross with height : width = 4 : 3.
Mittenzwey. 1880. Prob. 173-174, pp. 33 & 85; 1895?: 198-199, pp. 38 & 87; 1917:
198-199, pp. 35 & 84. The first is 3 Z pentominoes, L tetromino and L pentomino
to make a cross. The second is 4 Z pentominoes to make a (Greek) cross. (Also
entered in 6.F.5.)
Cassell's. 1881. P. 93: The magic cross. = Manson, 1911, p. 139. Same pattern as Les
Amusemens, but one end of the Zs is decidedly longer than the other and the middle
'square' of the Zs is decidedly not square. The Ls are approximately a pentomino and a
heptomino, But the middle 'square' of the Zs is almost a domino and that makes the Zs
into heptominoes, with the Ls being a hexomino and a nonomino.
S&B, p. 20, shows a 7 piece cross dissection, Jeu de La Croix, into 3 Zs, 2 Ls and 2
straights, from c1890. The Zs are pentominoes, with the centre 'square' lengthened a
bit. The Ls appear to be a heptomino and an octomino and the straights appear to be a
hexomino and a tetromino. Cf Hoffmann-Hordern for a version without the straight
pieces.
Handy Book for Boys and Girls. Showing How to Build and Construct All Kinds of Useful
Things of Life. Worthington, NY, 1892. Pp. 320-321: The cross puzzle. As in
Cassell's.
Hoffmann. 1893. Chap. III, no. 29: Another cross puzzle, pp. 103 & 136
= Hoffmann-Hordern, pp. 100-101, with photo. States that the two Ls are the same
shape, but the solution is as in Les Amusemens, with the Ls approximately a hexomino
and a heptomino. Hordern has corrected the problem statement. Photo on p. 100 shows
an ivory version, dated 1850-1900, of the same proportions. Hordern Collection, p. 65,
shows two wood versions, La Croix Brisée and Jeu de la Croix, dated 1880-1905, both
with Ls being approximately a heptomino and an octomino.
Benson. 1904. The Latin cross puzzle, p. 200. As in Hoffmann, but the solution is longer, as
in Les Amusemens.
Wehman. New Book of 200 Puzzles. 1908. Another cross puzzle, p. 32. As in Les
Amusemens, with the Ls being a pentomino and a hexomino.
S. Szabo. US Patent 1,263,960 -- Puzzle. Filed: 20 Oct 1917; patented: 23 Apr 1918. 1p +
1p diagrams. As in Les Amusemens, with even longer Ls, approximately a 10-omino
and an 11-omino.
6.F.4.
QUADRISECT AN L-TROMINO, ETC.
See also 6.AW.1 & 4.
Mittenzwey and Collins quadrisect a hollow square obtained by removing a 2 x 2 from
the centre of a 4 x 4.
Bile Beans quadrisects a 5 x 5 after deleting corners and centre.
Minguet. 1733. Pp. 114-115 (1755: 80; 1822: 133-134; 1864: 111-112). Quadrisect
L-tromino.
Alberti. 1747. Art. 30: Modo di dividere uno squadro di carta e di legno in quattro squadri
equali, p. ?? (131) & fig. 56, plate XVI, opp. p. 130.
Les Amusemens. 1749. P. xxx. L-tromino ("gnomon") into 4 congruent pieces.
SOURCES - page 184
Vyse. Tutor's Guide. 1771? Prob. 9, 1793: p. 305, 1799: p. 317 & Key p. 358. Refers to the
land as a parallelogram though it is drawn rectangular.
Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as
Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the
Essay of Games". F. 4v has an entry "8½ a Prob of figure" followed by the
L-tromino. 8½ b is the same with a mitre and there are other dissection problems
adjacent -- see 6.F.3, 6.AQ, 6.AW.1, 6.AY, so it seems clear that he knew this problem.
Jackson. Rational Amusement. 1821. Geometrical Puzzles, no. 3, pp. 23 & 83 & plate I,
fig. 2.
Manuel des Sorciers. 1825. Pp. 203-204, art. 20. ??NX. Quadrisect L-tromino.
Family Friend 2 (1850) 118 & 149. Practical Puzzle -- No. IV. Quadrisect L-tromino of land
with four trees.
Family Friend 3 (1850) 150 & 181. Practical puzzle, No. XV. 15/16 of a square with 10
trees to be divided equally. One tree is placed very close to another, cf Magician's Own
Book and Hoffmann, below.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868.
Mechanical puzzles, no. 8, p. 179 (1868: 190). Land in the shape of an L-tromino to be
cut into four congruent parts, each with a cherry tree.
Magician's Own Book. 1857.
Prob. 3: The divided garden, pp. 267 & 292. 15/16 of a square to be divided into five
(congruent) parts, each with two trees. The missing 1/16 is in the middle. One
tree is placed very close to another, cf Family Friend 3, above, and Hoffmann
below.
Prob. 22: Puzzle of the four tenants, pp. 273 & 296. Same as Parlour Pastime, but with
apple trees. (= Illustrated Boy's Own Treasury, 1860, No. 10, pp. 397 & 437.)
Prob. 28: Puzzle of the two fathers, pp. 275-276 & 298. Each father wants to divide
3/4 of a square. One has L-tromino, other has the mitre shape. See 6.AW.1.
Landells. Boy's Own Toy-Maker. 1858.
P. 144. = Magician's Own Book, prob. 3.
Pp. 148-149. = Magician's Own Book, prob. 27.
Book of 500 Puzzles. 1859.
Prob. 3: The divided garden, pp. 81 & 106. Identical to Magician's Own Book.
Prob. 22: Puzzle of the four tenants, pp. 87 & 110. Identical to Magician's Own Book.
Prob. 28: Puzzle of the two fathers, pp. 89-90 & 112. Identical to Magician's Own
Book. See also 6.AW.1.
Charades, Enigmas, and Riddles. 1860: prob. 28, pp. 59 & 63; 1862: prob. 29, pp. 135 &
141; 1865: prob. 573, pp. 107 & 154. Quadrisect L-tromino, attributed to Sir F.
Thesiger.
Boy's Own Conjuring book. 1860.
Prob. 3: The divided garden, pp. 229 & 255. Identical to Magician's Own Book.
Prob. 21: Puzzle of the four tenants, pp. 235 & 260. Identical to Magician's Own Book.
Prob. 27: Puzzle of the two fathers, pp. 237-238 & 262. Identical to Magician's Own
Book.
Illustrated Boy's Own Treasury. 1860. Prob. 21, pp. 399 & 439. 15/16 of a square to be
divided into five (congruent) parts, each with two trees. c= Magician's Own Book,
prob. 3.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 175, p. 88. L-tromino into four
congruent pieces, each with two trees. The problem is given in terms of the original
square to be divided into five parts, where the father gets a quarter of the whole in the
form of a square and the four sons get congruent pieces.
Hanky Panky. 1872. The divided orchards, p. 130. L-tromino into 4 congruent pieces, each
with two trees.
Boy's Own Book. The divided garden. 1868: 675. = Magician's Own Book, prob. 3.
Mittenzwey. 1880.
Prob. 192, pp. 36 & 89; 1895?: 217, pp. 40 & 91; 1917: 217, pp. 37 & 87. Cut 1 x 1
out of the centre of a 4 x 4. Divide the rest into five parts of equal area with four
being congruent. He cuts a 2 x 2 out of the centre, which has a 1 x 1 hole in it,
then divides the rest into four L-trominoes.
Prob. 213, pp. 38 & 90; 1895?: 238, pp. 42 & 92; 1917: 238, pp. 39 & 88. Usual
quadrisection of an L-tromino.
Prob. 214, pp. 38 & 90; 1895?: 239, pp. 42 & 92; 1917: 239, pp. 39 & 88. Square
SOURCES - page 185
garden with mother receiving 1/4 and the rest being divided into four congruent
parts.
Cassell's. 1881. P. 90: The divided farm. = Manson, 1911, pp. 136-137. = Magician's Own
Book, prob. 3.
Lemon. 1890.
The divided garden, no. 259, pp. 38 & 107. = Magician's Own Book, prob. 3.
Geometrical puzzle, no. 413, pp. 55 & 113 (= Sphinx, no. 556, pp. 76 & 116).
Quadrisect L-tromino.
Hoffmann. 1893. Chap. X, no. 41: The divided farm, pp. 352-353 & 391
= Hoffmann-Hordern, p. 250. = Magician's Own Book, prob. 3. [One of the trees is
invisible in the original problem, but Hoffmann-Hordern has added it, in a more
symmetric pattern than in Magician's Own Book.]
Loyd. Origin of a famous puzzle -- No. 18: An ancient puzzle. Tit-Bits 31 (13 Feb &
6 Mar 1897) 363 & 419. Nearly 50 years ago he was told of the quadrisection of 3/4
of a square, but drew the mitre shape instead of the L-tromino. See 6.AW.1.
Clark. Mental Nuts. 1897, no. 73; 1904, no. 31. Dividing the land. Quadrisect an
L-tromino. 1904 also has the mitre -- see 6.AW.1.
Benson. 1904. The farmer's puzzle, p. 196. Quadrisect an L-tromino.
Wehman. New Book of 200 Puzzles. 1908.
The divided garden, p. 17. = Magician's Own Book, prob. 3
Puzzle of the two fathers, p. 43. = Magician's Own Book, prob. 28.
Puzzle of the four tenants, p. 46. = Magician's Own Book, prob. 22.
Dudeney. Some much-discussed puzzles. Op. cit. in 2. 1908. Land in shape of an
L-tromino to be quadrisected. He says this is supposed to have been invented by Lord
Chelmsford (Sir F. Thesiger), who died in 1878 -- see Charades, Enigmas, and Riddles
(1860). But cf Les Amusemens.
M. Adams. Indoor Games. 1912. The clever farmer, pp. 23-25. Dissect L-tromino into four
congruent pieces.
Blyth. Match-Stick Magic. 1921. Dividing the inheritance, pp. 20-21. Usual quadrisection
of L-tromino set out with matchsticks.
Collins. Book of Puzzles. 1927. The surveyor's puzzle, pp. 2-3. Quadrisect 3/4 of a square,
except the deleted 1/4 is in the centre, so we are quadrisecting a hollow square -- cf
Mittenzwey,
The Bile Beans Puzzle Book. 1933.
No. 22: Paper squares. Quadrisect a P-pentomino into P-pentominoes. One solution
given, I find another. Are there more? How about quadrisecting into congruent
pentominoes? Which pentominoes can be quadrisected into four copies of
themself?
No. 41: Five lines. Consider a 5 x 5 square and delete the corners and centre.
Quadrisect into congruent pentominoes. One solution given. I find three more.
Are there more? One can extend this to consider quadrisecting the 5 x 5 with
just the centre removed into congruent hexominoes. I find seven ways.
Depew. Cokesbury Game Book. 1939. A plot of ground, p. 227. 3/4 of
XX
a square to be quadrisected, but the shape is as shown at the right.
XXX
X XX
XXXX
Ripley's Puzzles and Games. 1966. Pp. 18 & 19, item 8. Divide an L-tromino into eight
congruent pieces.
F. Göbel. Problem 1771: The L-shape dissection problem. JRM 22:1 (1990) 64-65. The
L-tromino can be dissected into 2, 3, or 4 congruent parts. Can it be divided into 5
congruent parts?
Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Apple-eating monsters, pp. 40
& 63. Trisect into equal parts, the shape consisting of a 2 x 4 rectangle with a 1 x 1
square attached to one of the central squares of the long side. [Actually, this can be
done with the square attached to any of the squares, though if it is attached to the end of
the long side, the resulting pieces are straight trominoes.]
6.F.5.
OTHER DISSECTIONS INTO POLYOMINOES
Catel. Kunst-Cabinet. 1790.
Das Zakk- und Hakenspiel, p. 10 & fig. 11 on plate 1. 4 Z-pentominoes and
SOURCES - page 186
4 L-tetrominoes make a 6 x 6 square.
Die zwolf Winkelhaken, p. 11 & fig. 26 on plate 1. 8 L-pentominoes and
4 L-hexominoes make a 8 x 8 square.
Bestelmeier. 1801. Item 61 -- Das Zakken und Hakkenspiel. As in Catel, p. 10, but
not as regularly drawn. Text copies some of Catel.
Manuel des Sorciers. 1825. Pp. 203-204, art. 20. ??NX Use four L-trominoes to make a
3 x 4 rectangle or a 4 x 4 square with four corners deleted.
Family Friend 3 (1850) 90 & 121. Practical puzzle -- No. XIII. 4 x 4 square, with 12 trees in
the corners, centres of sides and four at the centre of the square, to be divided into 4
congruent parts each with 3 trees. Solution uses 4 L-tetrominoes. The same problem
is repeated as Puzzle 17 -- Twelve-hole puzzle in (1855) 339 with solution in (1856) 28.
Magician's Own Book. 1857. Prob. 14: The square and circle puzzle, pp. 270 & 295. Same
as Family Friend. = Book of 500 Puzzles, 1859, prob. 14, pp. 84 & 109. = Boy's Own
Conjuring book, 1860, prob. 13, pp. 231-232 & 257. c= Illustrated Boy's Own
Treasury, 1860, prob. 8, pp. 396 & 437. c= Hanky Panky, 1872, A square of four
pieces, p. 117.
Landells. Boy's Own Toy-Maker. 1858. Pp. 146-147. Identical to Family Friend.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob. 584-2, pp. 286 & 404. 4 Z-pentominoes to make a Greek cross. (Also entered in
6.F.3.)
Prob. 584-3, pp. 286 & 404. 4 L-tetrominoes to make a square.
Prob. 584-5, pp. 286 & 404. 8 L-pentominoes and 4 L-hexominoes make a 8 x 8
square. Same as Catel, but diagram is inverted.
Prob. 584-7, pp. 287 & 405. 4 Z-pentominoes and 4 L-tetrominoes make a 6 x 6
square. Same as Catel, but diagram is inverted.
Mittenzwey. 1880.
Prob. 174, pp. 33 & 85; 1895?: 199, pp. 38 & 87; 1917: 199, pp. 35 & 84. 4 Z
pentominoes to make a (Greek) cross. (Also entered in 6.F.5.)
Prob. 186, pp. 35 & 88; 1895?: 211, pp. 40 & 90; 1917: 211, pp. 36 & 87. 4 x 4
square into 4 L-tetrominoes.
Prob. 187, pp. 35 & 88; 1895?: 212, pp. 40 & 90; 1917: 212, pp. 36 & 87. 6 x 6
square into 4 Z-pentominoes and 4 L-tetrominoes, as in Catel, p. 10.
Prob. 215, pp. 38 & 90; 1895?: 240, pp. 42 & 92; 1917: 240, pp. 39 & 88. Square
garden with 12 trees quadrisected into four L-tetrominoes.
S&B, p. 20, shows a 7 piece cross dissection into 3 Zs, 2 Ls and 2 straights, from c1890.
Hoffmann. 1893. Chap. X, no. 37: The orchard puzzle, pp. 350 & 390 = Hoffmann-Hordern,
pp. 247, with photo. Same as Family Friend 3. Photo on p. 247 shows St. Nicholas
Puzzle Card, © 1892 in the USA.
Tom Tit, vol 3. 1893. Les quatre Z et des quatre L, pp. 181-182. = K, No. 27: The four
Z's and the four L's, pp. 70-71. = R&A, Squaring the L's and Z's, p. 102. 6 x 6
square as in Catel, p. 10.
Sphinx. 1895. The Maltese cross, no. 181, pp. 28 & 103. Make a Maltese cross (actually a
Greek cross of five equal squares) from 4 P-pentominoes. Also: quadrisect a
P-pentomino.
Wehman. New Book of 200 Puzzles. 1908. The square and circle puzzle, p. 5. = Family
Friend.
Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY,
1909. The Zoltan's orchard, pp. 24-28 & 64. = Family Friend.
Anon. Prob. 84. Hobbies 31 (No. 799) (4 Feb 1911) 443. Use at least one each of: domino;
L-tetromino; P and X pentominoes to make the smallest possible square Due to
ending of this puzzle series, no solution ever appeared. I find numerous solutions for
5 x 5, 6 x 6, 8 x 8, of which the first is easily seen to be the smallest possibility.
A. Neely Hall. Carpentry & Mechanics for Boys. Lothrop, Lee & Shepard, Boston, nd
[1918]. The square puzzle, pp. 20-21. 7 x 7 square cut into 1 straight tromino,
1 L-tetromino and 7 L-hexominoes.
Collins. Book of Puzzles. 1927. The surveyor's puzzle, pp. 2-3. Quadrisect 3/4 of a square,
except the deleted 1/4 is in the centre, so we are quadrisecting a hollow square.
Arthur Mee's Children's Encyclopedia 'Wonder Box'. The Children's Encyclopedia appeared
in 1908, but versions continued until the 1950s. This looks like 1930s??
4 Z-pentominoes and 4 L-tetrominoes make a 6 x 6 square and a 4 x 9 rectangle.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. All square, pp. 42 &
SOURCES - page 187
129. Make a 6 x 6 square from the staircase hexomino, 2 Y-pentominoes, an
N-tetromino, an L-tetromino and 3 T-tetrominoes. None of the pieces is turned over
in the solution, though this restriction is not stated.
6.G. SOMA CUBE
Piet Hein invented the Soma Cube in 1936. (S&B, pp. 40-41.) ??Is there any patent??
M. Gardner. SA (Sep 1958) = 2nd Book, Chap 6.
Richard K. Guy. Loc. cit. in 5.H.2, 1960. Pp. 150-151 discusses cubical solutions -- 234
found so far. He proposes the 'bath' shape -- a 5 x 3 x 2 cuboid with a 3 x 1 x 1 hole
in the top layer. In a 1985 letter, he said that O'Beirne had introduced the Soma to him
and his family. in 1959 and they found 234 solutions before Mike Guy went to
Cambridge -- see below.
P. Hein, et al. Soma booklet. Parker Bros., 1969, 56pp. Asserts there are 240 simple
solutions and 1,105,920 total solutions, found by J. H. Conway & M. J. T. Guy with a
a computer (but cf Gardner, below) and by several others. [There seem to be several
versions of this booklet, of various sizes.]
Thomas V. Atwater, ed. Soma Addict. 4 issues, 1970-1971, produced by Parker Brothers.
(Gardner, below, says only three issues appeared.) ??NYS -- can anyone provide a set
or photocopies??
M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. States there are 240 solutions for the cube,
obtained by many programs, but first found by J. H. Conway & M. J. T. Guy in 1962,
who did not use a computer, but did it by hand "one wet afternoon". Richard Guy's
1985 letter notes that Mike Guy had a copy of the Guy family's 234 solutions with him.
SOMAP ??NYS -- ??details. (Schaaf III 52)
Winning Ways, 1982, II, 802-803 gives the SOMAP.
Jon Brunvall et al. The computerized Soma Cube. Comp. & Maths. with Appl. 12B:1/2
(1986 [Special issues 1/2 & 3/4 were separately printed as: I. Hargittai, ed.;
Symmetry -- Unifying Human Understanding; Pergamon, 1986.] 113-121. They cite
Gardner's 2nd Book which says the number of solutions is unknown and they use a
computer to find them.
6.G.1.
OTHER CUBE DISSECTIONS
See also 6.N, 6.U.2, 6.AY.1 and 6.BJ. The predecessors of these puzzles seem to be the
binomial and trinomial cubes showing (a+b)3 and (a+b+c)3. I have an example of the latter
from the late 19C. Here I will consider only cuts parallel to the cube faces -- cubes with cuts
at angles to the faces are in 6.BJ. Most of the problems here involve several types of piece -see 6.U.2 for packing with one kind of piece.
Catel. Kunst-Cabinet. 1790. Der algebraische Würfel, p. 6 & fig 50 on plate II. Shows a
binomial cube: (a + b)3 = a3 + 3a2b + 3ab2 + b3.
Bestelmeier. 1801. Item 309 is a binomial cube, as in Catel. "Ein zerschnittener Würfel, mit
welchem die Entstehung eines Cubus, dessen Seiten in 2 ungleiche Theile a + b
getheilet ist, gezeigt ist."
Hoffmann. 1893. Chap. III, no. 39: The diabolical cube, pp. 108 & 142 = HoffmannHordern, pp. 108-109, with photos. 6: 0, 1, 1, 1, 1, 1, 1, i.e. six pieces of volumes 2, 3,
4, 5, 6, 7. Photos on p. 108 shows Cube Diabolique and its box, by Watilliaux, dated
1874-1895.
J. G.-Mikusiński. French patent. ??NYS -- cited by Steinhaus.
H. Steinhaus. Mikusiński's Cube. Mathematical Snapshots. Not in Stechert, 1938, ed. OUP,
NY: 1950: pp. 140-142 & 263; 1960, pp. 179-181 & 326; 1969 (1983): pp. 168-169 &
303.
John Conway. In an email of 7 Apr 2000, he says he developed the dissection of the 3 x 3 x 3
into 3 1 x 1 x 1 and 6 1 x 2 x 2 in c1960 and then adapted it to the 5 x 5 x 5 into
3 1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and 13 1 x 2 x 4 and the 5 x 5 x 5 into
3 1 x 1 x 3 and 29 1 x 2 x 2. He says his first publication of it was in Winning Ways,
1982 (cf below).
Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970.
??NYS. 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2. [Jan de Geus has sent a
photocopy of some of this but it does not cover this topic.]
SOURCES - page 188
M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. Discusses Hoffmann's Diabolical Cube and
Mikusiński's cube. Says he has 8 solutions for the first and that there are just 2 for
the second. The Addendum reports that Wade E. Philpott showed there are just 13
solutions of the Diabolical Cube. Conway has confirmed this. Gardner briefly
describes the solutions. Gardner also shows the Lesk Cube, designed by Lesk Kokay
(Mathematical Digest [New Zealand] 58 (1978) ??NYS), which has at least 3
solutions.
D. A. Klarner. Brick-packing puzzles. JRM 6 (1973) 112-117. Discusses 3 x 3 x 3 into
3 1 x 1 x 1 and 6 1 x 2 x 2 attributed to Slothouber-Graatsma; Conway's 5 x 5 x 5
into 3 1 x 1 x 3 and 29 1 x 2 x 2; Conway's 5 x 5 x 5 into 3 1 x 1 x 3, 1 2 x 2 x 2,
1 1 x 2 x 2 and 13 1 x 2 x 4. Because of the attribution to Slothouber & Graatsma
and not knowing the date of Conway's work, I had generally attributed the 3 x 3 x 3
puzzle to them and Stewart Coffin followed this in his book. However, it now seems
that it really is Conway's invention and I must apologize for misleading people.
Leisure Dynamics, the US distributor of Impuzzables, a series of 6 3 x 3 x 3 cube
dissections identified by colours, writes that they were invented by Robert Beck,
Custom Concepts Inc., Minneapolis. However, the Addendum to Gardner, above, says
they were designed by Gerard D'Arcey.
Winning Ways. 1982. Vol. 2, pp. 736-737 & 801. Gives the 3 x 3 x 3 into 3 1 x 1 x 1 and
6 1 x 2 x 2 and the 5 x 5 x 5 into 3 1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and
13 1 x 2 x 4, which is called Blocks-in-a-Box. No mention of the other 5 x 5 x 5.
Mentions Foregger & Mather, cf in 6.U.2.
Michael Keller. Polycube update. World Game Review 4 (Feb 1985) 13. Reports results of
computer searches for solutions. Hoffmann's Diabolical Cube has 13; Mikusinski's
Cube has 2; Soma Cube has 240; Impuzzables: White -- 1; Red -- 1; Green -- 16;
Blue -- 8; Orange -- 30; Yellow -- 1142.
Michael Keller. Polyform update. World Game Review 7 (Oct 1987) 10-13. Says that Nob
Yoshigahara has solved a problem posed by O'Beirne: How many ways can
9 L-trominoes make a cube? Answer is 111. Gardner, Knotted, chap. 3, mentioned
this. Says there are solutions with n L-trominoes and 9-n straight trominoes for n  1
and there are 4 solutions for n = 0. Says the Lesk Cube has 4 solutions. Says Naef's
Gemini Puzzle was designed by Toshiaki Betsumiya. It consists of the 10 ways to join
two 1 x 2 x 2 blocks.
H. J. M. van Grol. Rik's Cube Kit -- Solid Block Puzzles. Analysis of all 3 x 3 x 3 unit solid
block puzzles with non-planar 4-unit and 5-unit shapes. Published by the author, The
Hague, 1989, 16pp. There are 3 non-planar tetracubes and 17 non-planar pentacubes.
A 3 x 3 x 3 cube will require the 3 non-planar tetracubes and 3 of the non-planar
pentacubes -- assuming no repeated pieces. He finds 190 subsets which can form
cubes, in 1 to 10 different ways.
Nob Yoshigahara. (Title in Japanese: (Puzzle in Wood)). H. Tokuda, Sowa Shuppan, Japan,
1987. Pp. 68-69 is a 3^3 designed by Nob -- 6: 01005.
6.G.2.
DISSECTION OF 63 INTO 33, 43 AND 53, ETC.
H. W. Richmond. Note 1672: A geometrical problem. MG 27 (No. 275) (Jul 1943) 142.
AND Note 1704: Solution of a geometrical problem (Note 1672). MG 28 (No. 278)
(Feb 1944) 31-32. Poses the problem of making such a dissection, then gives a solution
in 12 pieces: three 1 x 3 x 3; 4 x 4 x 4; four 1 x 5 x 5; 1 x 4 x 4; two 1 x 1 x 2 and a
V-pentacube.
Anon. [= John Leech, according to Gardner, below]. Two dissection problems, no. 2. Eureka
13 (Oct 1950) 6 & 14 (Oct 1951) 23. Asks for such a dissection using at most 10
pieces. Gives an 8 piece solution due to R. F. Wheeler. [Cundy & Rollett;
Mathematical Models; 2nd ed., pp. 203-205, say Eureka is the first appearance they
know of this problem. See Gardner, below, for the identity of Leech.]
Richard K. Guy. Loc. cit. in 5.H.2, 1960. Mentions the 8 piece solution.
J. H. Cadwell. Some dissection problems involving sums of cubes. MG 48 (No. 366)
(Dec 1964) 391-396. Notes an error in Cundy & Rollett's account of the Eureka
problem. Finds examples for 123 + 13 = 103 + 93 with 9 pieces and 93 = 83 + 63 + 13
with 9 pieces.
J. H. Cadwell. Note 3278: A three-way dissection based on Ramanujan's number. MG 54
(No. 390) (Dec 1970) 385-387. 7 x 13 x 19 to 103 + 93 and 123 + 13 using 12 pieces.
SOURCES - page 189
M. Gardner. SA (Oct 1973) c= Knotted, chap. 16. He says that the problem was posed by
John Leech. He gives Wheeler's initials as E. H. ?? He says that J. H. Thewlis found a
simpler 8-piece solution, further simplified by T. H. O'Beirne, which keeps the
4 x 4 x 4 cube intact. This is shown in Gardner. Gardner also shows an 8-piece
solution which keeps the 5 x 5 x 5 intact, due to E. J. Duffy, 1970. O'Beirne showed
that an 8-piece dissection into blocks is impossible and found a 9-block solution in
1971, also shown in Gardner.
Harry Lindgren. Geometric Dissections. Van Nostrand, Princeton, 1984. Section 24.1,
pp. 118-120 gives Wheeler's solution and admires it.
Richard K. Guy, proposer; editors & Charles H. Jepson [should be Jepsen], partial solvers.
Problem 1122. CM 12 (1987) 50 & 13 (1987) 197-198. Asks for such dissections
under various conditions, of which (b) is the form given in Eureka. Eight pieces is
minimal in one case and seems minimal in two other cases. Eleven pieces is best
known for the first case, where the pieces must be blocks, but this appears to be the
problem solved by O'Beirne in 1971, reported in Gardner, above.
Charles H. Jepsen. Additional comment on Problem 1122. CM 14 (1988) 204-206. Gives a
ten piece solution of the first case.
Chris Pile. Cube dissection. M500 134 (Aug 1993) 2-3. He feels the 1 x 1 x 2 piece
occurring in Cundy & Rollett is too small and he provides another solution with 8
pieces, the smallest of which contains 8 unit cubes. Asks how uniform the piece sizes
can be.
6.G.3.
DISSECTION OF A DIE INTO NINE 1 x 1 x 3
Hoffmann. 1893. Chap. III, no. 17: The "Spots" puzzle, pp. 98-99 & 130-131
= Hoffmann-Hordern, pp. 90-91, with photo. Says it is made by Wolff & Son. Photo
on p. 91 shows an example made by E. Wolff & Son, London.
Benson. 1904. The spots puzzle, pp. 203-204. As in Hoffmann.
Collins. Book of Puzzles. 1927. Pp. 131-134: The dissected die puzzle. The solution is
different than Hoffmann's.
Rohrbough. Puzzle Craft. 1932. P. 21 shows a dissected die, but with no text. The picture is
the same as in Hoffmann's solution.
Slocum. Compendium. Shows Diabolical Dice from Johnson Smith catalogue, 1935.
Harold Cataquet. The Spots puzzle revisited. CFF 33 (Feb 1994) 20-21. Brief discussion of
two versions.
David Singmaster. Comment on the "Spots" puzzle. 29 Sep 1994, 2pp. Letter in response to
the above. I note that there is no standard pattern for a die other than the opposite sides
adding to seven. There are 23 = 8 ways to orient the spots forming 2, 3, and 6. There
are two handednesses, so there are 16 dice altogether. (This was pointed out to me
perhaps 10 years before by Richard Guy and Ray Bathke. I have since collected
examples of all 16 dice.) However, Ray Bathke showed me Oriental dice with the two
spots of the 2 placed horizontal or vertically rather than diagonally, giving another 16
dice (I have 5 types), making 32 dice in all. A die can be dissected into 9 1 x 1 x 3
pieces in 6 ways if the layers have to alternate in direction, or in 21 ways in general. I
then pose a number of questions about such dissections.
6.G.4.
USE OF OTHER POLYHEDRAL PIECES
S&B. 1986. P. 42 shows Stewart Coffin's 'Pyramid Puzzle' using pieces made from truncated
octahedra and his 'Setting Hen' using pieces made from rhombic dodecahedra. Coffin
probably devised these in the 1960s -- perhaps his book has some details of the origins
of these ideas. ??check.
Mark Owen & Matthew Richards. A song of six splats. Eureka 47 (1987) 53-58. There are
six ways to join three truncated octahedra. For reasons unknown, these are called
3-splats. They give various shapes which can and which cannot be constructed from the
six 3-splats.
6.H. PICK'S THEOREM
Georg Pick. Geometrisches zur Zahlenlehre. Sitzungsberichte des deutschen
naturwissenschaftlich-medicinischen Vereines für Böhmen "Lotos" in Prag (NS) 19
SOURCES - page 190
(1899) 311-319. Pp. 311-314 gives the proof, for an oblique lattice. Pp. 318-319 gives
the extension to multiply connected and separated regions. Rest relates to number
theory. [I have made a translation of the material on Pick's Theorem.]
Charles Howard Hinton. The Fourth Dimension. Swan Sonnenschein & Co., London, 1906.
Metageometry, pp. 46-60. [This material is in Speculations on the Fourth Dimension,
ed. by R. v. B. Rucker; Dover, 1980, pp. 130-141. Rucker says the book was published
in 1904, so my copy may be a reprint??] In the beginning of this section, he draws
quadrilateral shapes on the square lattice and determines the area by counting points,
but he counts I + E/2 + C/4, which works for quadrilaterals but is not valid in general.
H. Steinhaus. O mierzeniu pól płaskich. Przegląd Matematyczno-Fizyczny 2 (1924) 24-29.
Gives a version of Pick's theorem, but doesn't cite Pick. (My thanks to A. Mąkowski
for an English summary of this.)
H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938, pp. 16-17 & 132. OUP, NY:
1950: pp. 76-77 & 260 (note 77); 1960: pp. 99-100 & 324 (note 95); 1969 (1983): pp.
96-97 & 301 (note 107). In 1938 he simply notes the theorem and gives one example.
In 1950, he outlines Pick's argument. He refers to Pick's paper, but in "Ztschr. d.
Vereins 'Lotos' in Prag". Steinhaus also cites his own paper, above.
J. F. Reeve. On the volume of lattice polyhedra. Proc. London Math. Soc. 7 (1957) 378-395.
Deals with the failure of the obvious form of Pick's theorem in 3-D and finds a valid
generalization.
Ivan Niven & H. S. Zuckerman. Lattice points and polygonal area. AMM 74 (1967)
1195-1200. Straightforward proof. Mention failure for tetrahedra.
D. W. De Temple & J. M. Robertson. The equivalence of Euler's and Pick's theorems.
MTr 67 (1974) 222-226. ??NYS.
W. W. Funkenbusch. From Euler's formula to Pick's formula using an edge theorem.
AMM 81 (1974) 647-648. Easy proof though it could be easier.
R. W. Gaskell, M. S. Klamkin & P. Watson. Triangulations and Pick's theorem. MM 49
(1976) 35-37. A bit roundabout.
Richard A. Gibbs. Pick iff Euler. MM 49 (1976) 158. Cites DeTemple & Robertson and
observes that both Pick and Euler can be proven from a result on triangulations.
John Reay. Areas of hex-lattice polygons, with short sides. Abstracts Amer. Math. Soc. 8:2
(1987) 174, #832-51-55. Gives a formula for the area in terms of the boundary and
interior points and the characteristic of the boundary, but it is an open question to
determine when this formula gives the actual area.
6.I.
SYLVESTER'S PROBLEM OF COLLINEAR POINTS
If a set of non-collinear points in the plane is such that the line through any two points
of the set contains a third point of the set, then the set is infinite.
J. J. Sylvester. Question 11851. The Educational Times 46 (NS, No. 383) (1 Mar 1893) 156.
H. J. Woodall & editorial comment. Solution to Question 11851. Ibid. (No. 385)
(1 May 1893) 231. A very spurious solution.
(The above two items appear together in Math. Quest. with their Sol. Educ. Times 59 (1893)
98-99.)
E. Melchior. Über Vielseite der projecktiven Ebene. Deutsche Math. 5 (1940) 461-475.
Solution, but in a dual form.
P. Erdös, proposer; R. Steinberg, solver & editorial comment giving solution of T. Grünwald
(later = T. Gallai). Problem 4065. AMM 50 (1943) 65 & 51 (1944) 169-171.
L. M. Kelly. (Solution.) In: H. S. M. Coxeter; A problem of collinear points; AMM 55
(1948) 26-28. Kelly's solution is on p. 28.
G. A. Dirac. Note 2271: On a property of circles. MG 36 (No. 315) (Feb 1952) 53-54.
Replace 'line' by 'circle' in the problem. He shows this is true by inversion. He asks for
an independent proof of the result, even for the case when two, three are replaced by
three, four.
D. W. Lang. Note 2577: The dual of a well-known theorem. MG 39 (No. 330) (Dec 1955)
314. Proves the dual easily.
H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Section 4.7: Sylvester's problem
of collinear points, pp. 65-66. Sketches history and gives Kelly's proof.
W. O. J. Moser. Sylvester's problem, generalizations and relatives. In his: Research
Problems in Discrete Geometry 1981, McGill University, Montreal, 1981. Section 27,
SOURCES - page 191
pp. 27-1 -- 27-14. Survey with 73 references. (This problem is not in Part 1 of the
1984 ed. nor in the 1986 ed.)
6.J.
FOUR BUGS AND OTHER PURSUIT PROBLEMS
The general problem becomes too technical to remain recreational, so I will not try to be
exhaustive here.
Arthur Bernhart.
Curves of pursuit. SM 20 (1954) 125-141.
Curves of pursuit -- II. SM 23 (1957) 49-65.
Polygons of pursuit. SM 24 (1959) 23-50.
Curves of general pursuit. SM 24 (1959) 189-206.
Extensive history and analysis. First article covers one dimensional pursuit, then
two dimensional linear pursuit. Second article deals with circular pursuit. Third
article is the 'four bugs' problem -- analysis of equilateral triangle, square, scalene
triangle, general polygon, Brocard points, etc. Last article includes such variants
as variable speed, the tractrix, miscellaneous curves, etc.
Mr. Ash, proposer; editorial note saying there is no solver. Ladies' Diary, 1748-47 = T.
Leybourn, II: 15-17, quest. 310, with 'Solution by ΦIΛΟΠΟΝΟΣ, taken from Turner's
Exercises, where this question was afterwards proposed and answered ...' A fly is
constrained to move on the periphery of a circle. Spider starts 30o away from the fly,
but walks across the circle, always aiming at the fly. If she catches the fly 180o from her
starting point, find the ratio of their speeds. ΦIΛΟΠΟΝΟΣ solves the more general
problem of finding the curve when the spider starts anywhere.
Carlile. Collection. 1793. Prob. CV, p. 62. A dog and a duck are in a circular pond of radius
40 and they swim at the same speed. The duck is at the edge and swims around the
circumference. The dog starts at the centre and always swims toward the duck, so the
dog and the duck are always on a radius. How far does the dog swim in catching the
duck? He simply gives the result as 20π. Letting R be the radius of the pond and V
be the common speed, I find the radius of the dog, r, is given by r = R sin Vt/R. Since
the angle, θ, of both the duck and the dog is given by θ = Vt/R, the polar equation of
the dog's path is r = R sin θ and the path is a semicircle whose diameter is the
appropriate radius perpendicular to the radius to the duck's initial position.
Cambridge Math. Tripos examination, 5 Jan 1871, 9 to 12. Problem 16, set by R. K. Miller.
Three bugs in general position, but with velocities adjusted to make paths similar and
keep the triangle similar to the original.
Lucas. (Problem of three dogs.) Nouvelle Correspondance Mathématique 3 (1877) 175-176.
??NYS -- English in Arc., AMM 28 (1921) 184-185 & Bernhart.
H. Brocard. (Solution of Lucas' problem.) Nouv. Corr. Math. 3 (1877) 280. ??NYS -English in Bernhart.
Pearson. 1907. Part II, no. 66: A duck hunt, pp. 66 & 172. Duck swims around edge of
pond; spaniel starts for it from the centre at the same speed.
A. S. Hathaway, proposer and solver. Problem 2801. AMM 27 (1920) 31 & 28 (1921)
93-97. Pursuit of a prey moving on a circle. Morley's and other solutions fail to deal
with the case when the velocities are equal. Hathaway resolves this and shows the prey
is then not caught.
F. V. Morley. A curve of pursuit. AMM 28 (1921) 54-61. Graphical solution of Hathaway's
problem.
R. C. Archibald [Arc.] & H. P. Manning. Remarks and historical notes on problems 19
[1894], 160 [1902], 273 [1909] & 2801 [1920]. AMM 28 (1921) 91-93.
W. W. Rouse Ball. Problems -- Notes: 17: Curves of pursuit. AMM 28 (1921) 278-279.
A. H. Wilson. Note 19: A curve of pursuit. AMM 28 (1921) 327.
Editor's note to Prob. 2 (proposed by T. A. Bickerstaff), National Mathematics Magazine
(1937/38) 417 cites Morley and Archibald and adds that some authors credit the
problem to Leonardo da Vinci -- e.g. MG (1930-31) 436 -- ??NYS
Nelson F. Beeler & Franklyn M. Branley. Experiments in Optical Illusion. Ill. by Fred H.
Lyon. Crowell, 1951, An illusion doodle, pp. 68-71, describes the pattern formed by
four bugs starting at the corners of a square, drawing the lines of sight at
(approximately) regular intervals. Putting several of the squares together, usually with
SOURCES - page 192
alternating directions of motion, gives a pleasant pattern which is now fairly common.
They call this 'Huddy's Doodle', but give no source.
J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. 'Lion and man',
pp. 135-136 (114-117). The 1986 ed. adds three diagrams and revises the text
somewhat. I quote from it. "A lion and a man in a closed circular arena have equal
maximum speeds. What tactics should the lion employ to be sure of his meal?" This
was "invented by R. Rado in the late thirties" and "swept the country 25 years later".
[The 1953 ed., says Rado didn't publish it.] The correct solution "was discovered by
Professor A. S. Besicovitch in 1952". [The 1953 ed. says "This has just been
discovered ...; here is the first (and only) version in print."]
C. C. Puckette. The curve of pursuit. MG 37 (No. 322) (Dec 1953) 256-260. Gives the
history from Bouguer in 1732. Solves a variant of the problem.
R. H. Macmillan. Curves of pursuit. MG 40 (No. 331) (Feb 1956) 1-4. Fighter pursuing
bomber flying in a straight line. Discusses firing lead and acceleration problems.
Gamow & Stern. 1958. Homing missiles. Pp. 112-114.
Howard D. Grossman, proposer; unspecified solver. Problem 66 -- The walk around. In:
L. A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959, pp. 40 &
203-205. Four bugs -- asserts Grossman originated the problem.
I. J. Good. Pursuit curves and mathematical art. MG 43 (No. 343) (Feb 1959) 34-35. Draws
tangent to the pursuit curves in an equilateral triangle and constructs various patterns
with them. Says a similar but much simpler pattern was given by G. B. Robison;
Doodles; AMM 61 (1954) 381-386, but Robison's doodles are not related to pursuit
curves, though they may have inspired Good to use the pursuit curves.
J. Charles Clapham. Playful mice. RMM 10 (Aug 1962) 6-7. Easy derivation of the distance
travelled for n bugs at corners of a regular n-gon. [I don't see this result in Bernhart.]
C. G. Paradine. Note 3108: Pursuit curves. MG 48 (No. 366) (Dec 1964) 437-439. Says
Good makes an error in Note 3079. He shows the length of the pursuit curve in the
equilateral triangle is ⅔ of the side and describes the curve as an equiangular spiral.
Gives a simple proof that the length of the pursuit curve in the regular n-gon is the side
divided by (1 - cos 2π/n).
M. S. Klamkin & D. J. Newman. Cyclic pursuit or "The three bugs problem". AMM 78
(1971) 631-639. General treatment. Cites Bernhart's four SM papers and some of the
history therein.
P. K. Arvind. A symmetrical pursuit problem on the sphere and the hyperbolic plane. MG 78
(No. 481) (Mar 1994) 30-36. Treats the n bugs problems on the surfaces named.
Barry Lewis. A mathematical pursuit. M500 170 (Oct 1999) 1-8. Starts with equilateral
triangular case, giving QBASIC programs to draw the curves as well as explicit
solutions. Then considers regular n-gons. Then considers simple pursuit, one beast
pursuing another while the other moves along some given path. Considers the path as a
straight line or a circle. For the circle, he asserts that the analytic solution was not
determined until 1926, but gives no reference.
6.K. DUDENEY'S SQUARE TO TRIANGLE DISSECTION
Dudeney. Weekly Dispatch (6 Apr, 20 Apr, 4 May, 1902) all p. 13.
Dudeney. The haberdasher's puzzle. London Mag. 11 (No. 64) (Nov 1903) 441 & 443.
(Issue with solution not found.)
Dudeney. Daily Mail (1 & 8 Feb 1905) both p. 7.
Dudeney. CP. 1907. Prob. 25: The haberdasher's puzzle, pp. 49-50 & 178-180.
Western Puzzle Works, 1926 Catalogue. No. 1712 -- unnamed, but shows both the square
and the triangle. Apparently a four piece puzzle.
M. Adams. Puzzle Book. 1939. Prob. C.153: Squaring a triangle, pp. 162 & 189. Asserts
that Dudeney's method works for any triangle, but his example is close to equilateral
and I recall that this has been studied and only certain shapes will work??
Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed.,
Educational Publishers, St. Louis, 1949. Pp. 40-41. Extends to dissecting a
quadrilateral to a specified triangle and gives a number of related problems.
6.L. CROSSED LADDERS
Two ladders are placed across a street, each reaching from the base of the house on one
SOURCES - page 193
side to the house on the other side.
The simple problem gives the heights a, b that the ladders reach on the walls. If the
height of the crossing is c, we easily get 1/c = 1/a + 1/b. NOTATION -- this problem will be
denoted by (a, b).
The more common and more complex problem is where the ladders have lengths a and
b, the height of their crossing is c and one wants the width d of the street. If the heights of
the ladder ends are x, y, this situation gives x2 - y2 = a2 - b2 and 1/x + 1/y = 1/c which
leads to a quartic and there seems to be no simple solution. NOTATION -- this will be
denoted (a, b, c).
Mahavira. 850. Chap. VII, v. 180-183, pp. 243-244. Gives the simple version with a
modification -- each ladder reaches from the top of a pillar beyond the foot of the other
pillar. The ladder from the top of pillar Y (of height y) extends by m beyond the
foot of pillar X and the ladder from the top of pillar X (of height x) reaches n
beyond the foot of pillar Y. The pillars are d apart. Similar triangles then yield:
(d+m+n)/c = (d+n)/x + (d+m)/y and one can compute the various distances along the
ground. He first does problems with m = n = 0, which are the simple version of the
problem, but since d is given, he also asks for the distances on the ground.
v. 181. (16, 16) with d = 16.
v. 182. (36, 20) with d = 12.
v. 183. x, y, d, m, n = 12, 15, 4, 1, 4.
Bhaskara II. Lilavati. 1150. Chap. VI, v. 160. In Colebrooke, pp. 68-69. (10, 15).
(= Bijaganita, ibid., chap. IV, v. 127, pp. 205-206.)
Fibonacci. 1202. Pp. 397-398 (S: 543-544) looks like a crossed ladders problem but is a
simple right triangle problem.
Pacioli. Summa. 1494. Part II.
F. 56r, prob. 48. (4, 6).
F. 60r, prob. 64. (10, 15).
Hutton. A Course of Mathematics. 1798? Prob. VIII, 1833: 430; 1857: 508. A ladder 40
long in a roadway can reach 33 up one side and, from the same point, can reach 21 up
the other side. How wide is the street? This is actually a simple right triangle problem.
Victor Katz reports that Hutton's problem, with values 60; 37, 23 appears in a notebook of
Benjamin Banneker (1731-1806).
Loyd. Problem 48: A jubilee problem. Tit-Bits 32 (21 Aug, 11 & 25 Sep 1897) 385, 439
& 475. Given heights of the ladder ends above ground and the width of the street, find
the height of the intersection. However one wall is tilted and the drawing has it covered
in decoration so one may interpret the tilt in the wrong way.
Jno. A. Hodge, proposer; G. B. M. Zerr, solver. Problem 131. SSM 8 (1908) 786 & 9
(1909) 174-175. (100, 80, 10).
W. V. N. Garretson, proposer; H. S. Uhler, solver. Problem 2836. AMM 27 (1920) & 29
(1922) 181. (40, 25, 15).
C. C. Camp, proposer; W. J. Patterson & O. Dunkel, solvers. Problem 3173. AMM 33
(1926) 104 & 34 (1927) 50-51. General solution.
Morris Savage, proposer; W. E. Batzler, solver. Problem 1194. SSM 31 (1931) 1000 & 32
(1932) 212. (100, 80, 10).
S. A. Anderson, proposer; Simon Vatriquant, solver. Problem E210. AMM 43 (1936) 242
& 642-643. General solution in integers.
C. R. Green, proposer; C. W. Trigg, solver. Problem 1498. SSM 37 (1937) 484 & 860-861.
(40, 30, 15). Trigg cites Vatriquant for smallest integral case.
A. A. Bennett, proposer; W. E. Buker, solver. Problem E433. AMM 47 (1940) 487 & 48
(1941) 268-269. General solution in integers using four parameters.
J. S. Cromelin, proposer; Murray Barbour, solver. Problem E616 -- The three ladders. AMM
51 (1944) 231 & 592. Ladders of length 60 & 77 from one side. A ladder from the
other side crosses them at heights 17 & 19. How long is the third ladder and how
wide is the street?
Geoffrey Mott-Smith. Mathematical Puzzles for Beginners and Enthusiasts. (Blakiston,
1946); revised ed., Dover, 1954. Prob. 103: The extension ladder, pp. 58-59 &
176-178. Complex problem with three ladders.
Arthur Labbe, proposer; various solvers. Problem 25 -- The two ladders. Sep 1947 [date
given in Graham's second book, cited at 1961]. In: L. A. Graham; Ingenious
Mathematical Problems and Methods; Dover, 1959, pp. 18 & 116-118. (20, 30, 8).
SOURCES - page 194
M. Y. Woodbridge, proposer and solver. Problem 2116. SSM 48 (1948) 749 & 49 (1949)
244-245. (60, 40, 15). Asks for a trigonometric solution. Trigg provides a list of early
references.
Robert C. Yates. The ladder problem. SSM 51 (1951) 400-401. Gives a graphical solution
using hyperbolas.
G. A. Clarkson. Note 2522: The ladder problem. MG 39 (No. 328) (May 1955) 147-148.
(20, 30, 10). Let A = (a2 - b2) and set x = A sec t, y = A tan t. Then
cos t + cot t = A and he gets a trigonometrical solution. Another method leads to
factoring the quartic in terms of a constant k whose square satisfies a cubic.
L. A. Graham. The Surprise Attack in Mathematical Problems. Dover, 1968. Problem 6:
Searchlight on crossed ladders, pp. 16-18. Says they reposed Labbe's Sep 1947 problem
in Jun 1961. Solution by William M. Dennis which is the same trigonometric method
as Clarkson.
H. E. Tester. Note 3036: The ladder problem. A solution in integers. MG 46 (No. 358) (Dec
1962) 313-314. A four parameter, incomplete, solution. He finds the example
(119, 70, 30).
A. Sutcliffe. Complete solution of the ladder problem in integers. MG 47 (No. 360) (May
1963) 133-136. Three parameter solution. First few examples are: (119, 70, 30);
(116, 100, 35); (105, 87, 35). Simpler than Anderson and Bennett/Buker.
Alan Sutcliffe, proposer; Gerald J. Janusz, solver. Problem 5323 -- Integral solutions of the
ladder problem. AMM 72 (1965) 914 & 73 (1966) 1125-1127. Can the distance f
between the tops of the ladders be integral? (80342, 74226, 18837) has x = 44758,
y = 32526, d = 66720, f = 67832. This is not known to be the smallest example.
Anon. A window cleaner's problem. Mathematical Pie 51 (May 1967) 399. From a point in
the road, a ladder can reach 30 ft up on one side and 40 ft up on the other side. If the
two ladder positions are at right angles, how wide is the road?
J. W. Gubby. Note 60.3: Two chestnuts (re-roasted). MG 60 (No. 411) (Mar 1976) 64-65.
1. Given heights of ladders as a, b, what is the height c of their intersection?
Solution: 1/c = 1/a + 1/b or c = ab/(a+b). 2. The usual ladder problem -- he finds a
quartic.
J. Jabłkowski. Note 61:11: The ladder problem solved by construction. MG 61 (No. 416)
(Jun 1977) 138. Gives a 'neusis' construction. Cites Gubby.
Birtwistle. Calculator Puzzle Book. 1978. Prob. 83, A second ladder problem, pp. 58-59 &
115-118. (15, 20, 6). Uses xy as a variable to simplify the quartic for numerical
solution and eventually gets 11.61.
See: Gardner, Circus, p. 266 & Schaaf for more references. ??follow up.
Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. The tangled ladders, pp. 43-44 & 116.
(30, 20, 10). Gives answer 12.311857... with no explanation.
6.L.1. LADDER OVER BOX
A ladder of length L is placed to just clear a box of width w and height h at the base
of a wall. How high does the ladder reach? Denote this by (w, h, L). Letting x be the
horizontal distance of the foot and y be the vertical distance of the top of the ladder,
measured from the foot of the wall, we get x2 + y2 = L2 and (x-w)(y-h) = wh, which gives a
quartic in general. But if w = h, then use of x + y as a variable reduces the quartic to a
quadratic. In this case, the idea is old -- see e.g. Simpson.
The question of determining shortest ladder which can fit over a box of width w and
height h is the same as determining the longest ladder which will pass from a corridor of
width w into another corridor of width h. See Huntington below and section 6.AG.
Simpson. Algebra. 1745. Section XVIII, prob. XV, p. 250 (1790: prob. XIX, pp. 272-273).
"The Side of the inscribed Square BEDF, and the Hypotenuse AC of a right-angled
Triangle ABC being given; to determine the other two Sides of the Triangle AB and
BC." Solves "by considering x + y as one Quantity".
Pearson. 1907. Part II, no. 102: Clearing the wall, p. 103. For (15, 12, 52), the ladder
reaches 48.
D. John Baylis. The box and ladder problem. MTg 54 (1971) 24. (2, 2, 10). Finds the
quartic which he solves by symmetry. Editorial note in MTg 57 (1971) 13 says several
people wrote to say that use of similar triangles avoids the quartic.
SOURCES - page 195
Birtwistle. Math. Puzzles & Perplexities. 1971. The ladder and the box problem, pp. 44-45.
= Birtwistle; Calculator Puzzle Book; 1978; Prob. 53: A ladder problem, pp. 37 &
96-98. (3, 3, 10). Solves by using x + y - 6 as a variable.
Monte Zerger. The "ladder problem". MM 60:4 (1987) 239-242. (4, 4, 16). Gives a
trigonometric solution and a solution via two quadratics.
Oliver D. Anderson. Letter. MM 61:1 (1988) 63. In response to Zerger's article, he gives a
simpler derivation.
Tom Heyes. The old box and ladder problem -- revisited. MiS 19:2 (Mar 1990) 42-43. Uses
a graphic calculator to find roots graphically and then by iteration.
A. A. Huntington. More on ladders. M500 145 (Jul 1995) 2-5. Does usual problem, getting a
quartic. Then finds the shortest ladder. [This turns out to be the same as the longest
ladder one can get around a corner from corridors of widths w and h, so this problem
is connected to 6.AG.]
David Singmaster. Integral solutions of the ladder over box problem. In preparation. Easily
constructs all the primitive integral examples from primitive Pythagorean triples. E.g.
for the case of a square box, i.e. w = h, if X, Y, Z is a primitive Pythagorean triple,
then the corresponding primitive solution has w = h = XY, x = X (X + Y),
y = Y (X + Y), L = Z (X + Y), and remarkably, x - h = X2, y - w = Y2.
6.M. SPIDER & FLY PROBLEMS
These involve finding the shortest distance over the surface of a cube or cylinder. I've
just added the cylindrical form -- see Dudeney (1926), Perelman and Singmaster. The shortest
route from a corner of a cube or cuboid to a diagonally opposite corner must date back several
centuries, but I haven't seen any version before 1937! I don't know if other shapes have been
done -- the regular (and other) polyhedra and the cone could be considered.
Two-dimensional problems are in 10.U.
Loyd. The Inquirer (May 1900). Gives the Cyclopedia problem. ??NYS -- stated in a letter
from Shortz.
Dudeney. Problem 501 -- The spider and the fly. Weekly Dispatch (14 & 28 Jun 1903) both
p. 16. 4 side version.
Dudeney. Breakfast table problems, No. 320 -- The spider and the fly. Daily Mail (18 &
21 Jan 1905) both p. 7. Same as the above problem.
Dudeney. Master of the breakfast table problem. Daily Mail (1 & 8 Feb 1905) both p. 7.
Interview with Dudeney in which he gives the 5 side version.
Ball. MRE, 4th ed., 1905, p. 66. Gives the 5 side version, citing the Daily Mail of
1 Feb 1905. He says he heard a similar problem in 1903 -- presumably Dudeney's first
version. In the 5th ed., 1911, p. 73, he attributes the problem to Dudeney.
Dudeney. CP. 1907. Prob. 75: The spider and the fly, pp. 121-122 & 221-222. 5 side
version with discussion of various generalizations.
Dudeney. The world's best problems. 1908. Op. cit. in 2. P. 786 gives the five side version.
Sidney J. Miller. Some novel picture puzzles -- No. 6. Strand Mag. 41 (No. 243) (Mar 1911)
372 & 41 (No. 244) (Apr 1911) 506. Contest between two snails. Better method uses
four sides, similar to Dudeney's version, but with different numbers.
Loyd. The electrical problem. Cyclopedia, 1914, pp. 219 & 368 (= MPSL2, prob. 149,
pp. 106 & 169 = SLAHP: Wiring the hall, pp. 72 & 114). Same as Dudeney's first,
four side, version. (In MPSL2, Gardner says Loyd has simplified Dudeney's 5 side
problem. More likely(?) Loyd had only seen Dudeney's earlier 4 side problem.)
Dudeney. MP. 1926. Prob. 162: The fly and the honey, pp. 67 & 157. (= 536, prob. 325,
pp. 112 & 313.) Cylindrical problem.
Perelman. FFF. 1934. The way of the fly. 1957: Prob. 68, pp. 111-112 & 117-118; 1979:
Prob. 72, pp. 136 & 142-144. MCBF: Prob. 72, pp. 134 & 141-142. Cylindrical form,
but with different numbers and arrangement than Dudeney's MP problem.
Haldeman-Julius. 1937. No. 34: The louse problem, pp. 6 & 22. Room 40 x 20 x 10 with
louse at a corner wanting to go to a diagonally opposite corner. Problem sent in by J. R.
Reed of Emmett, Idaho. Answer is 50!
M. Kraitchik. Mathematical Recreations, 1943, op. cit. in 4.A.2, chap. 1, prob. 7, pp. 17-21.
Room with 8 equal routes from spider to fly. (Not in his Math. des Jeux.)
Sullivan. Unusual. 1943. Prob. 10: Why not fly? Find shortest route from a corner of a cube
to the diagonally opposite corner.
SOURCES - page 196
William R. Ransom. One Hundred Mathematical Curiosities. J. Weston Walch, Portland,
Maine, 1955. The spider problem, pp. 144-146. There are three types of path, covering
3, 4 and 5 sides. He determines their relative sizes as functions of the room
dimensions.
Birtwistle. Math. Puzzles & Perplexities. 1971.
Round the cone, pp. 144 & 195. What is the shortest distance from a point P around a
cone and back to P? Answer is "An ellipse", which doesn't seem to answer the
question. If the cone has height H, radius R and P is l from the apex, then
the slant height L is (R2 + H2), the angle of the opened out cone is θ = 2πR/L
and the required distance is 2l sin θ/2.
Spider circuit, pp. 144 & 198. Spider is at the midpoint of an edge of a cube. He wants
to walk on each of the faces and return. What is his shortest route? Answer is
"A regular hexagon. (This may be demonstrated by putting a rubber band around
a cube.)"
David Singmaster. The spider spied her. Problem used as: More than one way to catch a fly,
The Weekend Telegraph (2 Apr 1984). Spider inside a glass tube, open at both ends,
goes directly toward a fly on the outside. When are there two equally short paths? Can
there be more than two shortest routes?
Yoshiyuki Kotani has posed the following general and difficult problem. On an a x b x c
cuboid, which two points are furthest apart, as measured by an ant on the surface? Dick
Hess has done some work on this, but I believe that even the case of square crosssection is not fully resolved.
6.N. DISSECTION OF A 1 x 1 x 2 BLOCK TO A CUBE
W. F. Cheney, Jr., proposer; W. R. Ransom; A. H. Wheeler, solvers. Problem E4. AMM 39
(1932) 489; 40 (1933) 113-114 & 42 (1934) 509-510. Ransom finds a solution in 8
pieces; Wheeler in 7.
Harry Lindgren. Geometric Dissections. Van Nostrand, Princeton, 1964. Section 24.2, p.
120 gives a variant of Wheeler's solution.
Michael Goldberg. A duplication of the cube by dissection and a hinged linkage. MG 50
(No. 373) (Oct 1966) 304-305. Shows that a hinged version exists with 10 pieces.
Hanegraaf, below, notes that there are actually 12 pieces here.
Anton Hanegraaf. The Delian Altar Dissection. Polyhedral Dissections, Elst, Netherlands,
1989. Surveys the problem, gives a 6 piece solution and a 7 piece hinged solution.
6.O. PASSING A CUBE THROUGH AN EQUAL OR SMALLER CUBE -PRINCE RUPERT'S PROBLEM
The projection of a unit cube along a space diagonal is a regular hexagon of side 2/3.
The largest square inscribable in this hexagon has edge 6 - 2 = 1.03527618. By passing
the larger cube on a slant to the space diagonal, one can get the larger cube having edge
32/4 = 1.06066172.
There are two early attributions of this. Wallis attributes it to Prince Rupert, but
Hennessy says Philip Ronayne of Cork invented it. I have discovered a possible connection.
Prince Rupert of the Rhine (1619-1682), nephew of Charles I, was a major military figure of
his time, becoming commander-in-chief of Charles I's armies in the 1640s. In 1648-1649, he
was admiral of the King's fleet and was blockaded with 16 ships in Kinsale Harbor for 20
months. Kinsale is about 20km south of Cork.
Ronayne wrote an Algebra, of which only a second edition of 1727 is in the BL. Schrek
has investigated the family histories and says Ronayne lived in the early 18C. This would
seem to make him too young to have met Rupert. Perhaps Rupert invented the problem while
in Kinsale and this was conveyed to Ronayne some years later. Does anyone know the dates
of Ronayne or of the 1st ed (Schrek only located the BL example of the 2nd ed)? I cannot find
anything on him in Wallis, May, Poggendorff, DNB, but Google has turned up a reference to a
1917 history of the family which Schrek cites, but I have not yet tried to find this.
Hennessy's article says a little about Daniel Voster and details are in Wallis's . His
father, Elias (1682 - >1728) wrote an Arithmetic, of which Wallis lists 30 editions. The BL
lists one as late as 1829. The son, Daniel (1705 - >1760) was a schoolmaster and instrument
maker who edited later versions of his father's arithmetic. The 1750 History of Cork quoted
by Hennessy says the author had seen the cubes with Daniel. Hennessy conjectures that his
SOURCES - page 197
example was made specially, perhaps under the direction of a mathematician. It seems likely
that Daniel knew Ronayne and made this example for him.
John Wallis. Perforatio cubi, alterum ipsi aequalem recipiens. (De Algebra Tractatus; 1685;
Chap. 109) = Opera Mathematica, vol. II, Oxford, 1693, pp. 470-471, ??NYS. Cites
Rupert as the source of the equal cube version. (Latin and English in Schrek.) Scriba,
below, found an errata slip in Wallis's copy of his Algebra in the Bodleian. This
corrects the calculations, but was published in the Opera, p. 695.
Ozanam-Montucla. 1778. Percer un cube d'une ouverture, par laquelle peut passer un autre
cube égal au premier. Prob. 30 & fig. 53, plate 7, 1778: 319-320; 1803: 315-316;
1814: 268-269. Prob. 29, 1840: 137. Equal cubes with diagonal movement.
J. H. van Swinden. Grondbeginsels der Meetkunde. 2nd ed., Amsterdam, 1816, pp. 512-513,
??NYS. German edition by C. F. A. Jacobi, as: Elemente der Geometrie, Friedrich
Frommann, Jena, 1834, pp. 394-395. Cites Rupert and Wallis and gives a simple
construction, saying Nieuwland has found the largest cube which can pass through a
cube.
Peter Nieuwland. (Finding of maximum cube which passes through another). In: van
Swinden, op. cit., pp. 608-610; van Swinden-Jacobi, op. cit. above, pp. 542-544, gives
Nieuwland's proof.
Cundy and Rollett, p. 158, give references to Zacharias (see below) and to Cantor, but Cantor
only cites Hennessy.
H. Hennessy. Ronayne's cubes. Phil. Mag. (5) 39 (Jan-Jun 1895) 183-187. Quotes, from
Gibson's 'History of Cork', a passage taken from Smith's 'History of Cork', 1st ed., 1750,
vol. 1, p. 172, saying that Philip Ronayne had invented this and that a Daniel Voster had
made an example, which may be the example owned by Hennessy. He gives no
reference to Rupert. He finds the dimensions.
F. Koch & I. Reisacher. Die Aufgabe, einen Würfel durch einen andern durchzuschieben.
Archiv Math. Physik (3) 10 (1906) 335-336. Brief solution of Nieuwland's problem.
M. Zacharias. Elementargeometrie und elementare nicht-Euklidische Geometrie in
synthetischer Behandlung. Encyklopädie der Mathematischen Wissenschaften.
Band III, Teil 1, 2te Hälfte. Teubner, Leipzig, 1914-1931. Abt. 28: Maxima und
Minima. Die isoperimetrische Aufgabe. Pp. 1133-1134. Attributes it to Prince Rupert,
following van Swinden. Cites Wallis & Ronayne, via Cantor, and Nieuwland, via van
Swinden.
U. Graf. Die Durchbohrung eines Würfels mit einem Würfel. Zeitschrift math. naturwiss.
Unterricht 72 (1941) 117. Nice photos of a model made at the Technische Hochschule
Danzig. Larger and better versions of the same photos can be found in: W. Lietzmann
& U. Graf; Mathematik in Erziehung und Unterricht; Quelle & Meyer, Leipzig, 1941,
vol. 2, plate 3, opp. p. 168, but I can't find any associated text for it.
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 12: Curios [sic] cubes, p. 14. First
says it can be done with equal cubes and then a larger can pass through a smaller.
Claims that the larger cube can be about 1.1, but this is due to an error -- he thinks the
hexagon has the same diameter as the cube itself.
H. D. Grossman, proposer; C. S. Ogilvy & F. Bagemihl, solvers. Problem E888 -- Passing a
cube through a cube of same size. AMM 56 (1949) 632 ??NYS & 57 (1950) 339.
Only considers cubes of the same size, though Bagemihl's solution permits a slightly
larger cube. No references.
D. J. E. Schrek. Prince Rupert's problem and its extension by Pieter Nieuwland. SM 16
(1950) 73-80 & 261-267. Historical survey, discussing Rupert, Wallis, Ronayne, van
Swinden & Nieuwland. Says Ronayne is early 18C.
M. Gardner. SA (Nov 1966) = Carnival, pp. 41-54. The largest square inscribable in a cube
is the cross section of the maximal hole through which another cube can pass.
Christoph J. Scriba. Das Problem des Prinzen Ruprecht von der Pfalz. Praxis der
Mathematik 10 (1968) 241-246. ??NYS -- described by Scriba in an email to HM
Mailing List, 20 Aug 1999. Describes the correction to Wallis's work and considers the
problem for the tetrahedron and octahedron.
6.P. GEOMETRICAL VANISHING
Gardner. MM&M. 1956. Chap. 7 & 8: Geometrical Vanishing -- Parts I & II, pp. 114-155.
Best extensive discussion of the subject and its history.
SOURCES - page 198
Gardner. SA (Jan 1963) c= Magic Numbers, chap. 3. Discusses application to making an
extra bill and Magic Numbers adds citations to several examples of people trying it and
going to jail.
Gardner. Advertising premiums to beguile the mind: classics by Sam Loyd, master
puzzle-poser. SA (Nov 1971) = Wheels, Chap. 12. Discusses several forms.
S&B, p. 144, shows several versions.
6.P.1.
PARADOXICAL DISSECTIONS OF THE CHESSBOARD BASED
ON FIBONACCI NUMBERS
Area 63 version: AWGL, Dexter, Escott, White, Loyd, Ahrens, Loyd Jr., Ransom.
(W. Leybourn. Pleasure with Profit. 1694. ?? I cannot recall the source of this reference and
think it may be an error. I have examined the book and find nothing relevant in it.)
Loyd. Cyclopedia, 1914, pp. 288 & 378. 8 x 8 to 5 x 13 and to an area of 63. Asserts Loyd
presented the first of these in 1858. Cf Loyd Jr, below.
O. Schlömilch. Ein geometrisches Paradoxon. Z. Math. Phys., 13 (1868) 162. 8 x 8 to
5 x 13. (This article is only signed Schl. Weaver, below, says this is Schlömilch, and
this seems right as he was a co-editor at the time. Coxeter (SM 19 (1953) 135-143)
says it is V. Schlegel, apparently confusing it with the article below.) Doesn't give any
explanation, leaving it as a student exercise.
F. J. Riecke. Op. cit. in 4.A.1. Vol. 3, 1873. Art. 16: Ein geometrisches Paradoxon. Quotes
Schlömilch and explains the paradox.
G. H. Darwin. Messenger of Mathematics 6 (1877) 87. 8 x 8 to 5 x 13 and generalizations.
V. Schlegel. Verallgemeinerung eines geometrischen Paradoxons. Z. Math. Phys. 24 (1879)
123-128 & Plate I. 8 x 8 to 5 x 13 and generalizations.
Mittenzwey. 1880. Prob. 299, pp. 54 & 105; 1895?: 332, pp. 58 & 106-107; 1917: 332, pp.
53 & 101. 8 x 8 to 5 x 13. Clear explanation.
The Boy's Own Paper. No. 109, vol. III (12 Feb 1881) 327. A puzzle. 8 x 8 to 5 x 13
without answer.
Richard A. Proctor. Some puzzles. Knowledge 9 (Aug 1886) 305-306. "We suppose all the
readers ... know this old puzzle." Describes and explains 8 x 8 to 5 x 13. Gives a
different method of cutting so that each rectangle has half the error -- several
typographical errors.
Richard A. Proctor. The sixty-four sixty-five puzzle. Knowledge 9 (Oct 1886) 360-361.
Corrects the above and explains it in more detail.
Will Shortz has a puzzle trade card with the 8 x 8 to 5 x 13 version, c1889.
Ball. MRE, 1st ed., 1892, pp. 34-36. 8 x 8 to 5 x 13 and generalizations. Cites Darwin and
describes the examples in Ozanam-Hutton (see Ozanam-Montucla in 6.P.2). In the 5th
ed., 1911, p. 53, he changes the Darwin reference to Schlömilch. In the 7th ed., 1917,
he only cites the Ozanam-Hutton examples.
Clark. Mental Nuts. 1897, no. 33; 1904, no. 41; 1916, no. 43. Four peculiar drawings. 8 x
8 to 5 x 13.
Carroll-Collingwood. 1899. Pp. 316-317 (Collins: 231 and/or 232 (lacking in my copy))
= Carroll-Wakeling II, prob. 7: A geometrical paradox, pp. 12 & 7. 8 x 8 to 5 x 13.
Carroll may have stated this as early as 1888. Wakeling says the papers among which
this was found on Carroll's death are now in the Parrish Collection at Princeton
University and suggests Schlömilch as the earliest version.
AWGL (Paris). L'Echiquier Fantastique. c1900. Wooden puzzle of 8 x 8 to 5 x 13 and to
area 63. ??NYS -- described in S&B, p. 144.
Walter Dexter. Some postcard puzzles. Boy's Own Paper (14 Dec 1901) 174-175. 8 x 8 to
5 x 13 and to area 63.
C. A. Laisant. Initiation Mathématique. Georg, Geneva & Hachette, Paris, 1906. Chap. 63:
Un paradoxe: 64 = 65, pp. 150-152.
Wm. F. White. In the mazes of mathematics. A series of perplexing puzzles. III. Geometric
puzzles. The Open Court 21 (1907) 241-244. Shows 8 x 8 to 5 x 13 and a two-piece
11 x 13 to area 145.
E. B. Escott. Geometric puzzles. The Open Court 21 (1907) 502-505. Shows 8 x 8 to area
63 and discusses the connection with Fibonacci numbers.
William F. White. Op. cit. in 5.E. 1908. Geometric puzzles, pp. 109-117. Partly based on
above two articles. Gives 8 x 8 to 5 x 13 and to area 63. Gives an extension which
SOURCES - page 199
turns 12 x 12 into 8 x 18 and into area 144, but turns 23 x 23 into 16 x 33 and
into area 145. Shows a puzzle of Loyd: three-piece 8 x 8 into 7 x 9.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. 5 x 5 into four pieces that make a
3 x 8.
M. Adams. Indoor Games. 1912. Is 64 equal to 65? Pp. 345-346 with fig. on p. 344.
Loyd. Cyclopedia. 1914. See entry at 1858.
W. Ahrens. Mathematische Spiele. Teubner, Leipzig. 3rd ed., 1916, pp. 94-95 & 111-112.
The 4th ed., 1919, and 5th ed., 1927, are identical with the 3rd ed., but on different
pages: pp. 101-102 & pp. 118-119. Art. X. 65 = 64 = 63 gives 8 x 8 to 5 x 13 and
to area 63. The area 63 case does not appear in the 2nd ed., 1911, which has Art. V.
64 = 65, pp. 107 & 118-119 and this material is not in the 1st ed. of 1907.
Tom Tit?? In Knott, 1918, but I can't find it in Tom Tit. No. 3: The square and the rectangle:
64 = 65!, pp. 15-16. Clearly explained.
Hummerston. Fun, Mirth & Mystery. 1924. A puzzling paradox, pp. 44 & 185. Usual 8 x 8
to 5 x 13, but he erases the chessboard lines except for the cells the cuts pass through,
so one way has 16 cells, the other way has 17 cells. Reasonable explanation.
Collins. Book of Puzzles. 1927. A paradoxical puzzle, pp. 4-5. 8 x 8 to 5 x 13. Shades the
unit cells that the lines pass through and sees that one way has 16 cells, the other way
has 17 cells, but gives only a vague explanation.
Loyd Jr. SLAHP. 1928. A paradoxical puzzle, pp. 19-20 & 90. Gives 8 x 8 to 5 x 13. "I
have discovered a companion piece ..." and gives the 8 x 8 to area 63 version. But cf
AWGL, Dexter, etc. above.
W. Weaver. Lewis Carroll and a geometrical paradox. AMM 45 (1938) 234-236. Describes
unpublished work in which Carroll obtained (in some way) the generalizations of the
8 x 8 to 5 x 13 in about 1890-1893. Weaver fills in the elementary missing
arguments.
W. R. Ransom, proposer; H. W. Eves, solver. Problem E468. AMM 48 (1941) 266 & 49
(1942) 122-123. Generalization of the 8 x 8 to area 63 version.
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 23: Summat for nowt?, pp. 27-28.
8 x 8 to 5 x 13, clearly drawn.
Warren Weaver. Lewis Carroll: Mathematician. Op. cit. in 1. 1956. Brief mention of 8 x 8
to 5 x 13. John B. Irwin's letter gives generalizations to other consecutive triples of
Fibonacci numbers (though he doesn't call them that). Weaver's response cites his 1938
article, above.
6.P.2. OTHER TYPES
In several early examples, the authors appear unaware that area has vanished!
Pacioli. De Viribus. c1500. Ff. 189v - 191r. Part 2. LXXIX. Do(cumento). un tetragono
saper lo longare con restregnerlo elargarlo con scortarlo (a tetragon knows lengthening
and contraction, enlarging with shortening ??) = Peirani 250-252. Convert a 4 x 24
rectangle to a 3 x 32 using one cut into two pieces. Pacioli's
description is cryptic but seems to have two cuts, making
d
c
three pieces. There is a diagram at the bottom of f. 190v, badly
k
f e
redrawn on Peirani 458. Below this is a inserted note which Peirani
252 simply mentions as difficult to read, but can make sense. The
g
points are as laid out at the right. abcd is the original 4 x 24
h a o b
rectangle. g is one unit up from a and e is one unit down from c.
Cut from c to g and from e parallel to the base, meeting cg at f. Then move cdg
to fkh and move fec to hag. Careful rereading of Pacioli seems to show he is using a
trick! He cuts from e to f to g. then turns over the upper piece and slides it along so
that he can continue his cut from g to h, which is where f to c is now. This gives
three pieces from a single cut! Pacioli clearly notes that the area is conserved.
Although not really in this topic, I have put it here as it seems to be a predecessor
of this topic and of 6.AY.
Sebastiano Serlio. Libro Primo d'Architettura. 1545. This is the first part of his Architettura,
5 books, 1537-1547, first published together in 1584. I have seen the following
editions.
With French translation by Jehan Martin, no publisher shown, Paris, 1545, f. 22.r.
??NX
SOURCES - page 200
1559. F. 15.v.
Francesco Senese & Zuane(?) Krugher, Venice, 1566, f. 16.r. ??NX
Jacomo de'Franceschi, Venice, 1619, f. 16.r.
Translated into Dutch by Pieter Coecke van Aelst as: Den eerstē vijfsten boeck
van architecturē; Amsterdam, 1606. This was translated into English as: The Five
Books of Architecture; Simon Stafford, London, 1611 = Dover, 1982. The first Booke,
f. 12v.
3 x 10 board is cut on a diagonal and slid to form a 4 x 7 table with 3 x 1 left
over, but he doesn't actually put the two leftover pieces together nor notice the area
change!
Pietro Cataneo. L'Architettura di Pietro Cataneo Senese. Aldus, Venice, 1567. ??NX. Libro
Settimo.
P. 164, prop. XXVIIII: Come si possa accresciere una stravagante larghezza. Gives a
correct version of Serlio's process.
P. 165, prop. XXX: Falsa solutione del Serlio. Cites p. xxii of Serlio. Carefully
explains the error in Serlio and says his method is "insolubile, & mal pensata".
Schwenter. 1636. Part 15, ex. 14, p. 541: Mit einem länglichten schmahlen Brett / für ein
bräites Fenster einen Laden zu machen. Cites Gualtherus Rivius, Architectur.
Discusses Serlio's dissection as a way of making a 4 x 7 from a 3 x 10 but doesn't
notice the area change.
Gaspar Schott. Magia Universalis. Joh. Martin Schönwetter, Bamberg, Vol. 3, 1677.
Pp. 704-708 describes Serlio's error in detail, citing Serlio. ??NX of plates.
I have a vague reference to the 1723 ed. of Ozanam, but I have not seen it in the 1725 ed. -this may be an error for the 1778 ed. below.
Minguet. 1755. Pp. not noted -- ??check (1822: 145-146; 1864: 127-128). Same as Hooper.
Not in 1733 ed.
Vyse. Tutor's Guide. 1771? Prob. 8, 1793: p. 304, 1799: p. 317 & Key p. 358. Lady has a
table 27 square and a board 12 x 48. She cuts the board into two 12 x 24 rectangles
and cuts each rectangle along a diagonal. By placing the diagonals of these pieces on
the sides of her table, she makes a table 36 square. Note that 362 = 1296 and 272 +
12 x 48 = 1305. Vyse is clearly unaware that area has been created. By dividing all
lengths by 3, one gets a version where one unit of area is lost. Note that 4, 8, 9 is
almost a Pythagorean triple.
William Hooper. Rational Recreations. 1774. Op. cit. in 4.A.1. Vol. 4, pp. 286-287:
Recreation CVI -- The geometric money. 3 x 10 cut into four pieces which make a
2 x 6 and a 4 x 5. (The diagram is shown in Gardner, MM&M, pp. 131-132.) (I
recently saw that an edition erroneously has a 3 x 6 instead of a 2 x 6 rectangle. This
must be the 1st ed. of 1774, as it is correct in my 2nd ed. of 1782.)
Ozanam-Montucla. 1778. Transposition de laquelle semble résulter que le tout peut être égal
à la partie. Prob. 21 & fig. 127, plate 16, 1778: 302-303 & 363; 1803: 298-299 & 361;
1814: 256 & 306; 1840: omitted. 3 x 11 to 2 x 7 and 4 x 5. Remarks that M. Ligier
probably made some such mistake in showing 172 = 2 x 122 and this is discussed
further on the later page.
E. C. Guyot. Nouvelles Récréations Physiques et Mathématiques. Nouvelle éd. La Librairie,
Rue S. André-des-Arcs[sic], Paris, Year 7 [1799]. Vol. 2, Deuxième récréation: Or
géométrique -- construction, pp. 41-42 & plate 6, opp. p. 37. Same as Hooper.
Manuel des Sorciers. 1825. Pp. 202-203, art. 19. ??NX Same as Hooper.
The Boy's Own Book. The geometrical money. 1828: 413; 1828-2: 419; 1829 (US): 212;
1855: 566-567; 1868: 669. Same as Hooper.
Magician's Own Book. 1857. Deceptive vision, pp. 258-259. Same as Hooper. = Book of
500 Puzzles, 1859, pp. 72-73.
Illustrated Boy's Own Treasury. 1860. Optics: Deceptive vision, p. 445. Same as Hooper.
Identical to Book of 500 Puzzles.
Wemple & Company (New York). The Magic Egg Puzzle. ©1880. S&B, p. 144.
Advertising card, the size of a small postcard, but with ads for Rogers Peet on the back.
Starts with 9 eggs. Cut into four rectangles and reassemble to make 6, 7, 8, 10, 11, 12
eggs.
R. March & Co. (St. James's Walk, Clerkenwell). 'The Magical Egg Puzzle', nd [c1890]. (I
have a photocopy.) Four rectangles which produce 6, 7, ..., 12 eggs. Identical to the
Wemple version, but with Wemple's name removed. I only have a photocopy of the
front of this and I don't know what's on the back. I also have a photocopy of the
SOURCES - page 201
instructions.
Loyd. US Patent 563,778 -- Transformation Picture. Applied: 11 Mar 1896; patented:
14 Jul 1896. 1p + 1p diagrams. Simple rotating version using 8 to 7 objects.
Loyd. Get Off the Earth. Puzzle notices in the Brooklyn Daily Eagle (26 Apr - 3 May 1896),
printing individual Chinamen. Presenting all of these at an office of the newspaper gets
you an example of the puzzle. Loyd ran discussions on it in his Sunday columns until
3 Jan 1897 and he also sold many versions as advertising promotions. S&B, p. 144,
shows several versions.
Loyd. Problem 17: Ye castle donjon. Tit-Bits 31 (6 & 27 Feb & 6 & 20 Mar 1897) 343,
401, 419 & 455. = Cyclopedia, 1914, The architect's puzzle, pp. 241 & 372. 5 x 25
to area 124.
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. Discusses and shows Get Off the Earth.
Ball. MRE, 4th ed., 1905, pp. 50-51: Turton's seventy-seven puzzle. Additional section
describing Captain Turton's 7 x 11 to 7 x 11 with one projecting square, using
bevelled cuts. This is dropped from the 7th ed., 1917.
William F. White. 1907 & 1908. See entries in 6.P.1.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Gives "Get Off the Earth" on p. 785.
Loyd. Teddy and the Lions. Gardner, MM&M, p. 123, says he has seen only one example,
made as a promotional item for the Eden Musee in Manhattan. This has a round disc,
but two sets of figures -- 7 natives and 7 lions which become 6 natives and 8 lions.
Dudeney. A chessboard fallacy. The Paradox Party. Strand Mag. 38 (No. 228) (Dec 1909)
676 (= AM, prob. 413, pp. 141 & 247). (There is a solution in Strand Mag. 39
(No. 229) (Jan 1910) ??NYS.) 8 x 8 into 3 pieces which make a 9 x 7.
Fun's Great Baseball Puzzle. Will Shortz gave this out at IPP10, 1989, as a colour photocopy,
433 x 280 mm (approx. A3). ©1912 by the Press Publishing Co (The New York
World). I don't know if Fun was their Sunday colour comic section or what. One has to
cut it diagonally and slide one part along to change from 8 to 9 boys.
Loyd. The gold brick puzzle. Cyclopedia, 1914, pp. 32 & 342 (= MPSL1, prob. 24, pp. 22 &
129). 24 x 24 to 23 x 25.
Loyd. Cyclopedia. 1914. "Get off the earth", p. 323. Says over 10 million were sold. Offers
prizes for best answers received in 1909.
Loyd Jr. SLAHP. 1928. "Get off the Earth" puzzle, pp. 5-6. Says 'My "Missing Chinaman
Puzzle"' of 1896. Gives a simple and clear explanation.
John Barnard. The Handy Boy's Book. Ward, Lock & Co., London, nd [c1930?]. Some
interesting optical illusions, pp. 310-311. Shows a card with 11 matches and a diagonal
cut so that sliding it one place makes 10 matches.
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 24: A chessboard fallacy, pp. 28-29.
8 x 8 cut with a diagonal of a 8 x 7 region, then pieces slid and a triangle cut off and
moved to the other end to make a 9 x 7. Clear illustration.
Mel Stover. From 1951, he devised a number of variations of both Get off the Earth (perhaps
the best is his Vanishing Leprechaun) and of Teddy and the Lions (6 men and 4 glasses
of beer become 5 men and 5 glasses). I have examples of some of these from Stover
and I have looked at his notebooks, which are now with Mark Setteducati. See
Gardner, MM&M, pp. 125-128.
Gardner. SA (May 1961) c= NMD, chap. 11. Mentioned in Workout, chap. 27. Describes
his adaptation of a principle of Paul Curry to produce The Disappearing Square puzzle,
where 16 or 17 pieces seem to make the same square. The central part of the 17 piece
version consists of five equal squares in the form of a Greek cross. The central part of
the 16 piece version has four of the squares in the shape of a square. This has since
been produced in several places.
Ripley's Puzzles and Games. 1966. P. 60. Asserts that when you cut a 2½ x 4½ board into
six right triangles with legs 1½ and 2½, then they assemble into an equilateral triangle
of edge 5. This has an area loss of about 4%.
John Fisher. John Fisher's Magic Book. Muller, London, 1968.
Financial Wizardry, pp. 18-19. 7 x 8 region with £ signs marking the area. A line
cuts off a triangle of width 7 and height 2 at the top. The rest of the area is
divided by a vertical into strips of widths 4 and 3, with a small rectangle 3 by 1
cut from the bottom of the width 3 strip. When the strips are exchanged, one unit
of area is lost and one £ sign has vanished.
Try-Angle, pp. 126-127. This is one of Curry's triangles -- see Gardner, MM&M,
p. 147.
SOURCES - page 202
Alco-Frolic!, pp. 148-149. This is a form of Stover's 6 & 4 to 5 & 5 version.
D. E. Knuth. Disappearances. In: The Mathematical Gardner; ed. by David Klarner; Prindle,
Weber & Schmidt/Wadsworth, 1981. P. 264. An eight line poem which rearranges to a
seven line poem.
Dean Clark. A centennial tribute to Sam Loyd. CMJ 23:5 (Nov 1992) 402-404. Gives an
easy circular version with 11 & 12 astronauts around the earth and a 15 & 16 face
version with three pieces, a bit like the Vanishing Leprechaun.
6.Q. KNOTTING A STRIP TO MAKE A REGULAR PENTAGON
Urbano d'Aviso. Trattato della Sfera e Pratiche per Uso di Essa. Col modeo di fare la figura
celeste, opera cavata dalli manoscritti del. P. Bonaventura Cavalieri. Rome, 1682.
??NYS cited by Lucas (1895) and Fourrey.
Dictionary of Representative Crests. Nihon Seishi Monshō Sōran (A Comprehensive Survey
of Names and Crests in Japan), Special issue of Rekishi Dokuhon (Readings in
History), Shin Jinbutsu Oraisha, Tokyo, 1989, pp. 271-484. Photocopies of relevant
pages kindly sent by Takao Hayashi.
Crests 3504 and 3506 clearly show a strip knotted to make a pentagon. 3507 has
two such knots and 3508 has five. I don't know the dates, but most of these crests are
several centuries old.
Lucas. RM2, 1883, pp. 202-203.
Tom Tit.
Vol. 2, 1892. L'Étoile à cinq branches, pp. 153-154. = K, no. 5: The pentagon and the
five pointed star, pp. 20-21. He adds that folding over the free end and holding
the knot up to the light shows the pentagram.
Vol. 3, 1893. Construire d'un coup de poing un hexagone régulier, pp. 159-161.
= K, no. 17: To construct a hexagon by finger pressure, pp. 49-51. Pressing an
appropriate size Möbius strip flat gives a regular hexagon.
Vol. 3, 1893. Les sept pentagones, pp. 165-166. = K, no. 19: The seven pentagons,
pp. 54-55. By tying five pentagons in a strip, one gets a larger pentagon with a
pentagonal hole in the middle.
Somerville Gibney. So simple! The hexagon, the enlarged ring, and the handcuffs. The
Boy's Own Paper 20 (No. 1012) (4 Jun 1898) 573-574. As in Tom Tit, vol. 3, pp. 159161.
Lucas. L'Arithmétique Amusante. 1895. Note IV: Section II: Les Jeux de Ruban, Nos. 1 &
2: Le nœud de cravate & Le nœud marin, pp. 220-222. Cites d'Aviso and says he does
both the pentagonal and hexagonal knots, but Lucas only shows the pentagonal one.
E. Fourrey. Procédés Originaux de Constructions Géométriques. Vuibert, Paris, 1924. Pp.
113 & 135-139. Cites Lucas and cites d'Aviso as Traité de la Sphère and says he gives
the pentagonal and hexagonal knots. Fourrey shows and describes both, also giving the
pictures on his title page.
F. V. Morley. A note on knots. AMM 31 (1924) 237-239. Cites Knott's translation of Tom
Tit. Says the process generalizes to (2n+3)-gons by using n loops. Gets even-gons by
using two strips. Discusses using twisted strips.
Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed.,
Educational Publishers, St. Louis, 1949. Pp. 64-65 gives square (a bit trivial),
pentagon, hexagon, heptagon and octagon. Even case need two strips.
Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, pp. 16-17:
Polygons constructed by tying paper knots. Shows how to tie square, pentagon,
hexagon, heptagon and octagon.
James K. Brunton. Polygonal knots. MG 45 (No. 354) (Dec 1961) 299-302. All regular
n-gons, n > 4, can be obtained, except n = 6 which needs two strips. Discusses which
can be made without central holes.
Marius Cleyet-Michaud. Le Nombre d'Or. Presses Universitaires de France, Paris, 1973. On
pp. 47-48, he calls this the 'golden knot' (Le "nœud doré") and describes how to make it.
6.R. GEOMETRIC FALLACIES
General surveys of such fallacies can be found in the following. See also: 6.P, 10.A.1.
These fallacies are actually quite profound as the first two point out some major gaps in
Euclid's axioms -- the idea of a point being inside a triangle really requires notions of order of
SOURCES - page 203
points on a line and even the idea of continuity, i.e. the idea of real numbers.
Ball. MRE. 1st ed., 1892, pp. 31-34, two examples, discussed below. 3rd ed., 1896,
pp. 39-46 = 4th ed., 1905, pp. 41-48, seven examples. 5th ed., 1911, pp. 44-52 =
11th ed., 1939, pp. 76-84, nine example.
Walther Lietzmann. Wo steckt der Fehler? Teubner, Stuttgart, (1950), 3rd ed., 1953.
(Strens/Guy has 3rd ed., 1963(?).) (There are 2nd ed, 1952??; 5th ed, 1969; 6th ed,
1972. MG 54 (1970) 182 says the 5th ed appears to be unchanged from the 3rd ed.)
Chap. B: V, pp. 87-99 has 18 examples.
(An earlier version of the book, by Lietzmann & Trier, appeared in 1913, with
2nd ed. in 1917. The 3rd ed. of 1923 was divided into two books: Wo Steckt der
Fehler? and Trugschlüsse. There was a 4th ed. in 1937. The relevant material would
be in Trugschlüsse, but I have not seen any of the relevant books, though E. P. Northrop
cites Lietzmann, 1923, three times -- ??NYS.)
E. P. Northrop. Riddles in Mathematics. 1944. Chap. 6, 1944: 97-116, 232-236 & 249-250;
1945: 91-109, 215-219 & 230-231; 1961: 98-115, 216-219 & 229. Cites Ball,
Lietzmann (1923), and some other individual items.
V. M. Bradis, V. L. Minkovskii & A. K. Kharcheva. Lapses in Mathematical Reasoning.
(As: [Oshibki v Matematicheskikh Rassuzhdeniyakh], 2nd ed, Uchpedgiz, Moscow,
1959.) Translated by J. J. Schorr-Kon, ed. by E. A. Maxwell. Pergamon & Macmillan,
NY, 1963. Chap. IV, pp. 123-176. 20 examples plus six discussions of other examples.
Edwin Arthur Maxwell. Fallacies in Mathematics. CUP, (1959), 3rd ptg., 1969. Chaps. II-V,
pp. 13-36, are on geometric fallacies.
Ya. S. Dubnov. Mistakes in Geometric Proofs. (2nd ed., Moscow?, 1955). Translated by
Alfred K. Henn & Olga A. Titelbaum. Heath, 1963. Chap 1-2, pp. 5-33. 10 examples.
А. Г. Конфорович. [A. G. Konforovich]. (Математичні Софізми і Парадокси
[Matematichnī Sofīzmi ī Paradoksi] (In Ukrainian). Радянська Школа [Radyans'ka
Shkola], Kiev, 1983.) Translated into German by Galina & Holger Stephan as:
Konforowitsch, Andrej Grigorjewitsch; Logischen Katastrophen auf der Spur –
Mathematische Sophismen und Paradoxa; Fachbuchverlag, Leipzig, 1990. Chap. 4:
Geometrie, pp. 102-189 has 68 examples, ranging from the type considered here up
through fractals and pathological curves.
S. L. Tabachnikov. Errors in geometrical proofs. Quantum 9:2 (Nov/Dec 1998) 37-39 & 49.
Shows: every triangle is isosceles (6.R.1); the sum of the angles of a triangle is 180o
without use of the parallel postulate; a rectangle inscribed in a square is a square;
certain approaching lines never meet (6.R.3); all circles have the same circumference
(cf Aristotle's Wheel Paradox in 10.A.1); the circumference of a wheel is twice its
radius; the area of a sphere of radius R is π2R2.
6.R.1.EVERY TRIANGLE IS ISOSCELES
This is sometimes claimed to have been in Euclid's lost Pseudaria (Fallacies).
Ball. MRE, 1st ed., 1892, pp. 33-34. On p. 32, Ball refers to Euclid's lost Fallacies and
presents this fallacy and the one in 6.R.2: "I do not know whether either of them has
been published previously." In the 3rd ed., 1896, pp. 42-43, he adds the heading: To
prove that every triangle is isosceles. In the 5th ed., 1911, p. 45, he adds a note that he
believes these two were first published in his 1st ed. and notes that Carroll was
fascinated by them and they appear in The Lewis Carroll Picture Book (= CarrollCollingwood) -- see below.
Mathesis (1893). ??NYS. [Cited by Fourrey, Curiosities Geometriques, p. 145. Possibly
Mathesis (2) 3 (Oct 1893) 224, cited by Ball in MRE, 3rd ed, 1896, pp. 44-45, cf in
Section 6.R.4.]
Carroll-Collingwood. 1899. Pp. 264-265 (Collins: 190-191). = Carroll-Wakeling II,
prob. 27: Every triangle has a pair of equal sides!, pp. 43 & 27. Every triangle is
isosceles. Carroll may have stated this as early as 1888. Wakeling's solution just
suggests making an accurate drawing. Carroll-Gardner, p. 65, mentions this and says it
was not original with Carroll.
Ahrens. Mathematische Spiele. Teubner. Alle Dreiecke sind gleichschenklige. 2nd ed.,
1911, chap. X, art. VI, pp. 108 & 119-120. 3rd ed., 1916, chap. IX, art. IX, pp. 92-93 &
109-111. 4th ed., 1919 & 5th ed., 1927, chap IX, art. IX, pp. 99-101 & 116-118.
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Call Mr. Euclid -- No. 15: To prove all
SOURCES - page 204
triangles are equilateral, pp. 16-17. Clear exposition of the fallacy.
See Read in 6.R.4 for a different proof of this fallacy.
6.R.2.A RIGHT ANGLE IS OBTUSE
Ball. MRE, 1st ed., 1892, pp. 32-33. See 6.R.1. In the 3rd ed., 1896, pp. 40-41, he adds the
heading: To prove that a right angle is equal to an angle which is greater than a right
angle.
Mittenzwey. 1895?. Prob. 331, pp. 58 & 106; 1917: 331, pp. 53 & 101.
Carroll-Collingwood. 1899. Pp. 266-267 (Collins 191-192). An obtuse angle is sometimes
equal to a right angle. Carroll-Gardner, p. 65, mentions this and says it was not original
with Carroll.
H. E. Licks. 1917. Op. cit. in 5.A. Art. 82, p. 56.
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Call Mr. Euclid -- No. 16: To prove one
right angle greater than another right angle, pp. 18-19. "Here again, if you take the
trouble to draw an accurate diagram, you will find that the "construction" used for the
alleged proof is impossible."
E. A. Maxwell. Note 2121: That every angle is a right angle. MG 34 (No. 307) (Feb 1950)
56-57. Detailed demonstration of the error.
6.R.3.LINES APPROACHING BUT NOT MEETING
Proclus. 5C. A Commentary on the First Book of Euclid's Elements. Translated by Glenn R.
Morrow. Princeton Univ. Press, 1970. Pp. 289-291. Gives the argument and tries to
refute it.
van Etten/Henrion/Mydorge. 1630. Part 2, prob. 7: Mener une ligne laquelle aura inclination
à une autre ligne, & ne concurrera jamais contre l'Axiome des paralelles, pp. 13-14.
Schwenter. 1636. To be added.
Ozanam-Montucla. 1778. Paradoxe géométrique des lignes .... Prob. 70 & fig. 116-117,
plate 13, 1778: 405-407; 1803: 411-413; 1814: 348-350. Prob. 69, 1840: 180-181.
Notes that these arguments really produce a hyperbola and a conchoid. Hutton adds that
a great many other examples might be found.
E. P. Northrop. Riddles in Mathematics. 1944. 1944: 209-211 & 239; 1945: 195-197 &
222; 1961: 197-198 & 222. Gives the 'proof' and its fallacy, with a footnote on p. 253
(1945: 234; 1961: 233) saying the argument "has been attributed to Proclus."
Jeremy Gray. Ideas of Space. OUP, 1979. Pp. 37-39 discusses Proclus' arguments in the
context of attempts to prove the parallel postulate.
6.R.4.OTHERS
Ball. MRE, 3rd ed, 1896, pp. 44-45. To prove that, if two opposite sides of a quadrilateral
are equal, the other two sides must be parallel. Cites Mathesis (2) 3 (Oct 1893) 224 -??NYS
Cecil B. Read. Mathematical fallacies & More mathematical fallacies. SSM 33 (1933)
575-589 & 977-983. There are two perpendiculars from a point to a line. Part of a
line is equal to the whole line. Every triangle is isosceles (uses trigonometry). Angle
trisection (uses a marked straightedge).
P. Halsey. Class Room Note 40: The ambiguous case. MG 43 (No. 345) (Oct 1959)
204-205. Quadrilateral ABCD with angle A = angle C and AB = CD. Is this a
parallelogram?
6.S. TANGRAMS, ET AL.
GENERAL HISTORIES.
Hoffmann. 1893. Chap III, pp. 74-90, 96-97, 111-124 & 128 = Hoffmann-Hordern, pp. 62-79
& 86-87 with several photos. Describes Tangrams and Richter puzzles at some length.
Lots of photos in Hordern. Photos on pp. 67, 71, 75, 87 show Richter's: Anchor
(1890-1900, = Tangram), Tormentor (1898), Pythagoras (1892), Cross Puzzle (1892),
Circular Puzzle (1891), Star Puzzle (1899), Caricature (1890-1900, = Tangram) and
four non-Richter Tangrams in Tunbridge ware, ivory, mother-of-pearl and tortoise shell.
SOURCES - page 205
Hordern Collection, pp. 45-57 & 60, (photos on pp. 46, 49, 50, 52, 54, 56, 60) shows
different Richter versions of Tormentor (1880-1900), Pythagoras (1880-1900), Circular
Puzzle (1880-1900), Star Puzzle (1880-1900) and has a wood non-Richter version
instead of the ivory version in the last photo.
Ronald C. Read. Tangrams -- 330 Puzzles. Dover, 1965. The Introduction, pp. 1-6, is a
sketch of the history. Will Shortz says this is the first serious attempt to counteract the
mythology created by Loyd and passed on by Dudeney. Read cannot get back before
the early 1800s and notes that most of the Loyd myth is historically unreasonable.
However, Read does not pursue the early 1800s history in depth and I consider van der
Waals to be the first really serious attempt at a history of the subject.
Peter van Note. Introduction. IN: Sam Loyd; The Eighth Book of Tan; (Loyd & Co., 1903);
Dover, 1968, pp. v-viii. Brief debunking of the Loyd myth.
Jan van der Waals. History & Bibliography. In: Joost Elffers; Tangram; (1973), Penguin,
1976. Pp. 9-27 & 29-31. Says the Chinese term "ch'i ch'ae" dates from the Chu era
(-740/-330), but the earliest known Chinese book is 1813. The History reproduces
many pages from early works. The Bibliography cites 8 versions of 4 Chinese books
(with locations!) from 1813 to 1826 and 18 Western books from 1805 to c1850. The
1805, and several other references, now seem to be errors.
S&B. 1986. Pp. 22-33 discusses loculus of Archimedes, Chie no Ita, Tangrams and Richter
puzzles.
Alberto Milano. Due giochi di società dell'inizio dell'800. Rassegna di Studi e di Notizie 23
(1999) 131-177. [This is a publication by four museums in the Castello Sforzesco,
Milan: Raccolta delle Stampe Achille Bertarelli; Archivio Fotografico; Raccolte d'Arte
Applicata; Museo degli Strumenti Musicali. Photocopy from Jerry Slocum.] This
surveys early books on tangrams, some related puzzles and the game of bell and
hammer, with many reproductions of TPs and problems.
Jerry Slocum. The Tangram Book. (With Jack Botermans, Dieter Gebhardt, Monica Ma,
Xiaohe Ma, Harold Raizer, Dic Sonneveld and Carla van Spluntern.) ©2001 (but the
first publisher collapsed), Sterling, 2003. This is the long awaited definitive history of
the subject! It will take me sometime to digest and summarize this, but a brief
inspection shows that much of the material below needs revision!
Recent research by Jerry Slocum, backed up by The Admired Chinese Puzzle, indicates
that the introduction of tangrams into Europe was done by a person or persons in Lord
Amherst's 1815-1817 embassy to China, which visited Napoleon on St. Helena on its
return voyage. If so, then the conjectural dating of several items below needs to be
amended. I have amended my discussion accordingly and marked such dates with ??.
Although watermarking of paper with the correct date was a legal requirement at the
time, paper might have been stored for some time before it was printed on, so
watermark dates only give a lower bound for the date of printing. I have seen several
further items dated 1817, but it is conceivable that some material may have been sent
back to Europe or the US a few years earlier -- cf Lee.
On 2 Nov 2003, I did the following brief summary of Slocum's work in a letter to an
editor. I've made a few corrections and added a citation to the following literature.
Tangrams. The history of this has now been definitely established in Jerry Slocum's
new book: The Tangram Book; ©2001 (but the first publisher collapsed), Sterling,
2003. This history has been extremely difficult to unravel because Sam Loyd
deliberately obfuscated it in 1903, claiming the puzzle went back to 2000 BC, because
the only previous attempt at a history had many errors, and because much of the
material doesn't survive, or only a few examples survive. The history covers a wide
range in both time and location, as evidenced by the presence of seven co-authors from
several countries.
Briefly, the puzzle, in the standard form, dates from about 1800, in China. It is
attributed to Yang-cho-chü-shih, but this is a pseudonym, meaning 'dim-witted recluse',
and no copies of his work are known. The oldest known example of the game is one
dated 1802 in a museum near Philadelphia -- see Lee, below. The oldest known book
on the puzzle had a preface by Sang-hsia-ko [guest under the mulberry tree] dated June
1813 and a postscript by Pi-wu-chü-shih dated July 1813. This is only known from a
Japanese facsimile of it made in 1839. This book was republished, with a book of
solutions, in two editions in 1815 -- one with about four problems per page, the other
SOURCES - page 206
with about eleven. The latter version was the ancestor of many 19C books, both in
China and the west. Another 2 volume version appeared later in 1815. Sang-hsia-ko
explicitly says "The origin of the Tangram lies within the Pythagorean theorem".
In 1816, several ships brought copies of the eleven problems per page books to the US,
England and Europe. The first western publication of the puzzle is in early 1817 when
J. Leuchars of 47 Piccadilly registered a copyright and advertised sets for sale. But the
craze was really set off by the publication of The Fashionable Chinese Puzzle and its
Key by John and Edward Wallis and John Wallis Jr in March 1817. This included a
poem with a note that the game was "the favourite amusement of Ex-Emperor
Napoleon". This went through many printings, with some (possibly the first) versions
having nicely coloured illustrations. By the end of the year, there were many other
books, including examples in France, Italy and the USA.
Dic Sonneveld, one of the co-authors of Slocum's book, managed to locate the tangram
and books that had belonged to Napoleon in the Château de Malmaison, outside Paris,
but there is no evidence that Napoleon spent much time playing with it. St. Helena was
a regular stop for ships in the China trade. Napoleon is recorded as having bought a
chess set from one ship and several notables are recorded as having presented Napoleon
with gifts of Chinese objects. A diplomatic letter of Jan 1817 records sending an
example of the game from St. Helena to Prince Metternich, but this example has not
been traced.
The first American book was Chinese Philosophical and Mathematical Trangram by
James Coxe, appearing in Philadelphia in August 1817. The word 'trangram' meaning
'an odd, intricately contrived thing' according to Johnson's Dictionary, was essentially
obsolete by 1817, but was still in some use in the US. The earliest known use of the
word 'tangram' is in Thomas Hill's Geometrical Puzzles for the Young, Boston, 1848.
One suspects that he was influenced by Coxe's book, but he may have known that
'T'ang' is the Cantonese word for 'Chinese'. Hill later became President of Harvard
University and was an active promoter and inventor of games for classroom use. In
1864, the word was in Webster's Dictionary.
However, the above is the story of the seven-piece tangram that we know today. There
is a long background to this, dating back to the 3rd century BC, when Archimedes wrote
a letter to Eratosthenes describing a fourteen piece puzzle, known as the Stomachion or
Loculus of Archimedes. The few surviving texts are not very clear and there are two
interpretations -- in one the standard arrangement of the pieces is a square and in the
other it is a rectangle twice as wide as high. There are six (at least) references to the
puzzle in the classical world, the last being in the 6th century. The puzzle was used to
make a monstrous elephant, a brutal boar, a ship, a sword, etc., etc. The puzzle then
disappears, and no form of it appears in the Arabic world, which has always surprised
me, given the Arabic interest in patterns.
Further, several eastern predecessors of the tangrams are known. The earliest is a
Japanese version of 1742 by Ganriken (or Granreiken) which has seven pieces,
attributed (as were many things) to Sei Shonagon, a 10th century courtesan famous for
her ingenuity. By the end of the 18th century, three other dissection/arrangement
puzzles appeared in Japan, with 15, 19 and 19 pieces, including some semi-circles. An
1804 print by Utamaro shows courtesans playing with some version of the puzzle -only two copies of this print have been located.
But the basic puzzle idea has its roots in Chinese approaches to the Theorem of
Pythagoras and similar geometric proofs by dissection and rearrangement which date back to
the 3rd century (and perhaps earlier). But the tangram did not develop directly from these
ideas. From the 12th century, there was a Chinese tradition of making "Banquet Tables" in
the form of several pieces that could be arranged in several ways. The first known Chinese
book on furniture, by Huang Po-ssu in 1194, describes a Banquet Table formed of seven
rectangular pieces: two long, two medium and three short. In 1617, Ko Shan described
'Butterfly Wing" tables with 13 pieces, including isosceles right triangles, right trapeziums and
isosceles trapeziums. In 1856, a Chinese scholar noted the resemblance of these tables with
the tangram and a modern Chinese historian of mathematics has observed that half of the
butterfly arrangement can be easily transformed into the tangrams. No examples of these
tables have survived, but tables (and serving dishes) in the tangram pattern exist and are
probably still being made in China.
SPECIFIC ITEMS
SOURCES - page 207
Kanchusen. Wakoku Chiekurabe. 1727. Pp. 9 & 28-29: a simple dissection puzzle with
8 pieces. The problem appears to consist of a mitre comprising ¾ of a unit square;
4 isosceles right triangles of hypotenuse 1 and 3 isosceles right triangles of side ½,
but the solution shows that all the triangles are the same size, say having hypotenuse 1,
and the mitre shape is actually formed from a rectangle of size 1 x 2.
"Ganriken" [pseud., possibly of Fan Chu Sen]. Sei Shōnagon Chie-no-Ita (The Ingenious
Pieces by Sei Shōnagon.) (In Japanese). Kyoto Shobo, Aug 1742, 18pp, 42 problems
and solutions. Reproduced in a booklet, ed. by Kazuo Hanasaki, Tokyo, 1984, as
pp. 19-36. Also reproduced in a booklet, transcribed into modern Japanese, with
English pattern names and an English abstract, by Shigeo Takagi, 1989. This uses a set
of seven pieces different than the Tangram. S&B, p. 22, shows these pieces. Sei
Shōnagon (c965-c1010) was a famous courtier, author of The Pillow Book and
renowned for her intelligence. The Introduction is signed Ganriken. S&B say this is
probably Fan Chu Sen, but Takagi says the author's real name is unknown.
Utamaro. Interior of an Edo house, from the picture-book: The Edo Sparrows (or Chattering
Guide), 1786. Reproduced in B&W in: J. Hillier; Utamaro -- Colour Prints and
Paintings; Phaidon Press, Oxford, (1961), 2nd ed., 1979, p. 27, fig. 15. I found this
while hunting for the next item. This shows two women contemplating some pieces but
it is hard to tell if it is a tangram-type puzzle, or if perhaps they are cakes. Hiroko and
Mike Dean tell me that they are indeed cooking cakes.
Utamaro. Woodcut. 1792. Shows two courtesans working on a tangram puzzle. Van der
Waals dated this as 1780, but Slocum has finally located it, though he has only been
able to find two copies of it! The courtesans are clearly doing a tangram-like puzzle
with 12(?) pieces -- the pieces are a bit piled up and one must note that one of the
courtesans is holding a piece. They are looking at a sheet with 10 problem figures on it.
Early 19C books from China -- ??NYS -- cited by Needham, p. 111.
Jean Gordon Lee. Philadelphians and the China Trade 1784-1844. Philadelphia Museum of
Art, 1984, pp. 122-124. (Photocopy from Jerry Slocum.) P. 124, item 102, is an ivory
tangram in a cardboard box, inscribed on the bottom of the box: F. Waln April 4th 1802.
Robert Waln was a noted trader with China and this may have been a present for his
third son Francis (1799-1822). This item is in the Ryerss Museum, a city museum in
Philadelphia in the country house called Burholme which was built by one of Robert
Waln's sons-in-law.
A New Invented Chinese Puzzle, Consisting of Seven Pieces of Ivory or Wood, Viz.
5 Triangles, 1 Rhomboid, & 1 Square, which may be so placed as to form the Figures
represented in the plate. Paine & Simpson, Boro'. Undated, but the paper is
watermarked 1806. This consists of two 'volumes' of 8 pages each, comprising 159
problems with no solutions. At the end are bound in a few more pages with additional
problems drawn in -- these are direct copies of plates 21, 26, 22, 24, and 28 (with two
repeats from plate 22) of The New and Fashionable Chinese Puzzle, 1817. Bound in
plain covers. This is in Edward Hordern's collection and he provided a photocopy.
Dalgety also has a copy.
Ch'i Ch'iao t'u ho-pi (= Qiqiao tu hebi) (Harmoniously combined book of tangram problems
OR Seven clever pieces). 1813. (Bibliothek Leiden 6891, with an 1815 edition at
British Library 15257 d 13.) van der Waals says it has 323 examples. The 1813 seems
to be the earliest Chinese tangram book of problems, with the 1815 being the solutions.
Slocum says there was a solution book in 1815 and that the problem book had a preface
by Sang-hsia K'o (= Sang-xia-ke), which was repeated in the solution book with the
same date. Milano mentions this, citing Read and van der Waals/Elffers, and says an
example is on the BL. A version of this appears to have been the book given to
Napoleon and to have started the tangram craze in Europe. I have now received a
photocopy from Peter Rasmussen & Wei Zhang which is copied from van der Waals'
copy from BL 15257 d 13. It has a cover, 6 preliminary pages and 28 plates with 318
problems. The pages are larger than the photocopies of 1813/1815 versions in the BL
that Slocum gave me, which have 334 problems on 86 pages, but I see these are from
15257 d 5 and 14. I have a version of the smaller page format from c1820s which has
334 problems on 84pp, apparently lacking its first sheet. The problems are not
numbered, but given Chinese names. They are identical to those appearing in Wallis's
Fashionable Chinese Puzzle, below, except the pages are in different order, two pages
are inverted, Wallis replaces Chinese names by western numbers and draws the figures
SOURCES - page 208
a bit more accurately. Wallis skips one number and adds four new problems to get 323
problems - van der Waals seems to have taken 323 from Wallis.
Shichi-kou-zu Gappeki [The Collection of Seven-Piece Clever Figures]. Hobunkoku
Publishing, Tokyo, 1881. This is a Japanese translation of an 1813 Chinese book
"recognized as the earliest of existing Tangram book", apparently the previous item.
[The book says 1803, but Jerry Slocum reports this is an error for 1813!] Reprinted,
with English annotations by Y. Katagiri, from N. Takashima's copy, 1989. 129
problems (but he counts 128 because he omits one after no. 124), all included in my
version of the previous item, no solutions.
Anonymous. A Grand Eastern Puzzle. C. Davenporte & Co. Registered on 24 Feb 1817,
hence the second oldest English (and European?) tangram book [Slocum, p. 71.] It is
identical to Ch'i Ch'iao t'u ho-pi, 1815, above, except that plates 25 and 27 have been
interchanged. It appears to be made by using Chinese pages and putting a board cover
on it. On the front cover is the only English text:
A
Grand Eastern Puzzle
---------THE following Chineze Puzzle is recommended
to the Nobility, Gentry, and others, being superior to
any hitherto invented for the Amusement of the Juvenile
World, to whom it will afford unceasing recreation and
information; being formed on Geometrical principles, it
may not be considered as trifling to those of mature
years, exciting interest, because difficult and instructive,
imperceptibly leading the mind on to invention and perseverence. -- The Puzzle consists of five triangles, a
square, and a rhomboid, which may be placed in upwards
of THREE HUNDRED and THIRTY Characters, greatly resembling MEN, BEASTS, BIRDS, BOATS, BOTTLES, GLASSES, URNS, &c. The whole being the unwearied exertion
of many years study and application of one of the Literati of China, and is now offered to the Public for their
patronage and support.
ENTERED AT STATIONERS HALL
---Published and sold by
C. DAVENPORTE and Co.
No. 20, Grafton Street, East Euston Square.
The Fashionable Chinese Puzzle. Published by J. & E. Wallis, 42, Skinner Street and J.
Wallis Junr, Marine Library, Sidmouth, nd [Mar 1817]. Photocopy from Jerry Slocum.
This has an illustrated cover, apparently a slip pasted onto the physical cover. This
shows a Chinese gentleman holding a scroll with the title. There is a pagoda in the
background, a bird hovering over the scroll and a small person in the foreground
examining the scroll. Slocum's copy has paper watermarked 1816.
PLUS
A Key to the New and Fashionable Chinese Puzzle, Published by J. and E. Wallis, 42, Skinner
Street, London, Wherein is explained the method of forming every Figure contained in
That Pleasing Amusement. Nd [Mar 1817]. Photocopy from the Bodleian Library,
Oxford, catalogue number Jessel e.1176. TP seems to made by pasting in the cover slip
and has been bound in as a left hand page. ALSO a photocopy from Jerry Slocum. In
the latter copy, the apparent TP appears to be a paste down on the cover. The latter
copy does not have the Stanzas mentioned below. Slocum's copy has paper
watermarked 1815; I didn't check this at the Bodleian.
NOTE. This is quite a different book than The New and Fashionable Chinese
Puzzle published by Goodrich in New York, 1817.
SOURCES - page 209
Bound in at the beginning of the Fashionable Chinese Puzzle and the Bodleian
copy of the Key is: Stanzas, Addressed to Messrs. Wallis, on the Ingenious Chinese
Puzzle, Sold by them at the Juvenile Repository, 42, Skinner Street. In the Key, this is
on different paper than the rest of the booklet. The Stanzas has a footnote referring to
the ex-Emperor Napoleon as being in a debilitated state. (Napoleon died in 1821,
which probably led to the Bodleian catalogue's date of c1820 for the entire booklet - but
see below. Then follow 28 plates with 323 numbered figures (but number 204 is
skipped), solved in the Key. In the Bodleian copy of the Key, these are printed on stiff
paper, on one side of each sheet, but arranged as facing pairs, like Chinese booklets.
[Philip A. H. Brown; London Publishers and Printers c. 1800-1870; British
Library, 1982, p. 212] says the Wallis firm is only known to have published under the
imprint J. & E. Wallis during 1813 and Ruth Wallis showed me another source giving
1813?-1814. This led me to believe that the booklets originally appeared in 1813 or
1814, but that later issues or some owner inserted the c1820 sheet of Stanzas, which
was later bound in and led the Bodleian to date the whole booklet as c1820. Ruth
Wallis showed me a source that states that John Wallis (Jun.) set up independently of
his father at 186 Strand in 1806 and later moved to Sidmouth. Finding when he moved
to Sidmouth might help date this publication more precisely, but it may be a later
reissue. However, Slocum has now found the book advertised in the London Times in
Mar 1817 and says this is the earliest Western publication of tangrams, based on the
1813/1815 Chinese work. Wallis also produced a second book of problems of his own
invention and some copies seem to be coloured.
In AM, p. 43, Dudeney says he acquired the copy of The Fashionable Chinese
Puzzle which had belonged to Lewis Carroll. He says it was "Published by J. and E.
Wallis, 42 Skinner Street, and J. Wallis, Jun., Marine Library, Sidmouth" and quotes
the Napoleon footnote, so this copy had the Stanzas included. This copy is not in the
Strens Collection at Calgary which has some of Dudeney's papers.
Van der Waals cites two other works titled The Fashionable Chinese Puzzle. An
1818 edition from A. T. Goodridge [sic], NY, is in the American Antiquarian Society
Library (see below) and another, with no details given, is in the New York Public
Library. Could the latter be the Carroll/Dudeney copy?
Toole Stott 823 is a copy with the same title and imprint as the Carroll/Dudeney
copy, but he dates it c1840. This version is in two parts. Part I has 1 leaf text + 26 col.
plates -- it seems clear that col. means coloured, a feature that is not mentioned in any
other description of this book -- perhaps these were hand-coloured by an owner.
Unfortunately, he doesn't give the number of puzzles. I wonder if the last two plates are
missing from this?? Part II has 1 leaf text + 32 col. plates, giving 252 additional
figures. The only copy cited was in the library of J. B. Findlay -- I have recently bought
a copy of the Findlay sale catalogue, ??NYR.
Toole Stott 1309 is listed with the title: Stanzas, .... J. & F. [sic] Wallis ... and
Marine Library, Sidmouth, nd [c1815]. This has 1 leaf text and 28 plates of puzzles, so
it appears that the Stanzas have been bound in and the original cover title slip is lost or
was not recognised by Toole Stott. The date of c1815 is clearly derived from the
Napoleon footnote but 1817 would have been more reasonable, though this may be a
later reissue. Again only one copy is cited, in the library of Leslie Robert Cole.
Plates 1-28 are identical to plates 1-28 of The Admired Chinese Puzzle, but in
different order. The presence of the Chinese text in The Admired Chinese Puzzle made
me think the Wallis version was later than it.
Comparison of the Bodleian booklet with the first 27 plates of Giuoco Cinese,
1818?, reveals strong similarities. 5 plates are essentially identical, 17 plates are
identical except for one, two or three changes and 3 plates are about 50% identical. I
find that 264 of the 322 figures in the Wallis booklet occur in Giuoco Cinese, which is
about 82%. However, even when the plates are essentially identical, there are often
small changes in the drawings or the layout.
Some of the plates were copied by hand into the Hordern Collection's copy of A
New Invented Chinese Puzzle, c1806??.
The Admired Chinese Puzzle A New & Correct Edition From the Genuine Chinese Copy.
C. Taylor, Chester, nd [1817]. Paper is clearly watermarked 1812, but the Prologue
refers to the book being brought from China by someone in Lord Amherst's embassy to
China, which took place in 1815-1817 and which visited Napoleon on St. Helena on its
SOURCES - page 210
return. Slocum dates this to after 17 Aug 1817, when Amherst's mission returned to
England and this seems to be the second western book on tangrams. Not in
Christopher, Hall, Heyl or Toole Stott -- Slocum says there is only one copy known in
England! It originally had a cover with an illustration of two Chinese, titled The
Chinese Puzzle, and one of the men holds a scroll saying To amuse and instruct. The
Chinese text gives the title Ch'i ch'iao t'u ho pi (Harmoniously combined book of
tangram problems). I have a photocopy of the cover from Slocum. Prologue facing TP;
TP; two pp in Chinese, printed upside down, showing the pieces; 32pp of plates
numbered at the upper left (sometimes with reversed numbers), with problems labelled
in Chinese, but most of the characters are upside down! The plates are printed with two
facing plates alternating with two facing blank pages. Plate 1 has 12 problems, with
solution lines lightly indicated. Plates 2 - 28 contain 310 problems. Plates 29-32
contain 18 additional "caricature Designs" probably intended to be artistic versions of
some of the abstract tangram figures. The Prologue shows faint guide lines for the
lettering, but these appear to be printed, so perhaps it was a quickly done copperplate.
The text of the Prologue is as follows.
This ingenious geometrical Puzzle was introduced into this Kingdom
from China.
The following sheets are a correct Copy from the Chinese Publication, brought to
England by a Gentleman of high Rank in the suit [sic] of Lord Amherst's late Embassy.
To which are added caricature Designs as an illustration, every figure being
emblematical of some Being or Article known to the Chinese.
The plates are identical to the plates in The Fashionable Chinese Puzzle above, but in
different order and plate 4 is inverted and this version is clearly upside down.
Sy Hall. A New Chinese Puzzle, The Above Consists of Seven Pieces of Ivory or Wood, viz.
5 Triangles, 1 Rhomboid, and 1 Square, which will form the 292 Characters, contained
in this Book; Observing the Seven pieces must be used to form each Character. NB.
This Edition has been corrected in all its angles, with great care and attention.
Engraved by Sy Hall, 14 Bury Street, Bloomsbury. 31 plates with 292 problems.
Slocum, the Hordern Collection and BL have copies. I have a photocopy from a version
from Slocum which has no date but is watermarked 1815. Slocum's recent book [The
Tangram Book, pp. 74-75] shows a version of the book with the publisher's name as
James Izzard and a date of 1817. Sy probably is an abbreviation of Sydney (or possibly
Stanley?).
(The BL copy is watermarked IVY MILL 1815 and is bound with a large folding
Plate 2 by Hall, which has 83 tinted examples with solution lines drawn in (by hand??),
possibly one of four sheets giving all the problems in the book. However there is no
relationship between the Plate and the book -- problems are randomly placed and often
drawn in different orientation. I have a photocopy of the plate on two A3 sheets and a
copy of a different plate with 72 problems, watermarked J. Green 1816.)
A New Chinese Puzzle. Third Edition: Universally allowed to be the most correct that has
been published. 1817. Dalgety has a copy.
A New Chinese Puzzle Consisting of Seven Pieces of Ivory or Wood, The Whole of which
must be used, and will form each of the CHARACTERS. J. Buckland, 23 Brook Street,
Holborn, London. Paper watermarked 1816. (Dalgety has a copy, ??NYS.)
Miss D. Lowry. A Key to the Only Correct Chinese Puzzle Which has Yet Been Published,
with above a Hundred New Figures. No. 1. Drawn and engraved by Miss Lowry.
Printed by J. Barfield, London, 1817. The initial D. is given on the next page. Edward
Hordern's collection has a copy.
W. Williams. New Mathematical Demonstrations of Euclid, rendered clear and familiar to the
minds of youth, with no other mathematical instruments than the triangular pieces
commonly called the Chinese Puzzle. Invented by Mr. W. Williams, High Beech
Collegiate School, Essex. Published by the author, London, 1817. [Seen at BL.]
Enigmes Chinoises. Grossin, Paris, 1817. ??NYS -- described and partly reproduced in
Milano. Frontispiece facing the TP shows an oriental holding a banner which has the
pieces and a few problems on it. This is a small book, with five or six figures per page.
The figures seem to be copied from the Fashionable Chinese Puzzle, but some figures
are not in that work. Milano says this is cited as the first French usage of the term
'tangram', but this does not appear in Milano's photos and it is generally considered that
Loyd introduced the word in the 1850s. Milano's phrasing might be interpreted as
SOURCES - page 211
saying this is the first French work on tangrams.
Chinesische-Raethsel. Produced by Daniel Sprenger with designs by Matthaeus Loder,
Vienna, c1818. ??NYS -- mentioned by Milano.
Chinesisches Rätsel. Enigmes chinoises. Heinrich Friedrich Muller (or Mueller), Vienna,
c1810??. ??NYS (van der Waals). This is probably a German edition of the above and
should be dated 1817 or 1818. However, Milano mentions a box in the Historisches
Museum der Stadt Wien, labelled Grosse Chinesische Raethsel, produced by Mueller
and dated 1815-1820.
Passe-temps Mathématique, ou Récréation à l'ile Sainte-Hélène. Ce jeu qui occupé à qu'on
prétend, les loisirs du fameux exilé à St.-Hélène. Briquet, Geneva, 1817. 21pp. [Copy
advertised by Interlibrum, Vaduz, in 1990.]
The New and Fashionable Chinese Puzzle. A. T. Goodrich & Co., New York, 1817. TP, 1p
of Stanzas (seems like there should be a second page??), 32pp with 346 problems.
Slocum has a copy.
[Key] to the Chinese Philosophical Amusements. A. T. Goodrich & Co., New York, 1817.
TP, 2pp of stanzas (the second page has the Napoleon footnote and a comment which
indicates it is identical to the material in the problem book), Index to the Key to the
Chinese Puzzle, 80pp of solutions as black shapes with white spacing. Slocum has a
copy.
NOTE. This is quite a different book than The Fashionable Chinese Puzzle
published in London by Wallis in 1817.
Slocum writes: "Although the Goodrich problem book has the same title as the
British book by Wallis and Goodrich has the "Stanzas" poem (except for the first 2
paragraphs which he deleted) the problem books have completely different layouts and
Goodrich's solution book largely copies Chinese books."
Il Nuovo e Dilettevole Giuoco Chinese. Bardi, Florence, 1817. ??NYS -- mentioned by
Milano.
Buonapartes Geliefkoosste Vermaack op St. Helena, op Chineesch Raadsel. 1er Rotterdam by
J. Harcke. Prijs 1 - 4 ??. 2e Druck te(?) Rotterdam. Ter Steendrukkery van F. Scheffers
& Co. Nanco Bordewijk has recently acquired this and Slocum has said it is a
translation of one of the English items in c1818. I have just a copy of the cover, and it
uses many fancy letters which I don't guarantee to have read correctly.
Recueil des plus jolis Jeux de Sociéte, dans lequel on trouve les gravures d'un grand nombre
d'énigmes chinoises, et l'explication de ce nouveau jeu. Chez Audot, Librairie, Paris,
1818. Pp. 158-162: Le jeu des énigmes chinoises. This is a short introduction, saying
that the English merchants in Japan have sent it back to their compatriots and it has
come from England to France. This is followed by 11 plates. The first three are
numbered. The first shows the pieces formed into a rectangle. The others have 99
problems, with 7 shown solved (all six of those on plate 2 and one (the square) on the
10th plate.)
Das grosse chinesische Rätselspiel für die elegante Welt. Magazin für Industrie (Leipzig)
(1818). ??NYS (van der Waals). Jerry Slocum informs me that 'Magazin' here denotes
a store, not a periodical, and that this is actually a game version with a packet of 50
cards of problems, occurring in several languages, from 1818. I have acquired a set of
the cards which lacks one card (no. 17), in a card box with labels in French and Dutch
pasted on. One side has: Nouvelles / ENIGMES / Chinoises / en Figures et en
Paysages with a dancing Chinaman below. The other side has: Chineesch /
Raadselspel, / voor / de Geleerde Waereld / in / 50 Beelaachlige / Figuren. with two
birds below. Both labels are printed in red, with the dancing Chinaman having some
black lines. The cards are 82 x 55 mm and are beautifully printed with coloured
pictures of architectonic, anthropomorphic and zoomorphic designs in appropriate
backgrounds. The first card has four shapes, three of which show the solution with
dotted lines. All other cards have just one problem shape. The reverses have a simple
design. Slocum says the only complete set he has seen is in the British Library. I have
scanned the cards and the labels.
Gioco cinese chiamato il rompicapo. Milan, 1818. ??NYS (van der Waals). Fratelli Bettali,
Milan, nd, of which Dalgety has a copy.
Al Gioco Cinese Chiamato Il Rompicapo Appendice di Figure Rappresentanti ... Preceduta da
un Discorso sul Rompicapo e sulla Cina intitolato Passatempo Preliminare scritto
dall'Autore Firenze All'Insegna dell'Ancora 1818. 64pp + covers. The cover or TP has
an almond shape with the seven shapes inside. Pp. 3-43 are text -- the Passatempo
SOURCES - page 212
Preliminare and an errata page. 12 plates. The first is headed Alfabeto in fancy Gothic.
Plates 1-3 give the alphabet (J and W are omitted). Plate 4 has the positive digits.
Plates 5-12 have facing pages giving the names of the figures (rather orientalized) and
contain 100 problems. Hence a total of 133 problems, no solutions. The Hordern
collection has a copy and I have a photocopy from it. This has some similarities to
Giuoco Cinese. Described and partly reproduced in Milano.
Al Gioco Cinese chiamato il Rompicapo Appendice. Pietro & Giuseppe Vallardi, Milan,
1818. Possibly another printing of the item above. ??NYS -- described in Milano, who
reproduces plates 1 & 2, which are identical to the above item, but with a simpler
heading. Milano says the plates are identical to those in the above item.
Nuove e Dilettevole Giuoco Chinese. Milano presso li Frat. Bettalli Cont. del Cappello N.
4031. Dalgety has a copy. It is described and two pages are reproduced in Milano from
an example in the Raccolta Bertarelli. Milano dates it as 1818. Cover illustration is the
same as The Fashionable Chinese Puzzle, with the text changed. But it is followed by
some more text: Questa ingegnosa invenzione è fondata sopra principi Geometrici, e
consiste in 7 pezzi cioè 5. triangoli, un quadrato ed un paralellogrammo i quali possono
essere combinati in modo da formare piu di 300 figure curiose. The second photo
shows a double page identical to pp. 3-4 of The Fashionable Chinese Puzzle, except that
the page number on p. 4 was omitted in printing and has been written in. (Quaritch's
catalogue 646 (1947) item 698 lists this as Nuovo e dilettevole Giuoco Chinese, from
Milan, [1820?])
Nuove e Dilettevole Giuoco Chinese. Bologna Stamperia in pietra di Bertinazzi e Compag.
??NYS -- described and partly reproduced in Milano from an example in the Raccolta
Bertarelli. Identical to the above item except that it is produced lithographically, the
text under the cover illustration has been redrawn, the page borders, the page numbers
and the figure numbers are a little different. Milano's note 5 says the dating of this is
very controversial. Apparently the publisher changed name in 1813, and one author
claims the book must be 1810. Milano opts for 1813? but feels this is not consistent
with the above item. From Slocum's work and the examples above, it seems clear it
must be 1818?
Supplemento al nuovo giuoco cinese. Fratelli Bettalli, Milan, 1818. ??NYS -- described in
Milano, who says it has six plates and the same letters and digits as Al Gioco Cinese
Chiamato Il Rompicapo Appendice.
Giuoco Cinese Ossia Raccolta di 364. Figure Geometrica [last letter is blurred] formate con
un Quadrato diviso in 7. pezzi, colli quali si ponno formare infinite Figure diversi, come
Vuomini[sic], Bestie, Ucelli[sic], Case, Cocchi, Barche, Urne, Vasi, ed altre suppelletili
domestiche: Aggiuntovi l'Alfabeto, e li Numeri Arabi, ed altre nuove Figure. Agapito
Franzetti alle Convertite, Rome, nd [but 1818 is written in by hand]. Copy at the
Warburg Institute, shelf mark FMH 4050. TP & 30 plates. It has alternate openings
blank, apparently to allow you to draw in your solutions, as an owner has done in a few
cases. The first plate shows the solutions with dotted lines, otherwise there are no
solutions. There is no other text than on the TP, except for a florid heading Alfabeto
on plate XXVIII. The diagrams have no numbers or names. The upper part of the TP is
a plate of three men, intended to be Orientals, in a tent? The one on the left is standing
and cutting a card marked with the pieces. The man on the right is sitting at a low table
and playing with the pieces. He is seated on a box labelled ROMPI CAPO. A third
man is seated behind the table and watching the other seated man. On the ground are a
ruler, dividers and right angle. The Warburg does not know who put the date 1818 in
the book, but the book has a purchase note showing it was bought in 1913. James
Dalgety has the only other copy known. Sotheby's told him that Franzetti was most
active about 1790, but Slocum finds Sotheby's is no longer very definite about this. I
thought it possible that a page was missing at the beginning which gave a different form
of the title, but Dalgety's copy is identical to this one. Mario Velucchi says it is not
listed in a catalogue of Italian books published in 1800-1900. The letters and numbers
are quite different to those shown in Elffers and the other early works that I have seen,
but there are great similarities to The New and Fashionable Chinese Puzzle, 1817
(check which??), and some similarities to Al Gioco Cinese above. I haven't counted the
figures to verify the 364. Mentioned in Milano, based on the copy I sent to Dario Uri.
Jeu du Casse Tete Russe. 1817? ??NYS -- described and partly reproduced in Milano from
an example in the Raccolta Bertarelli but which has only four cards. Here the figures
are given anthropomorphic or architectonic shapes. There are four cards on one
SOURCES - page 213
coloured sheet and each card has a circle of three figures at the top with three more
figures along the bottom. Each card has the name of the game at the top of the circle
and "les secrets des Chinois dévoliés" and "casse tête russe" inside and outside the
bottom of the circle. The figures are quite different than in the following item.
Nuovo Giuoco Russo. Milano presso li Frat. Bettalli Cont. del Cappello. [Frat. is an
abbreviation of Fratelli (Brothers) and Cont. is an abbreviation of Contrade (road).]
Box, without pieces, but with 16 cards of problems (one being examples) and
instruction sheet (or leaflet). ??NYS - described by Milano with reproductions of the
box cover and four of the cards. This example is in the Raccolta Bertarelli. Box shows
a Turkish(?) man handing a box to another. On the first card is given the title and
publisher in French: Le Casse-Tête Russe Milan, chez les Fr. Bettalli, Rue du Chapeau.
The instruction sheet says that the Giuoco Chinese has had such success in the
principal cities of Europe that a Parisian publisher has conceived another game called
the Casse Tête Russe and that the Brothers Bettalli have hurried to produce it. Each
card has four problems where the figures are greatly elaborated into architectonic forms,
very like those in Metamorfosi, below. Undated, but Milano first gives 1815-1820, and
feels this is closely related to Metamorfosi and similar items, so he concludes that it is
1818 or 1819, and this seems to be as correct as present knowledge permits. The
figures are quite different than in the French version above.
Metamorfosi del Giuoco detto l'Enimma Chinese. Firenze 1818 Presso Gius. Landi Libraio
sul Canto di Via de Servi. Frontispiece shows an angel drawing a pattern on a board
which has the seven pieces at the top. The board leans against a plinth with the solution
for making a square shown on it. Under the drawing is A. G. inv. Milano reproduces
this plate. One page of introduction, headed Idea della Metamorfosi Imaginata
dell'Enimma Chinese. 100 shapes, some solved, then with elegant architectonic
drawings in the same shapes, signed Gherardesce inv: et inc: Milano identifies the artist
as Alessandro Gherardesca (1779-1852), a Pisan architect. See S&B, pp. 24-25.
Grand Jeu du Casse Tête Français en X. Pieces. ??NYS -- described and partly reproduced in
Milano, who says it comes from Paris and dates it 1818? The figures are
anthropomorphic and are most similar to those in Jeu du Casse Tete Russe.
Grande Giuocho del Rompicapo Francese. Milano presso Pietro e Giuseppe Vallardi
Contrada di S. Margherita No 401(? my copy is small and faint). ??NYS -- described
and partly reproduced in Milano, who dates it as 1818-1820. Identical problems as in
the previous item, but the figures have been redrawn rather than copied exactly.
Ch'i Ch'iao pan. c1820. (Bibliothek Leiden 6891; Antiquariat Israel, Amsterdam.) ??NYS
(van der Waals).
Le Veritable Casse-tete, ou Enigmes chinoises. Canu Graveur, Paris, c1820. BL. ??NYS
(van der Waals).
L'unico vero Enimma Chinese Tradotto dall'originale, pubblicato a Londra, da J. Barfield.
Florence, [1820?]. [Listed in Quaritch's catalogue 646 (1947) item 699.)
A tangram appears in Pirnaisches Wochenblatt of 16 Dec 1820. ??NYS -- described in
Slocum, p. 60.
Ch'i Ch'iao ch'u pien ho-pi. After 1820. (Bibliothek Leiden 6891.) ??NYS (van der Waals).
476 examples.
Nouveau Casse-Tête Français. c1820 (according to van der Waals). Reproduced in van der
Waals, but it's not clear how the pages are assembled. Milano dates it a c1815 and
indicates it is 16 cards, but van der Waals looks like it may have been a booklet of 16
pp with TP, example page and end page. The 16 pp have 80 problems.
Jerry Slocum has sent 2 large pages with 58 figurative shapes which are clearly
the same pictures. The instructions are essentially the same, but are followed by rules
for a Jeu de Patience on the second page and there is a 6 x 6 table of words on the first
page headed "Morales trouvées dans les ruines de la célébres Ville de Persépolis ..."
which one has to assemble into moral proverbs. It looks like these are copies of folding
plates in some book of games.
Chinese Puzzle Georgina. A. & S. Josh Myers, & Co 144, Leadenhall Street, London.
Ganton Litho. 81 examples on 8 plates with elegant TP. Pages are one-sided sheets,
sewn in the middle, but some are upside down. Seen at BL (1578/4938).
Bestelmeier, 1823. Item 1278: Chinese Squares. It is not in the 1812 catalogue.
Slocum. Compendium. Shows the above Bestelmeier entry.
Anonymous. Ch'i ch'iao t'u ho pi (Harmoniously combined book of Tangram problems) and
Ch'i ch'iao t'u chieh (Tangram solutions). Two volumes of tangrams and solutions with
SOURCES - page 214
no title page, Chinese labels of the puzzles, in Chinese format (i.e. printed as long
sheets on thin paper, accordion folded and stitched with ribbon. Nd [c1820s??], stiff
card covers with flyleaves of a different paper, undoubtedly added later. 84 pages in
each volume, containing 334 problems and solutions. With ownership stamp of a
cartouche enclosing EWSHING, probably a Mr. E. W. Shing. Slocum says this is a
c1820s reprint of the earliest Chinese tangram book which appeared in 1813 & 1815.
This version omits the TP and opening text. I have a photocopy of the opening material
from Slocum. The original problem book had a preface by Sang-hsia K'o, which was
repeated in the solution book with the same date. Includes all the problems of Shichikou-zu Gappeki, qv.
New Series of Ch'i ch'iau puzzles. Printed by Lou Chen-wan, Ch'uen Liang, January 1826.
??NYS. (Copy at Dept. of Oriental Studies, Durham Univ., cited in R. C. Bell;
Tangram Teasers.)
Neues chinesisches Rätselspiel für Kinder, in 24 bildlichen und alphabetischen Darstellungen.
Friese, Pirna. Van der Waals, copying Santi, gives c1805, but Slocum, p. 60, reports
that it first appears in Pirnaisches Wochenblatt of 19 Dec 1829, though there is another
tangram in the issue of 16 Dec 1820. ??NYS.
Child. Girl's Own Book. 1833: 85; 1839: 72; 1842: 156. "Chinese Puzzles -- These consist
of pieces of wood in the form of squares, triangles, &c. The object is to arrange them
so as to form various mathematical figures."
Anon. Edo Chiekata (How to Learn It??) (In Japanese). Jan 1837, 19pp, 306 problems.
(Unclear if this uses the Tangram pieces.) Reprinted in the same booklet as Sei
Shōnagon, on pp. 37-55.
A Grand Eastern Puzzle. C. Davenport & Co., London. Nd. ??NYS (van der Waals).
(Dalgety has a copy and gives C. Davenporte (??SP) and Co., No. 20, Grafton Street,
East Euston Square. Chinese pages dated 1813 in European binding with label bearing
the above information.)
Augustus De Morgan. On the foundations of algebra, No. 1. Transactions of the Cambridge
Philosophical Society 7 (1842) 287-300. ??NX. On pp. 289, he says "the well-known
toy called the Chinese Puzzle, in which a prescribed number of forms are given, and a
large number of different arrangements, of which the outlines only are drawn, are to be
produced."
Crambrook. 1843. P. 4, no. 4: Chinese Puzzle. Chinese Books, thirteen numbers. Though
not illustrated, this seems likely to be the Tangrams -- ??
Boy's Own Book. 1843 (Paris): 439.
No. 19: The Chinese Puzzle. Instructions give five shapes and say to make one copy of
some and two copies of the others. As written, this has two medium sized
triangles instead of two large ones, though it is intended to be the tangrams. 11
problem shapes given, no answers. Most of the shapes occur in earlier tangram
collections, particularly in A New Invented Chinese Puzzle. "The puzzle may be
purchased, ..., at Mr. Wallis's, Skinner street, Snow hill, where numerous books,
containing figures for this ingenious toy may also be obtained." = Boy's
Treasury, 1844, pp. 426-427, no. 16. It is also reproduced, complete with the
error, but without the reference to Wallis, as: de Savigny, 1846, pp. 355-356, no.
14: Le casse-tête chinois; Magician's Own Book, 1857, prob. 49, pp. 289-290;
Landells, Boy's Own Toy-Maker, 1858, pp. 139-140; Book of 500 Puzzles,
1859, pp. 103-104; Boy's Own Conjuring Book, 1860, pp. 251-252; Wehman,
New Book of 200 Puzzles, 1908, pp. 34-35.
No. 20: The Circassian puzzle. "This is decidedly the most interesting puzzle ever
invented; it is on the same principle, but composed of many more pieces than the
Chinese puzzle, and may consequently be arranged in more intricate figures. ..."
No pieces or problems are shown. In the next problem, it says: "This and the
Circassian puzzle are published by Mr. Wallis, Skinner-street, Snow-hill."
= Boy's Treasury, 1844, p. 427, no. 17. = de Savigny, 1846, p. 356, no. 15: Le
problème circassien, but the next problem omits the reference to Wallis.
Although I haven't recorded a Circassian puzzle yet -- cf in 6.S.2 -- I have just seen that
the puzzle succeeding The Chinese Puzzle in Wehman, New Book of 200
Puzzles, 1908, pp. 35-36, is called The Puzzle of Fourteen which might be the
Circassian puzzle. Taking a convenient size, this has two equilateral triangles of
edge 1 and four each of the following: a 30o-60o-90o triangle with edges
2, 1, 3; a parallelogram with angles 60o and 120o with edges 1 and 2; a
SOURCES - page 215
trapezium with base angles 60o and 60o, with lower and upper base edges 2 and
1, height 3/4 and slant edges 1/2 and 3/2. All 14 pieces make a rectangle
23 by 4.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob. 584-11, pp. 288 & 405: Chinesisches Verwandlungsspiel. Make a square with
the tangram pieces. Shows just five of the pieces, but correctly states which two
to make two copies of.
Prob. 584-16, pp. 289 & 406. Make an isosceles right triangle with the tangram pieces.
Prob. 584-18/25, pp. 289-291 & 407: Hieroglyphenspiele. Form various figures from
various sets of pieces, mostly tangrams, but the given shapes have bits of writing
on them so the assembled figure gives a word. Only one of the shapes is as in
Boy's Own Book.
Prob. 588, pp. 298 & 410: Etliche Knackmandeln. Another tangram problem like the
preceding, not equal to any in Boy's Own Book.
Adams & Co., Boston. Advertisement in The Holiday Journal of Parlor Plays and Pastimes,
Fall 1868. Details?? -- photocopy sent by Slocum. P. 6: Chinese Puzzle. The
celebrated Puzzle with which a hundred or more symmetrical forms can be made, with
book showing the designs. Though not illustrated, this seems likely to be the Tangrams
-- ??
Mittenzwey. 1880. Prob. 243-252, pp. 45 & 95-96; 1895?: 272-281, pp. 49 & 97-98; 1917:
272-282, pp. 45 & 92-93. Make a funnel, kitchen knife, hammer, hat with brim being
horizontal or hanging down or turned up, church, saw, dovecote, hatchet, square, two
equal squares.
J. Murray (editor of the OED). Two letters to H. E. Dudeney (9 Jun 1910 & 1 Oct 1910).
The first inquires about the word 'tangram', following on Dudeney's mention of it in his
"World's best puzzles" (op. cit. in 2). The second says that 'tan' has no Chinese origin;
is apparently mid 19C, probably of American origin; and the word 'tangram' first
appears in Webster's Dictionary of 1864. Dudeney, AM, 1917, p. 44, excerpts these
letters.
F. T. Wang & C.-S. Hsiung. A theorem on the tangram. AMM 49 (1942) 596-599. They
determine the 20 convex regions which 16 isosceles right triangles can form and hence
the 13 ones which the Tangram pieces can form.
Mitsumasa Anno. Anno's Math Games. (Translation of: Hajimete deau sugaku no ehon;
Fufkuinkan Shoten, Tokyo, 1982.) Philomel Books, NY, 1987. Pp. 38-43 & 95-96
show a simplified 5-piece tangram-like puzzle which I have not seen before. The
pieces are: a square of side 1; three isosceles right triangles of side 1; a right
trapezium with bases 1 and 2, altitude 1 and slant side 2. The trapezium can be
viewed as putting together the square with a triangle. 19 problems are set, with
solutions at the back.
James Dalgety. Latest news on oldest puzzles. Lecture to Second Meeting on the History of
Recreational Mathematics, 1 Jun 1996. 10pp. In 1998, he extracted the two sections on
tangrams and added a list of tangram books in his collection as: The origins of
Tangram; © 1996/98; 10pp. (He lists about 30 books, eight up to 1850.) In 1993, he
was buying tangrams in Hong Kong and asked what they called it. He thought they said
'tangram' but a slower repetition came out 'ta hau ban' and they wrote down the
characters and said it translates as 'seven lucky tiles'. He has since found the characters
in 19C Chinese tangram books. It is quite possible that Sam Loyd (qv under Murray,
above) was told this name and wrote down 'tangram', perhaps adjusted a bit after
thinking up Tan as the inventor.
At the International Congress on Mathematical Education, Seville, 1996, the Mathematical
Association gave out The 3, 4, 5 Tangram, a cut card tangram, but in a 6 x 8
rectangular shape, so that the medium sized triangle was a 3-4-5 triangle. I modified
this in Nov 1999, by stretching along a diagonal to form a rhombus with angles double
the angles of a 3-4-5 triangle, so that four of the triangles are similar to 3-4-5 triangles.
Making the small triangles be actually 3-4-5, all edges are integral. I made up 35
problems with these pieces. I later saw that Hans Wiezorke has mentioned this
dissection in CFF, but with no problems. I distributed this as my present at G4G4,
2000.
6.S.1. LOCULUS OF ARCHIMEDES
SOURCES - page 216
See S&B 22. I recall there is some dispute as to whether the basic diagram should be a square
or a double square.
E. J. Dijksterhuis. Archimedes. Munksgaard, Copenhagen, 1956; reprinted by Princeton
Univ. Press, 1987. Pp. 408-412 is the best discussion of this topic and supplies most of
the classical references.
Archimedes. Letter to Eratosthenes, c-250?. Greek palimpsest, c975, on MS no. 355, from
the Cloister of Saint Sabba (= Mar Saba), Jerusalem, then at Metochion of the Holy
Sepulchre, Constantinopole. [This MS disappeared in the confusion in Asia Minor in
the 1920s but reappeared in 1998 when it was auctioned by Christie's in New York for
c2M$. Hopefully, modern technology will allow a facsimile and an improved
transcription in the near future.] Described by J. L. Heiberg (& H. G. Zeuthen); Eine
neue Schrift des Archimedes; Bibliotheca Math. (3) 7 (1906-1907) 321-322. Heiberg
describes the MS, but only mentions the loculus. The text is in Heiberg's edition of
Archimedes; Opera; 2nd ed., Teubner, Leipzig, 1913, vol. II, pp. 415-424, where it has
been restored using the Suter MSS below. Heath only mentions the problem in passing.
Heiberg quotes Marius Victorinus, Atilius Fortunatianus and cites Ausonius and
Ennodius.
H. Suter. Der Loculus Archimedius oder das Syntemachion des Archimedes. Zeitschr. für
Math. u. Phys. 44 (1899) Supplement = AGM 9 (1899) 491-499. This is a collation
from two 17C Arabic MSS which describe the construction of the loculus. They are
different than the above MS. The German is included in Archimedes Opera II, 2nd ed.,
1913, pp. 420-424.
Dijksterhuis discusses both of the above and says that they are insufficient to determine what
was intended. The Greek seems to indicate that Archimedes was studying the
mathematics of a known puzzle. The Arabic shows the construction by cutting a
square, but the rest of the text doesn't say much.
Lucretius. De Rerum Natura. c-70. ii, 778-783. Quoted and discussed in H. J. Rose;
Lucretius ii. 778-83; Classical Review (NS) 6 (1956) 6-7. Brief reference to assembling
pieces into a square or rectangle.
Decimus Magnus Ausonius. c370. Works. Edited & translated by H. G. Evelyn White.
Loeb Classical Library, ??date. Vol. I, Book XVII: Cento Nuptialis (A Nuptial Cento),
pp. 370-393 (particularly the Preface, pp. 374-375) and Appendix, pp. 395-397. Refers
to 14 little pieces of bone which form a monstrous elephant, a brutal boar, etc. The
Appendix gives the construction from the Arabic version, via Heiberg, and forms the
monstrous elephant.
Marius Victorinus. 4C. VI, p. 100 in the edition of Keil, ??NYS. Given in Archimedes
Opera II, 2nd ed., 1913, p. 417. Calls it 'loculus Archimedes' and says it had 14 pieces
which make a ship, sword, etc.
Ennodius. Carmina: De ostomachio eburneo. c500. In: Magni Felicis Ennodii Opera; ed. by
F. Vogel, p. 340. In: Monumenta Germaniae Historica, VII (1885) 249. ??NYS.
Refers to ivory pieces to be assembled.
Atilius Fortunatianus. 6C. ??NYS Given in Archimedes Opera II, p. 417. Same comment as
for Marius Victorinus.
E. Fourrey. Curiositiés Géométriques. (1st ed., Vuibert & Nony, Paris, 1907); 4th ed.,
Vuibert, Paris, 1938. Pp. 106-109. Cites Suter, Ausonius, Marius Victorinus, Atilius
Fortunatianus.
Collins. Book of Puzzles. 1927. The loculus of Archimedes, pp. 7-11. Pieces made from a
double square.
6.S.2. OTHER SETS OF PIECES
See Hoffmann & S&B, cited at the beginning of 6.S, for general surveys.
See Bailey in 6.AS.1 for an 1858 puzzle with 10 pieces and The Sociable and Book of
500 Puzzles, prob. 10, in 6.AS.1 for an 11 piece puzzle.
There are many versions of this idea available and some are occasionally given in JRM.
The Richter Anchor Stone puzzles and building blocks were inspired by Friedrich
Froebel (or Fröbel) (1782-1852), the educational innovator. He was the inventor of
Kindergartens, advocated children's play blocks and inspired both the Richter Anchor Stone
Puzzles and Milton Bradley. The stone material was invented by Otto Lilienthal (1848-1896)
SOURCES - page 217
(possibly with his brother Gustav) better known as an aviation pioneer -- they sold the patent
and their machines to F. Adolph Richter for 1000 marks. The material might better be
described as a kind of fine brick which could be precisely moulded. Richter improved the
stone and began production at Rudolstadt, Thüringen, in 1882; the plant closed in 1964.
Anchor was the company's trademark. He made at least 36 puzzles and perhaps a dozen sets
of building blocks which were popular with children, architects, engineers, etc. The
Deutsches Museum in Munich has a whole room devoted to various types of building blocks
and materials, including the Anchor blocks. The Speelgoed Museum 'Op Stelten' (Sint
Vincentiusstraat 86, NL-4902 (or 4901) GL Oosterhout, Noord-Brabant, The Netherlands;
tel: 0262 452 825; fax: 0262 452 413) has a room of Richter blocks and some puzzles. There
was an Anker Museum in the Netherlands (Stichting Ankerhaus (= Anker Museum);
Opaalstraat 2-4 (or Postf. 1061), NL-2400 BB Alphen aan den Rijn, The Netherlands;
tel: 01720-41188) which produced replacement parts for Anker stone puzzles. Modern
facsimiles of the building sets are being produced at Rudolstadt.
In 1996 I noticed the ceiling of the room to the south of the Salon of the Ambassadors in the
Alcazar of Seville. This 15C? ceiling was built by workmen influenced by the Moorish
tradition and has 112 square wooden panels in a wide variety of rectilineal patterns.
One panel has some diagonal lines and looks like it could be used as a 10 piece
tangram-like puzzle. Consider a 4 x 4 square. Draw both diagonal lines, then at two
adjacent corners, draw two lines making a unit square at these corners. At the other two
corners draw one of these two lines, namely the one perpendicular to their common
side. This gives six isosceles right triangles of edge 1; two pentagons with three right
angles and sides 1, 2, 1, 2, 2; two quadrilaterals with two right angles and sides
2, 1, 2, 22. Since geometric patterns and panelling are common features of Arabic
art, I wonder if there are any instances of such patterns being used for a tangram-like
puzzle?
Grand Jeu du Casse Tête Français en X. Pieces. ??NYS -- described and partly reproduced in
Milano, who says it comes from Paris and dates it 1818? The figures are
anthropomorphic and are most similar to those in Jeu du Casse Tete Russe.
Grande Giuocho del Rompicapo Francese. Milano presso Pietro e Giuseppe Vallardi
Contrada di S. Margherita No 401(? my copy is small and faint). ??NYS -- described
and partly reproduced in Milano, who dates it as 1818-1820. Identical problems as in
the previous item, but the figures have been redrawn rather than copied exactly.
Allizeau. Les Métamorphoses ou Amusemens Géometriques Dédiée aux Amateurs Par
Allizeau. A Paris chex Allizeau Quai Malaquais, No 15. ??NYS -- described and
partly reproduced in Milano. This uses 15 pieces and the problems tend to be
architectural forms, like towers.
Jackson. Rational Amusement. 1821. Geometrical Puzzles, nos. 20-27, pp. 27-29 & 88-89
& plate II, figs. 15-22. This is a set of 20 pieces of 8 shapes used to make a square, a
right triangle, three squares, etc.
Crambrook. 1843. P. 4, no. 1: Pythagorean Puzzle, with Book. Though not illustrated, this is
probably(??) the puzzle described in Hoffmann, below, which was a Richter Anchor
puzzle No. 12 of the same name and is still occasionally seen. See S&B 28.
Edward Hordern's collection has a Circassian Puzzle, c1870, with many pieces, but I didn't
record the shapes -- cf Boy's Own Book, 1843 (Paris), in section 6.S.
Mittenzwey. 1880.
Prob. 177-179, pp. 34 & 86; 1895?: 202-204, pp. 38-39 & 88; 1917: 202-204, pp. 35
& 84-85. Consider the ten piece version of dissecting 5 squares to one (6.AS.1).
Use the pieces to make:
a squat octagon, a house gable-end, a church (no solution), etc.;
two dissimilar rectangles;
three dissimilar parallelograms, two dissimilar trapezoids. Solution says
one can make many other shapes with these pieces, e.g. a trapezoid with parallel
sides in the proportion 9 : 11.
Prob. 181-184, pp. 34-35 & 87-88; 1895?: 206-209, pp. 39 & 89-90; 1917: 206-209,
pp. 36 & 85-86. Take six equilateral triangles of edge 2. Cut an equilateral
triangle of edge 1 from the corner of each of them, giving 12 pieces. Make a
hexagon in eight different ways [there are many more -- how many??] and three
tangram-like shapes.
Prob. 195-196, pp. 36 & 89; 1895?: 220-221, pp. 41 & 91; 1917: 220-221, pp. 37 &
SOURCES - page 218
87. Use four isosceles right triangles, say of leg 1, to make a square, a 1 x 4
rectangle and an isosceles right triangle.
Nicholas Mason. US Patent 232,140 - Geometrical Puzzle-Block. Applied: 13 May 1880;
patented 14 Sep 1880. 1p plus 2pp diagrams. Five squares, six units square, each cut
into four pieces in the same way. Start at the midpoint of a side and cut to an opposite
corner. (This is the same cut used to produce the ten piece 'Five Squares to One'
puzzle.) Cut again in the triangle just formed, from the same midpoint to a point one
unit from the right angle corner of the piece just made. This gives a right triangle of
sides 3, 1, 10 and a triangle of sides 5, 10, 345. Cut again from the same
midpoint across the trapezoidal piece made by the first cut, to a point five units from the
corner previously cut to. This gives a triangle of sides 5, 35, 210 and a right
trapezoid with sides 2,10, 1, 6, 3. This was produced as Hill's American Geometrical
Prize Puzzle in England ("Price, One Shilling.") in 1882. Harold Raizer produced a
facsimile version, with facsimile box label and instructions for IPP22. The instructions
have 20 problems to solve and the solutions have to be submitted by 1 May 1882.
Hoffmann. 1893. Chap. III, no. 3: The Pythagoras Puzzle, pp. 83-85 & 117-118
= Hoffmann-Hordern, pp. 69-72. This has 7 pieces and is quite like the Tangram -- see
comment under Crambrook. Photo on p. 71, with different version in Hordern
Collection, p. 50.
C. Dudley Langford. Note 1538: Tangrams and incommensurables. MG 25 (No. 266)
(Oct 1941) 233-235. Gives alternate dissections of the square and some hexagonal
dissections.
C. Dudley Langford. Note 2861: A curious dissection of the square. MG 43 (No. 345)
(Oct 1959) 198. There are 5 triangles whose angles are multiples of π/8 = 22½o. He
uses these to make a square.
See items at the end of 6.S.
6.T. NO THREE IN A LINE PROBLEM
See also section 6.AO.2.
Loyd. Problem 14: A crow puzzle. Tit-Bits 31 (16 Jan & 6 Feb 1897) 287 & 343.
= Cyclopedia, 1914, Crows in the corn, pp. 110 & 353. = MPSL1, prob. 114, pp. 113 &
163-164. 8 queens with no two attacking and no three in any line.
Dudeney. The Tribune (7 Nov 1906) 1. ??NX. = AM, prob. 317, pp. 94 & 222. Asks for a
solution with two men in the centre 2 x 2 square.
Loyd. Sam Loyd's Puzzle Magazine, January 1908. ??NYS. (Given in A. C. White; Sam
Loyd and His Chess Problems; 1913, op. cit. in 1; p. 100, where it is described as the
only solution with 2 pieces in the 4 central squares.)
Ahrens, MUS I 227, 1910, says he first had this in a letter from E. B. Escott dated 1 Apr
1909. (W. Moser, below, refers this to the 1st ed., 1900, but this must be due to his not
having seen it.)
C. H. Bullivant. Home Fun, 1910, op. cit. in 5.S. Part VI, Chap. IV: No. 2: Another draught
puzzle, pp. 515 & 520. The problem says "no three men shall be in a line, either
horizontally or perpendicularly". The solution says "no three are in a line in any
direction" and the diagram shows this is indeed true.
Loyd. Picket posts. Cyclopedia, 1914, pp. 105 & 352. = MPSL2, prob. 48, pp. 34 & 136.
2 pieces initially placed in the 4 central squares.
Blyth. Match-Stick Magic. 1921. Matchstick board game, p. 73. 6 x 6 version phrased as
putting "only two in any one line: horizontal, perpendicular, or diagonal." However, his
symmetric solution has three in a row on lines of slope 2.
King. Best 100. 1927. No. 69, pp. 28 & 55. Problem on the 6 x 6 board -- gives a
symmetric solution. Says "there are two coins on every row" including "diagonally
across it", but he has three in a row on lines of slope 2.
Loyd Jr. SLAHP. 1928. Checkers in rows, pp. 40 & 98. Different solution than in
Cyclopedia.
M. Adams. Puzzle Book. 1939. Prob. C.83: Stars in their courses, pp. 144 & 181. Same
solution as King, but he says "two stars in each vertical row, two in each horizontal row,
and two in each of the the two diagonals .... There must not be more than two stars in
the same straight line", but he has three in a row on lines of slope 2.
W. O. J. Moser & J. Pach. No-three-in-line problem. In: 100 Research Problems in Discrete
SOURCES - page 219
Geometry 1986; McGill Univ., 1986. Problem 23, pp. 23.1 -- 23.4. Survey with 25
references. Solutions are known on the n x n board for n  16 and for even n  26.
Solutions with the symmetries of the square are only known for n = 2, 4, 10.
6.U. TILING
6.U.1.PENROSE PIECES
R. Penrose. The role of aesthetics in pure and applied mathematical research. Bull. Inst.
Math. Appl. 10 (1974) 266-272.
M. Gardner. SA (Jan 1977). Extensively rewritten as Penrose Tiles, Chaps. 1 & 2.
R. Penrose. Pentaplexity. Eureka 39 (1978) 16-22. = Math. Intell. 2 (1979) 32-37.
D. Shechtman, I. Blech, D. Gratias & J. W. Cohn. Metallic phase with long-range
orientational order and no translational symmetry. Physical Rev. Letters 53:20
(12 Nov 1984) 1951-1953. Describes discovery of 'quasicrystals' having the symmetry
of a Penrose-like tiling with icosahedra.
David R. Nelson. Quasicrystals. SA 255:2 (Aug 1986) 32-41 & 112. Exposits the discovery
of quasicrystals. First form is now called 'Shechtmanite'.
Kimberly-Clark Corporation has taken out two patents on the use of the Penrose pattern for
quilted toilet paper as the non-repetition prevents the tissue from 'nesting' on the roll. In
Apr 1997, Penrose issued a writ against Kimberly Clark Ltd. asserting his copyright on
the pattern and demanding damages, etc.
John Kay. Top prof goes potty at loo roll 'rip-off'. The Sun (11 Apr 1997) 7.
Patrick McGowan. It could end in tears as maths boffin sues Kleenex over design. The
Evening Standard (11 Apr 1997) 5.
Kleenex art that ended in tears. The Independent (12 Apr 1997) 2.
For a knight on the tiles. Independent on Sunday (13 Apr 1997) 24. Says they exclusively
reported Penrose's discovery of the toilet paper on sale in Dec 1996.
D. Trull. Toilet paper plagiarism. Parascope, 1997 -- available on
www.noveltynet.org/content/paranormal/www.parascope.com/arti...
6.U.2.PACKING BRICKS IN BOXES
In two dimensions, it is not hard to show that a x b packs A x B if and only if
a divides either A or B; b divides either A or B; A and B are both linear combinations of
a and b. E.g. 2 x 3 bricks pack a 5 x 6 box.
See also 6.G.1.
Anon. Prob. 52. Hobbies 30 (No. 767) (25 Jun 1910) 268 & 283 & (No. 770) (16 Jul 1910)
328. Use at least one of each of 5 x 7, 5 x 10, 6 x 10 to make the smallest possible
square. Solution says to use 4, 4, 1, but doesn't show how. There are lots of ways to
make the assembly.
Manuel H. Greenblatt ( -1972, see JRM 6:1 (Winter 1973) 69). Mathematical
Entertainments. Crowell, NY, 1965. Construction of a cube, pp. 80-81. Can 1 x 2 x 4
fill 6 x 6 x 6? He asserts this was invented by R. Milburn of Tufts Univ.
N. G. de Bruijn. Filling boxes with bricks. AMM 76 (1969) 37-40. If a1 x ... x an fills
A1 x ... x An and b divides k of the ai, then b divides at least k of the Ai. He
previously presented the results, in Hungarian, as problems in Mat. Lapok 12,
pp. 110-112, prob. 109 and 13, pp. 314-317, prob. 119. ??NYS.
D. A. Klarner. Brick-packing puzzles. JRM 6 (1973) 112-117. General survey. In this he
mentions a result that I gave him -- that 2 x 3 x 7 fills a 8 x 11 x 21, but that the box
cannot be divided into two packable boxes. However, I gave him the case 1 x 3 x 4 in
5 x 5 x 12 which is the smallest example of this type. Tom Lensch makes fine
examples of these packing puzzles.
T. H. Foregger, proposer; Michael Mather, solver. Problem E2524 -- A brick packing
problem. AMM 82:3 (Mar 1975) 300 & 83:9 (Nov 1976) 741-742. Pack
41 1 x 2 x 4 bricks in a 7 x 7 x 7 box. One cannot get 42 such bricks into the box.
6.V. SILHOUETTE AND VIEWING PUZZLES
Viewing problems must be common among draughtsmen and engineers, but I haven't
SOURCES - page 220
seen many examples. I'd be pleased to see further examples.
2 silhouettes.
Circle & triangle -- van Etten, Ozanam, Guyot, Magician's Own Book (UK
version)
Circle & square -- van Etten
Circle & rhombus -- van Etten, Ozanam
Rectangle with inner rectangle & rectangle with notch -- Diagram Group.
3 silhouettes.
Circle, circle, circle -- Madachy
Circle, cross, square -- Shortz collection (c1884), Wyatt, Perelman
Circle, oval, rectangle -- van Etten, Ozanam, Guyot,
Magician's Own Book (UK version)
Circle, oval, square -- van Etten, Tradescant, Ozanam, Ozanam-Montucla,
Badcock, Jackson, Rational Recreations, Endless Amusement II, Young Man's
Book
Circle, rhombus, rectangle -- Ozanam, Alberti
Circle, square, triangle -- Catel, Bestelmeier, Jackson, Boy's Own Book,
Crambrook, Family Friend, Magician's Own Book, Book of 500 Puzzles,
Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Riecke, Elliott,
Mittenzwey, Tom Tit, Handy Book, Hoffmann, Williams, Wyatt, Perelman,
Madachy. But see Note below.
Square, tee, triangle -- Perelman
4 silhouettes.
Circle, square, triangle, rectangle with curved ends -- Williams
2 views.
Antilog, Ripley's, Diagram Group;
3 views.
Madachy, Ranucci,
For the classic Circle, Square, Triangle, version, the triangle cannot be not equilateral.
Consider a circle, rectangle, triangle version. If D is the diameter of the circle and H is the
height of the plug, then the rectangle has dimensions D x H and the triangle has base D and
side S, so S = (H2 + D2/4). Making the rectangle a square, i.e. H = D, makes S = D5/2,
while making the triangle equilateral, i.e. S = D, makes H = D3/2.
van Etten. 1624.
Prob. 22 (misnumbered 15 in 1626) (Prob. 20), pp. 19-20 & figs. opp. p. 16 (pp. 35-36):
2 silhouettes -- one circular, the other triangular, rhomboidal or square. (English
ed. omits last case.) The 1630 Examen says the author could have done better
and suggests: isosceles triangle, several scalene triangles, oval or circle, which
he says can be done with an elliptically cut cone and a scalene cone. I am not
sure I believe these. It seems that the authors are allowing the object to fill the
hole and to pass through the hole moving at an angle to the board rather than
perpendicularly as usually understood. In the English edition the Examination is
combined with that of the next problem.
Prob. 23 (21), pp. 20-21 & figs. opp. p. 16 (pp. 37-38): 3 silhouettes -- circle, oval and
square or rectangle. The 1630 Examen suggests: square, circle, several
parallelograms and several ellipses, which he says can be done with an elliptic
cylinder of height equal to the major diameter of the base. The English
Examination says "a solid colume ... cut Ecliptick-wise" -- ??
John II Tradescant (1608-1662). Musæum Tradescantianum: Or, A Collection of Rarities
Preserved at South-Lambeth neer London By John Tradescant. Nathaniel Brooke,
London, 1656. [Facsimile reprint, omitting the Garden List, Old Ashmolean Reprints I,
edited by R. T. Gunther, on the occasion of the opening of the Old Ashmolean Museum
as what has now become the Museum of the History of Science, Oxford. OUP, 1925.]
John I & II Tradescant were gardeners to nobility and then royalty and used their
connections to request naval captains to bring back new plants, curiosities and "Any
thing that Is strang". These were accumulated at his house and garden in south
Lambeth, becoming known as Tradescant's Ark, eventually being acquired by Elias
Ashmole and becoming the foundation of the Ashmolean Museum in Oxford. This
SOURCES - page 221
catalogue was prepared by Elias Ashmole and his friend Thomas Wharton, but they are
not named anywhere in the book. It was the world's first museum catalogue.
P. 37, last entry: "A Hollow cut in wood, that will fit a round, square and ovall
figure."
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. He says square, circle and triangle is in
a book in front of him dated 1674. I suspect this must be the 1674 English edition of
van Etten, but I don't find the problem in the English editions I have examined. Perhaps
Dudeney just meant that the idea was given in the 1674 book, though he is specifically
referring to the square, circle, triangle version.
Ozanam. 1725. Vol. II, prob. 58 & 59, pp. 455-458 & plate 25* (53 (note there is a second
plate with the same number)). Circle and triangle; circle and rhombus; circle, oval,
rectangle; circle, oval, square. Figures are very like van Etten. See Ozanam-Montucla,
1778.
Ozanam. 1725. Vol. IV. No text, but shown as an unnumbered figure on plate 15 (17).
3 silhouettes: circle, rhombus, rectangle.
Simpson. Algebra. 1745. Section XVIII, prob. XXIX, pp. 279-281. (1790: prob. XXXVII,
pp. 306-307. Computes the volume of an ungula obtained by cutting a cone with a
plane. Cf Riecke, 1867.
Alberti. 1747. No text, but shown as an unnumbered figure on plate XIIII, opp. p. 218 (112),
copied from Ozanam, 1725, vol IV. 3 silhouettes: circle, rhombus, rectangle.
Ozanam-Montucla. 1778. Faire passer un même corps par un trou quarré, rond & elliptique.
Prob. 46, 1778: 347-348; 1803: 345-346; 1814: 293. Prob. 45, 1840: 149-150. Circle,
ellipse, square.
Catel. Kunst-Cabinet. 1790. Die mathematischen Löcher, p. 16 & fig. 42 on plate II. Circle,
square, triangle.
E. C. Guyot. Nouvelles Récréations Physiques et Mathématiques. Op. cit. in 6.P.2. 1799.
Vol. 2, Quatrième récréation, p. 45 & figs. 1-4, plate 7, opp. p. 45. 2 silhouettes: circle
& triangle; 3 silhouettes: circle, oval, rectangle.
Bestelmeier.
1801. Item 536: Die 3 mathematischen Löcher. (See also the picture of Item 275, but
that text is for another item.) Square, triangle and circle.
1807. Item 1126: Tricks includes the square, triangle and circle.
Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. P. 14, no. 23: How to
make a Peg that will exactly fit three different kinds of Holes. "Let one of the holes be
circular, the other square, and the third an oval; ...." Solution is a cylinder whose height
equals its diameter.
Jackson. Rational Amusement. 1821. Geometrical Puzzles.
No. 16, pp. 26 & 86. Circle, square, triangle, with discussion of the dimensions: "a
wedge, except that its base must be a circle".
No. 29, pp. 30 & 89-90. Circle, oval, square.
Rational Recreations. 1824. Feat 19, p. 66. Circle, oval, square.
Endless Amusement II. 1826? P. 62: "To make a Peg that will exactly fit three different
kinds of Holes." Circle, oval, square. c= Badcock.
The Boy's Own Book. The triple accommodation. 1828: 419; 1828-2: 424; 1829 (US): 215;
1855: 570; 1868: 677. Circle, square and triangle.
Young Man's Book. 1839. Pp. 294-295. Circle, oval, square. Identical to Badcock.
Crambrook. 1843. P. 5, no. 16: The Mathematical Paradox -- the Circle, Triangle, and
Square. Check??
Family Friend 3 (1850) 60 & 91. Practical puzzle -- No. XII. Circle, square, triangle. This is
repeated as Puzzle 16 -- Cylinder puzzle in (1855) 339 with solution in (1856) 28.
Magician's Own Book. 1857. Prob. 21: The cylinder puzzle, pp. 273 & 296. Circle, square,
triangle. = Book of 500 Puzzles, 1859, prob. 21, pp. 87 & 110. = Boy's Own Conjuring
Book, 1860, prob. 20, pp. 235 & 260.
Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 42, pp. 403 & 442. Identical to
Magician's Own Book, with diagram inverted.
F. J. P. Riecke. Op. cit. in 4.A.1, vol. 1, 1867. Art. 33: Die Ungula, pp. 58-61. Take a
cylinder with equal height and diameter. A cut from the diameter of one base which
just touches the other base cuts off a piece called an ungula (Latin for claw). He
computes the volume as 4r3/3. He then makes the symmetric cut to produce the circle,
square, triangle shape, which thus has volume (2π - 8/3) r3. Says he has seen such a
shape and a board with the three holes as a child's toy. Cf Simpson, 1745.
SOURCES - page 222
Magician's Own Book (UK version). 1871. The round peg in the square hole: To pass a
cylinder through three different holes, yet to fill them entirely, pp. 49-50. Circle, oval,
rectangle; circle & (isosceles) triangle.
Alfred Elliott. Within-Doors. A Book of Games and Pastimes for the Drawing Room.
Nelson, 1872. [Toole Stott 251. Toole Stott 1030 is a 1873 ed.] No. 4: The cylinder
puzzle, pp. 27-28 & 30-31. Circle, square, triangle.
Mittenzwey. 1880. Prob. 257, pp. 46 & 97; 1895?: 286, pp. 50 & 99-100; 1917: 286, pp. 45
& 94-95. Circle, square, triangle.
Will Shortz has a puzzle trade card with the circle, cross, square problem, c1884.
Tom Tit, vol. 2. 1892. La cheville universelle, pp. 161-162. = K, no. 28: The universal plug,
pp. 72-73. = R&A, A versatile peg, p. 106. Circle, square, triangle.
Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. Pp. 238-242: Captain S's peg
puzzle. Circle, square, triangle.
Hoffmann. 1893. Chap. X, no. 20: One peg to fit three holes, pp. 344 & 381-382
= Hoffmann-Hordern, pp. 238-239, with photo. Circle, square, triangle. Photo on
p. 239 shows two examples: one simply a wood board and pieces; the other labelled
The Holes and Peg Puzzle, from Clark's Cabinet of Puzzles, 1880-1900, but this seems
to be just a card box with the holes.
Williams. Home Entertainments. 1914. The plug puzzle, pp. 103-104. Circle, square,
triangle and rectangle with curved ends. This is the only example of this four-fold form
that I have seen. Nice drawing of a board with the plug shown in each hole, except the
curve on the sloping faces is not always drawn down to the bottom.
E. M. Wyatt. Puzzles in Wood, 1928, op. cit. in 5.H.1.
The "cross" plug puzzle, p. 17. Square, circle and cross.
The "wedge" plug puzzle, p. 18. Square, circle and triangle.
Perelman. FMP. c1935? One plug for three holes; Further "plug" puzzles, pp. 339-340 &
346. 6 simple versions; 3 harder versions: square, triangle, circle; circle, square,
cross; triangle, square, tee. The three harder versions are also in FFF, 1957: probs. 6971, pp. 112 & 118-119; 1979: probs. 73-75, pp. 137 & 144 = MCBF: probs. 73-75,
pp. 134-135 & 142-143.
Anonymous [Antilog]. An elevation puzzle. Eureka 19 (Mar 1957) 11 & 19. Front and top
views are a square with a square inside it. What is the side view? Gives two solutions.
Anonymous. An elevation puzzle. Eureka 21 (Oct 1958) 7 & 29. Front is the lower half of a
circle. Plan (= top view) is a circle. What is the side view? Solution is a V shape, but
it ought to be the other way up! Nowadays, one can buy potato crisps (= potato chips)
in this shape.
Joseph S. Madachy. 3-D in 2-D. RMM 2 (Apr 1961) 51-53 & 3 (Jun 1961) 47. Discusses 3
view and 3 silhouette problems.
3 circular silhouettes, but not a sphere.
Square, circle, triangle.
Ernest R. Ranucci. Non-unique orthographic projections. RMM 14 (Jan-Feb 1964) 50.
3 views such that there are 10 different objects with these views.
Ripley's Puzzles and Games. 1966. Pp. 18-19, item 1. Same problem as Antilog, 1957.
Gives one solution.
Cedric A. B. Smith. Simple projections. MG 62 (No. 419) (Mar 1978) 19-25. This is about
how different projections affect one's recognition of what an object is. He starts with an
example with two views and the isometric projection which is very difficult to interpret.
He gives three other views, each of which is easily interpreted.
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley,
Middlesex, 1984. Problem 114, with Solution at the back of the book. Front view is a
rectangle with an interior rectangle. Side view is a rectangle with a rectangular notch
on front side. Solution is a short cylinder with a straight notch in it. This is a fairly
classic problem for engineers but I haven't seen it in print elsewhere.
Marek Penszko. Polish your wits -- 3: Loop the loop. Games 11:2 (Feb/Mar 1987) 28 & 58.
Draw lines on a glass cube to produce three given projections. Problem asks for all
three projections to be the same.
6.W. BURR PUZZLES
When assembled, a burr looks like three sticks crossing orthogonally, forming a 'star'
with six points at the vertices of an octahedron. Slocum says Wyatt [Puzzles in Wood, 1928,
SOURCES - page 223
op. cit. in 5.H.1] is the first to use the word 'burr'. Collins, Book of Puzzles, 1927, p. 135,
calls them "Cluster, Parisian or Gordian Knot Puzzles" and states: "it is believed that they
were first made in Paris, if, indeed, they were not invented there." Since about 1990, there has
been considerable development in new types of burr which use plates or boards rather than
sticks, or whose central volume is subdivided more (cf in 6.W.1).
See S&B, pp. 62-85.
See also 6.BJ.
6.W.1.
THREE PIECE BURR
Most of these have three pieces which are rectangular in cross-section (1 x 3 x 5) with
slots of the same size and some of the pieces have notches from the slot to the outside. When
one piece is pushed, it slides, revealing its notch. When placed properly, this allows a second
piece to slide off and out.
In the 1990s, a more elaborate type of three piece burr appeared. These have three
3 x 3 x 5 pieces which intersect in a central 3 x 3 x 3 region. Within this region, some of the
unit cubes are not present, which allows sliding of the pieces. Some versions of the puzzle
permit twisting of pieces though this usually requires a bit of rounding of edges and the actual
examples tend to break, so these are not as acceptable.
Crambrook. 1843. P. 5, no. 4: Puzzling Cross 3 pieces. This seems likely to be a three
piece burr, but perhaps is in 6.W.3 -- ?? It is followed by "Maltese Cross 6 pieces".
Edward Hordern's collection has examples in ivory from 1850-1900.
Hoffmann. 1893. Chap. III, no. 35: The cross-keys or three-piece puzzle, pp. 106 & 139
= Hoffmann-Hordern, pp. 104-105, with photo. One piece has an extra small notch
which does not appear in other versions where the dimensions are better chosen. I have
recently acquired an example which appears identical to the illustrations but does not
have the extra notch - this came from a Jaques puzzle box, c1900, and Dalgety has
several examples of such boxes with the solution, where the puzzle is named The Cross
Keys Puzzle (cf discussion at the beginning of Section 11). The photo on p. 105 is an
assembled version, with verbal instructions, by Jaques & Son, 1880-1895 (but Jaques
was producing them up to at least c1910). Hordern Collection, p. 67, shows Le Noeud
Mystérieux, 1880-1905, with a pictorial solution and this does not have the extra notch.
Benson. 1904. The cross keys puzzle, pp. 205-206.
Pearson. 1907. Part III, no. 56: The cross-keys, pp. 56 & 127-128.
Anon. A puzzle in wood. Hobbies 31 (No. 795) (7 Jan 1911) 345. Three piece burr with
small extra notch as in Hoffmann.
Anon. Woodwork Joints. Evans, London, (1918), 2nd ed., 1919. [I have also seen a 4th ed.,
1925, which is identical to the 2nd ed., except for advertising pages at the end.]
A mortising puzzle, pp. 197-199.
Collins. Book of Puzzles. 1927. Pp. 136-137: The cross-keys puzzle.
E. M. Wyatt. Three piece cross. Puzzles in Wood, 1928, op. cit. in 5.H.1, pp. 24-25.
Arthur Mee's Children's Encyclopedia 'Wonder Box'. The Children's Encyclopedia appeared
in 1908, but versions continued until the 1950s. This looks like 1930s?? 3-Piece
Mortise with thin pieces.
A. S. Filipiak. Burr puzzle. Mathematical Puzzles, 1942, op. cit. in 5.H.1, p. 101.
Dic Sonneveld seems to be the first to begin designing three piece burrs of the more elaborate
style, perhaps about 1985. Trevor Wood has made several examples for sale.
Bill Cutler. Email announcement to NOBNET on 27 Jan 1999. He has begun analysing the
newer style of three piece burr, excluding twist moves. His first stage has examined
cases where the centre cube of the central region is occupied and the piece this central
cube belongs to has no symmetry. He finds 202 x 109 assemblies (I'm not sure if this
is an exact figure) and there are 33 level-8 examples (i.e. where it takes 8 moves to
remove the first piece); 6674 level-7 examples; 73362 level-6 examples. He thinks
this is about 70% of the total and it is already about six times the number of cases
considered for the six piece burr (see 6.W.2).
Bill Cutler. Christmas letter of 4 Dec 1999. Says he has completed the above analysis and
found 25 x 1010 possibilities, which took 225 days on a workstation. The most
elaborate examples require 8 moves to get a piece out and there are 80 of these. He
used one for his IPP19 puzzle. He has a website with many of his results on burrs, etc.:
www.billcutlerpuzzles.com .
SOURCES - page 224
6.W.2.
SIX PIECE BURR = CHINESE CROSS
The usual form of these has six sticks, 2 x 2 x 6 (or 8), which have various notches in
them. In the 1990s, new forms were introduced, using plates or boards. One version makes
an open frame shape, something like a 3 x 3 x 3 chessboard. In the other, 1 x 4 x 6 boards
are paired side by side and the result looks like a classic six-piece burr with the end rectangle
divided lengthwise rather than crosswise. See also 6.W.7.
Jurgis Baltrušaitis. Anamorphoses ou magie artificielle des effets merveilleux. Olivier Perrin
Éditeur, Paris, 1969. On pp. 110-116 & 184 is a discussion of a 1698 engraving
"L'Académie des Sciences et des Beaux Arts" by Sébastien Leclerc (or Le Clerc). In the
right foreground is an object looking like a six piece burr. James Dalgety discusses this
in his Latest news on oldest puzzles; Lecture to Second Meeting on the History of
Recreational Mathematics, 1 Jun 1996. This image also exists in a large painted version
(950 x 480 mm) which is more precise and more legible in many details, so it is
supposed that the engraving was done in conjunction with the painting. Though it was
normal for a notable painting to be turned into an engraving, the opposite sometimes
happened and Leclerc was a famous engraver. The painter is unknown. The divisions
between the pairs of pieces of the 'burr' are pretty clear in the engraving, but two of
them are not visible in the painting. The 'burr' is also not quite correctly drawn, but all
in all, it seems pretty convincing. James Dalgety was the first to discover this picture
and he has a copy of the engraving, but has not been able to locate the painting, though
it was in the Bernard Monnier Collection exhibited at the Musée des Arts Decoratifs in
Paris in 1975/76.
Camille Frémontier. Sébastien Leclerc and the British Encyclopeaedists. Sphæra [Newsletter
of the Museum of the History of Science, Oxford] 6 (Aut 1997) 6-7. Discusses the
Leclerc engraving which was used as the frontispiece to several encyclopedias, the
earliest being Chambers Cyclopaedia of 1728.
Minguet. 1733. Pp. 103-105 (1755: 51-52; 1822: 122-124; 1864: 103-104). Pieces
diagrammed. One plain key piece.
Catel. Kunst-Cabinet. 1790. Die kleine Teufelsklaue, p. 10 & fig. 16 on plate I. Figure
shows it assembled and fails to draw one of the divisions between pieces. Description
says it is 6 pieces, 2 inches long, from plum wood and costs 3 groschen (worth about an
English penny of the time). (See also pp. 9-10, fig. 20 on plate I for Die grosse
Teufelsklaue -- the 'squirrelcage'.)
Bestelmeier. 1801. Item 147: Die kleine Teufelsklaue. (Note -- there is another item 147 on
the next plate.) Only shows it assembled. Brief text may be copying part of Catel. See
also the picture for item 1099 which looks like a six-piece burr included in a set of
puzzles. (See also Item 142: Die grosse Teufelsklaue.)
Edward Hordern's collection has examples, called The Oak of Old England, from c1840.
Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these
might be here or in 6.W.4 or 6.W.5 -- ??
Magician's Own Book. 1857. Prob. 1: The Chinese cross, pp. 266-267 & 291. One plain key
piece. Not the same as in Minguét.
Landells. Boy's Own Toy-Maker. 1858. Pp. 137-139. Identical to Magician's Own Book.
Book of 500 Puzzles. 1859. 1: The Chinese cross, pp. 80-81 & 105. Identical to Magician's
Own Book.
A. F. Bogesen (1792-1876). In the Danish Technical Museum, Helsingør (= Elsinore) are a
number of wooden puzzles made by him, including a 6 piece burr, a 12 piece burr, an
Imperial Scale? and a complex (trick??) joint.
Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 23: The Chinese Cross, pp. 399
& 439. Identical to Magician's Own Book, except one diagram in the solution omits
two labels.
Boy's Own Conjuring Book. 1860. Prob. 1: The Chinese cross, pp. 228 & 254. Identical to
Magician's Own Book.
Hoffmann. 1893. Chap. III, no. 36: The nut (or six-piece) puzzle, pp. 106 & 139-140
= Hoffmann-Hordern, pp. 104-106. Different pieces than in Minguét and Magician's
Own Book.
Dudeney. Prob. 473 -- Chinese cross. Weekly Dispatch (23 Nov & 7 Dec 1902), both p. 13.
"There is considerable variety in the manner of cutting out the pieces, and though the
SOURCES - page 225
puzzle has been given in some of the old books, I have purposely presented it in a form
that has not, I believe, been published."
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the "Chinese Cross," a puzzle of
undoubted Oriental origin that was formerly brought from China by travellers as a
curiosity, but for a long time has had a steady sale in this country."
Wehman. New Book of 200 Puzzles. 1908. The Chinese cross, pp. 40-41. = Magician's
Own Book.
Dudeney. The world's best puzzles. 1908. Op. cit. in 2. P. 779 shows a '"Chinese Cross"
which ... is of great antiquity.'
Oscar W. Brown. US Patent 1,225,760 -- Puzzle. Applied: 27 Jun 1916; patented:
15 May 1917. 3pp + 1p diagrams. Coffin says this is the earliest US patent, with
several others following soon after.
Anon. Woodwork Joints, 1918, op. cit. in 6.W.1. Eastern joint puzzle, pp. 196-197:
Two versions using different pieces. Six-piece joint puzzle, pp. 199-200. Another
version.
Western Puzzle Works, 1926 Catalogue. No. 86: 6 piece Wood Block. Several other possible
versions -- see 6.W.7.
E. M. Wyatt. Six-piece burr. Puzzles in Wood, 1928, op. cit. in 5.H.1, pp. 27-28. Describes
17 versions from 13 types of piece.
A. S. Filipiak. Mathematical Puzzles, 1942, op. cit. in 5.H.1, pp. 79-87. 73 versions from 38
types of piece.
William H. [Bill] Cutler. The six-piece burr. JRM 10 (1977-78) 241-250. Complete,
computer assisted, analysis, with help from T. H. O'Beirne and A. C. Cross. Pieces are
considered as 'notchable' if they can be made by a sequence of notches, which are
produced by two saw cuts and then chiselling out the space between them. Otherwise
viewed, notches are what could be produced by a wide cutter or router. There are 25
of these which can occur in solutions. (In 1994, he states that there are a total of 59
notchable pieces and diagrams all of them.) One can also have more general pieces
with 'right-angle notches' which would require four chisel cuts -- e.g. to cut a single
1 x 1 x 1 piece out of a 2 x 2 x 8 rod. Alternatively, one can glue cubes into notches.
There are 369 which can occur in solutions. (In 1994, he states that there are 837
pieces which produce 2225 different oriented pieces, and he lists them all.) He only
considers solid solutions -- i.e. ones where there are no internal holes. He finds and
lists the 314 'notchable' solutions. There are 119,979 general solutions.
C. Arthur Cross. The Chinese Cross. Pentangle, Over Wallop, Hants., UK, 1979. Brief
description of the solutions in the general case, as found by Cutler and Cross.
S&B, p. 83, describes holey burrs.
W. H. [Bill] Cutler. Christmas letter, 1987. Sketches results of his (and other's) search for
holey burrs with notchable pieces.
Bill Cutler. Holey 6-Piece Burr! Published by the author, Palatine, Illinois. (1986); with
addendum, 1988, 48pp. He is now permitting internal holes. Describes holey burrs
with notchable pieces, particularly those with multiple moves to release the first piece.
Bill Cutler. A Computer Analysis of All 6-Piece Burrs. Published by the author, ibid., 1994.
86pp. Sketches complete history of the project. (I have included a few details in the
description of his 1977/78 article, above.) In 1987, he computed all the notchable holey
solutions, using about 2 months of PC AT time, finding 13,354,991 assemblies giving
7.4 million solutions. Two of these were level 10 -- i.e. they require 10 moves to
remove the first piece (or pieces), but the highest level occurring for a unique solution
was 5. After that he started on the general holey burrs and estimated it would take 400
years of PC AT time -- running at 8 MHz. After some development, the actual time
used was about 62.5 PC AT years, but a lot of this was done on by Harry L. Nelson
during idle time on the Crays at Lawrence Livermore Laboratories, and faster PCs
became available, so the whole project only took about 2½ years, being completed in
Aug 1990 and finding 35,657,131,235 assemblies. He hasn't checked if all assemblies
come apart fully, but he estimates there are 5.75 billion solutions. He estimates the
project used 45 times the computing power used in the proof of the Four Color Theorem
and that the project would only take two weeks on the eight RS6000 workstations he
now supervises. Some 70,000 high-level solutions were specifically saved and can be
obtained on disc from him. The highest level found was 12 and the highest level for a
unique solution was 10. See 6.W.1 for a continuation of this work. He has a website
with many of his results on burrs, etc.: www.billcutlerpuzzles.com .
SOURCES - page 226
Bill Cutler & Frans de Vreugd. Information leaflet accompanying their separate IPP22
puzzles, 2002. In 2001, they did an analysis of six-board burrs, of the type where the
boards are paired side by side. There are 4096 possible such boards, but only 219
usable boards occur. They looked at all combinations of six of these and found
14,563,061,989 assemblies. Of these, the highest level found was 13.
6.W.3.
THREE PIECE BURR WITH IDENTICAL PIECES
See S&B, p. 66.
Crambrook. 1843. P. 5, no. 4: Puzzling Cross 3 pieces. This seems likely to be a three
piece burr, but perhaps is in 6.W.1 -- ?? It is followed by "Maltese Cross 6 pieces".
Wilhelm Segerblom. Trick wood joining. SA (1 Apr 1899) 196.
6.W.4.
DIAGONAL SIX PIECE BURR = TRICK STAR
This version often looks like a stellated rhombic dodecahedron. It has two basic forms,
one with a key piece; the other with all pieces identical, which assembles as two groups of
three.
See S&B, p. 78.
Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these
belong here or in 6.W.2 or 6.W.5 -- ??
The Youth's Companion. 1875. [Mail order catalogue.] Reproduced in: Joseph J. Schroeder,
Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books,
Northfield, Illinois, 1977?, p. 19. Star Puzzle. The picture does not show which form it
is. Slocum's Compendium also shows this.
Samuel P. Chandler. US Patent 393,816 -- Puzzle. Applied: 9 Mar 1888; patented: 23 Apr
1888. 1p + 1p diagrams. Coffin says this is the earliest version, but it is more complex
than usual, with 12 pieces, and has a key piece.
John S. Pinnell. US Patent 774,197 -- Puzzle. Applied: 9 Oct 1902; patented: 8 Nov 1904.
2pp + 2pp diagrams. Coffin notes that this extends the idea to 102 pieces!
William E. Hoy. US Patent 766,444 -- Puzzle-Ball. Applied: 16 Oct 1902; patented:
2 Aug 1904. 2pp + 2pp diagrams. Spherical version with a key piece.
George R. Ford. US Patent 779,121 -- Puzzle. Applied: 16 May 1904; patented: 3 Jan 1905.
1p + 1p diagrams. With square rods, all identical. He shows assembly by inserting a
last piece rather than joining two groups of three.
Anon. Simple wood puzzle. Hobbies 31 (No. 786) (5 Nov 1910) 127. With key piece.
E. M. Wyatt. Woodwork puzzles. Industrial Arts Magazine 12 (1923) 326-327. Version
with a key piece and square rods.
Collins. Book of Puzzles. 1927. The bonbon or nut puzzle, pp. 137-139.
Iffland Frères (Lausanne). Swiss Patent 245,402 -- Zusammensetzspiel. Received:
19 Nov 1945; granted: 15 Nov 1946; published: 1 Jul 1947. 2pp + 1p diagrams.
Stellated rhombic dodecahedral version with a key piece. (Coffin says this is the first to
use this shape, although Slocum has a version c1875.)
6.W.5.
SIX PIECE BURR WITH IDENTICAL PIECES
One form has six identical pieces and all move outward or inward together. Another
form with flat notched pieces has one piece with an extra notch or an extended notch which
allows it to fit in last, either by sliding or twisting, but this is not initially obvious. This form
is sometimes made with equal pieces so that it can only be assembled by force, perhaps after
steaming, and it then makes an unopenable money box. This might be considered under
11.M.
Edward Hordern's collection has a version with one piece a little smaller than the rest from
c1800.
Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these
belong here or in 6.W.2 or 6.W.4 -- ??
C. Baudenbecher catalogue, c1850s. Op. cit. in 6.W.7. This has an example of the six equal
flat pieces making an unopenable(?) money box.
SOURCES - page 227
F. Chasemore. Some mechanical puzzles. In: Hutchison; op. cit. in 5.A; 1891, chap. 70,
part 1, pp. 571-572. Item 5: The puzzle box, p. 572. Six U pieces make a uniformly
expanding cubical box.
Hoffmann. 1893. Chap. III, no.33: The bonbon nut puzzle, pp. 104 & 138
= Hoffmann-Hordern, pp. 102-103, with photo. One piece has an extra notch to
simplify the assembly. Photo on p. 103 shows an example, almost certainly by Jaques
& Son, 1860-1895.
Burnett Fallow. How to make a puzzle money-box. The Boy's Own Paper 15 (No. 755)
(1 Jul 1893) 638. Equal flat notched pieces forced together to make an unopenable box.
Burnett Fallow. How to make a puzzle picture-frame. The Boy's Own Paper 16 (No. 815)
(25 Aug 1894) 749. Each corner has the same basic forced construction as used in the
puzzle money-box.
Benson. 1904. The bonbon nut puzzle, p. 204.
Bartl. c1920. Several versions on p. 306.
Western Puzzle Works, 1926 Catalogue. Last page shows 20 Chinese Wood Block Puzzles,
High Grade. Some of these are of the present type.
Collins. Book of Puzzles. 1927. The bonbon or nut puzzle, pp. 137-139. As in Hoffmann.
Iona & Robert Opie and Brian Alderson. Treasures of Childhood. Pavilion (Michael Joseph),
London, 1989. P. 158 shows a "cluster puzzle which Professor Hoffman [sic] names
the 'Nut (or Six-piece) Puzzle', but which is usually called 'The Maltese Puzzle'."
6.W.6.
ALTEKRUSE PUZZLE
William Altekruse. US Patent 430,502 -- Block-Puzzle. Applied: 3 Apr 1890; patented:
17 Jun 1890. 1p + 1p diagrams. Described in S&B, p. 72. The standard version has 12
pieces, but variations discovered by Coffin have 14, 36 & 38 pieces.
Western Puzzle Works, 1926 Catalogue. No. 112: 12 piece Wood Block. Possibly Altekruse.
6.W.7. OTHER BURRS
See also 6.BJ for other 3D dissections. I have avoided repeating items, so 6.BJ should
also be consulted if you are reading this section.
Catel. Kunst-Cabinet. 1790. Die grosse Teufelsklaue, pp. 9-10 & fig. 20 on plate I. 24 piece
'squirrel cage'. Cost 16 groschen.
Bestelmeier. 1801. Item 142: Die grosse Teufelsklaue. The 'squirrelcage', identical to Catel,
with same drawing, but reversed. Text may be copying some of Catel.
C. Baudenbecher, toy manufacturer in Nuremberg. Sample book or catalogue from c1850s.
Baudenbecher was taken over by J. W. Spear & Sons in 1919 and the catalogue is now
in the Spear's Game Archive, Ware, Hertfordshire. It comprises folio and double folio
sheets with finely painted illustrations of the firm's products. One whole folio page
shows about 20 types of wooden interlocking puzzles, including most of the types
mentioned elsewhere in this section and in 6.W.5 and 6.BJ. Until I get a picture, I can't
be more specific.
The Youth's Companion. 1875. [Mail order catalogue.] Reproduced in: Joseph J. Schroeder,
Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books,
Northfield, Illinois, 1977?, p. 19. Shows a 'woodchuck' type puzzle, called White
Wood Block Puzzle, from The Youth's Companion, 1875. I can't see how many pieces
it has: 12 or 18?? Slocum's Compendium also shows this.
Slocum. Compendium. Shows: "Mystery", Magic "Champion Puzzle" and "Puzzle of
Puzzles" from Bland's Catalogue, c1890.
The first looks like a 6 piece burr with circular segments added to make it look like a
ball. So it may be a 6 piece burr in disguise. See also Hoffmann, Chap. III,
no. 38, pp. 107-108 & 141-142 = Hoffmann-Hordern, pp. 106-108 = Benson,
p. 205.
The second is a six piece puzzle, but the pieces are flattish and it may be of the type
described in 6.W.5.
The third is complex, with perhaps 18 pieces.
Bartl. c1920. Several versions on pp. 306-307, including some that are in 6.W.5 and some
'Chinese block puzzles'.
Western Puzzle Works, 1926 Catalogue. Shows a number of burrs and similar puzzles.
SOURCES - page 228
No. 86: 6 piece Wood Block.
No. 112: 12 piece Wood Block. Possibly Altekruse.
No. 212: 11 piece Wood Block
The last page shows 20 Chinese Wood Block Puzzles, High Grade. Some of these are
burrs.
Collins. Book of Puzzles. 1927. Other cluster puzzles, pp. 139-142. Describes and
illustrates: The cluster; The cluster of clusters; The gun cluster; The point cluster;
The flat cluster; The cluster (or secret) table; The barrel; The Ball; The football. All
of these have a key piece.
Jan van de Craats. Das unmögliche Escher-puzzle. (Taken from: De onmogelijke Escherpuzzle; Pythagoras (Amsterdam) (1988).) Alpha 6 (or: Mathematik Lehren / Heft 55 -??) (1992) 12-13. Two Penrose tribars made into an impossible 5-piece burr.
6.X. ROTATING RINGS OF POLYHEDRA
Generally, these have edge to edge joints. 'Jacob's ladder' joints are used by Engel -- see
11.L for other forms of this joint.
I am told these may appear in Fedorov (??NYS).
Max Brückner. Vielecke und Vielfläche. Teubner, Leipzig, 1900. Section 162, pp. 215-216
and Tafel VIII, fig. 4. Describes rings of 2n tetrahedra joined edge to edge, called
stephanoids of the second order. The figure shows the case n = 5.
Paul Schatz. UK Patent 406,680 -- Improvements in or relating to Boxes or Containers.
Convention date (Germany): 10 Dec 1931; application date (in UK): 19 Jul 1932;
accepted: 19 Feb 1934. 6pp + 6pp diagrams. Six and four piece rings of prisms which
fold into a box.
Paul Schatz. UK Patent 411,125 -- Improvements in Linkwork comprising Jointed Rods or
the like. Convention Date (Germany): 31 Aug 1931; application Date (in UK):
31 Aug 1932; accepted: 31 May 1934. 3p + 6pp diagrams. Rotating rings of six
tetrahedra and linkwork versions of the same idea, similar to Flowerday's Hexyflex.
Ralph M. Stalker. US Patent 1,997,022 -- Advertising Medium or Toy. Applied: 27 Apr
1933; patented: 9 Apr 1935. 3pp + 2pp diagrams. "... a plurality of tetrahedron
members or bodies flexibly connected together." Shows six tetrahedra in a ring and an
unfolded pattern for such objects. Shows a linear form with 14 tetrahedra of decreasing
sizes.
Sidney Melmore. A single-sided doubly collapsible tessellation. MG 31 (No. 294) (1947)
106. Forms a Möbius strip of three triangles and three rhombi, which is basically a
flexagon (cf 6.D). He sees it has two distinct forms, but doesn't see the flexing
property!! He describes how to extend these hexagons into a tessellation which has
some resemblance to other items in this section.
Alexander M. Shemet. US Patent 2,688,820 -- Changeable Display Amusement Device.
Applied: 25 Jul 1950; patented: 14 Sep 1954. 2pp + 2pp diagrams. Basically a rotating
ring of six tetrahedra, but says 'at least six'. Gives an unfolded version or net for
making it and a mechanism for flexing it continually. Cites Stalker.
Wallace G. Walker invented his "IsoAxis" ® in 1958 while a student at Cranbrook Academy
of Art, Michigan. This is approximately a ring of ten tetrahedra. He obtained a US
Patent for it in 1967 -- see below. In 1973(?) he sent an example to Doris
Schattschneider who soon realised that the basic idea was a ring of tetrahedra and that
Escher tessellations could be adapted to it. They developed the idea into "M. C. Escher
Kaleidocycles", published by Ballantine in 1977 and reprinted several times since.
Douglas Engel. Flexahedrons. RMM 11 (Oct 1962) 3-5. These have 'Jacob's ladder' hinges,
not edge-to-edge hinges. He says he invented these in Fall, 1961. He formed rings of
4, 6, 7, 8 tetrahedra and used a diagonal joining to make rings of 4 and 6 cubes.
Wallace G. Walker. US Patent 3,302,321 -- Foldable Structure. Filed: 16 Aug 1963; issued:
7 Feb 1967. 2pp + 6pp diagrams.
Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Solid
Flexagons, pp. 81-84. Based on Engel, but only gives the ring of 6 tetrahedra.
D. Engel. Flexing rings of regular tetrahedra. Pentagon 26 (Spring 1967) 106-108. ??NYS -cited in Schaaf II 89 -- write Engel.
Paul Bethell. More Mathematical Puzzles. Encyclopædia Britannica International, London,
1967. The magic ring, pp. 12-13. Gives diagram for a ten-tetrahedra ring, all tetrahedra
SOURCES - page 229
being regular.
Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970.
??NYS. Presents versions of the flexing cubes and the 'Shinsei Mystery'. [Jan de Geus
has sent a photocopy of some of this but it does not cover this topic.]
Jan Slothouber. Flexicubes -- reversible cubic shapes. JRM 6 (1973) 39-46. As above.
Frederick George Flowerday. US Patent 3,916,559 -- Vortex Linkages. Filed: 12 Aug 1974
(23 Aug 1973 in UK); issued: 4 Nov 1975. Abstract + 2pp + 3pp diagrams. Mostly
shows his Hexyflex, essentially a six piece ring of tetrahedra, but with just four edges of
each tetrahedron present. He also shows his Octyflex which has eight pieces. Text
refers to any even number  6.
Naoki Yoshimoto. Two stars in a cube (= Shinsei Mystery). Described in Japanese in:
Itsuo Sakane; A Museum of Fun; Asahi Shimbun, Tokyo, 1977, pp. 208-210. Shown
and pictured as Exhibit V-1 with date 1972 in: The Expanding Visual World -- A
Museum of Fun; Exhibition Catalogue, Asahi Shimbun, Tokyo, 1979, pp. 102 &
170-171. (In Japanese). ??get translated??
Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 63-66.
Describes Walkers IsoAxis and rotating rings of six and eight tetrahedra.
6.Y. ROPE ROUND THE EARTH
The first few examples illustrate what must be the origin of the idea in more
straightforward situations.
Lucca 1754. c1330. F. 8r, pp. 31-32. This mentions the fact that a circumference increases
by 44/7 times the increase in the radius.
Muscarello. 1478.
Ff. 932-93v, p. 220. A circular garden has outer circumference 150 and the wall is
3½ thick. What is the inner circumference? Takes π as 22/7.
F. 95r, p. 222. The internal circumference of a tower is 20 and its wall is 3 thick.
What is the outer circumference? Again takes π as 22/7.
Pacioli. Summa. 1494. Part II, f. 55r, prob. 33. Florence is 5 miles around the inside. The
wall is 3½ braccia wide and the ditch is 14 braccia wide -- how far is it around the
outside? Several other similar problems.
William Whiston. Edition of Euclid, 1702. Book 3, Prop. 37, Schol. (3.). ??NYS -- cited by
"A Lover" and Jackson, below.
"A Lover of the Mathematics." A Mathematical Miscellany in Four Parts. 2nd ed., S. Fuller,
Dublin, 1735. The First Part is: An Essay towards the Probable Solution of the Forty
five Surprising PARADOXES, in GORDON's Geography, so the following must have
appeared in Gordon. Part I, no. 73, p. 56. "'Tis certainly Matter of Fact, that three
certain Travellers went a Journey, in which, Tho' their Heads travelled full twelve Yards
more than their Feet, yet they all return'd alive, with their Heads on."
Carlile. Collection. 1793. Prob. XXV, p. 17. Two men travel, one upright, the other
standing on his head. Who "sails farthest"? Basically he compares the distance
travelled by the head and the feet of the first man. He notes that this argument also
applies to a horse working a mill by walking in a circle; the outside of the horse travels
about six times the thickness of the horse further than the inside on each turn.
Jackson. Rational Amusement. 1821. Geographical Paradoxes, no. 54, pp. 46 & 115-116.
"It is a matter of fact, that three certain travellers went on a journey, in which their
heads travelled full twelve yards more than their feet; and yet, they all returned alive
with their heads on." Solution says this is discussed in Whiston's Euclid, Book 3, Prop.
37, Schol. (3.). [This first appeared in 1702.]
K. S. Viwanatha Sastri. Reminiscences of my esteemed tutor. In: P. K. Srinivasan, ed.;
Ramanujan Memorial Volumes: 1: Ramanujan -- Letters and Reminiscences;
2: Ramanujan -- An Inspiration; Muthialpet High School, Number Friends Society, Old
Boys' Committee, Madras, 1968. Vol. 1, pp. 89-93. On p. 93, he relates that this was a
favourite problem of his tutor, Srinivasan Ramanujan. Though not clearly dated, this
seems likely to be c1908-1910, but may have been up to 1914. "Suppose we prepare a
belt round the equator of the earth, the belt being 2π feet longer, and if we put the belt
round the earth, how high will it stand? The belt will stand 1 foot high, a substantial
height."
Dudeney. The paradox party. Strand Mag. 38 (No. 228) (Dec 1909) 673-674 (= AM, p. 139).
SOURCES - page 230
Anon. Prob. 58. Hobbies 30 (No. 773) (6 Aug 1910) 405 & (No. 776) (27 Aug 1910) 448.
Double track circular railway, five miles long. Move all rails outward one foot. How
much more material is needed? Solution notes the answer is independent of the length.
Ludwig Wittgenstein was fascinated by the problem and used to pose it to students. Most
students felt that adding a yard to the rope would raise it from the earth by a negligible
amount -- which it is, in relation to the size of the earth, but not in relation to the yard.
See: John Lenihan; Science in Focus; Blackie, 1975, p. 39.
Ernest K. Chapin. Loc. cit. in 5.D.1. 1927. Prob. 5, p. 87 & Answers p. 7. A yard is added
to a band around the earth. Can you raise it 5 inches? Answer notes the size of the
earth is immaterial.
Collins. Book of Puzzles. 1927. The globetrotter's puzzle, pp. 68-69. If you walk around the
equator, how much farther does your head go?
Abraham. 1933. Prob. 33 -- A ring round the earth, pp. 12 & 24 (9 & 112).
Perelman. FMP. c1935?? Along the equator, pp. 342 & 349. Same as Collins.
Sullivan. Unusual. 1943.
Prob. 20: A global readjustment. Take a wire around the earth and insert an extra 40 ft
into it -- how high up will it be?
Prob. 23: Getting ahead. If you walk around the earth, how much further does your
head go than your feet?
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Things are seldom what they seem -No. 42a, 43, 44, pp. 50-51. 42a and 43 ask how much the radius increases for a yard
gain of circumference. No. 44 asks if we add a yard to a rope around the earth and then
tauten it by pulling outward at one point, how far will that point be above the earth's
surface?
Richard I Hess. Puzzles from Around the World. The author, 1997. (This is a collection of
117 puzzles which he published in Logigram, the newsletter of Logicon, in 1984-1994,
drawn from many sources. With solutions.) Prob. 28. Consider a building 125 ft wide
and a rubber band stretched around the earth. If the rubber band has to stretch an extra
10 cm to fit over the building, how tall is the building? He takes the earth's radius as
20,902,851 ft. He gets three trigonometric equations and uses iteration to obtain
85.763515... ft.
Erwin Brecher & Mike Gerrard. Challenging Science Puzzles. Sterling, 1997. [Reprinted
by Goodwill Publishing House, New Delhi, India, nd [bought in early 2000]]. Pp. 3839 & 77. The M25 is a large ring road around London. A man commutes from the
south to the north and finds the distance is the same if goes by the east or the west, so he
normally goes to the east in the morning and to the west in the evening. Recalling that
the English drive on the left, he realised that his right wheels were on the outside in
both journeys and he worried that they were wear out sooner. So he changed and drove
both ways by the east. But he then worried whether the wear on the tires was the same
since the evening trip was on the outer lanes of the Motorway.
6.Z. LANGLEY'S ADVENTITIOUS ANGLES
Let ABC be an isosceles triangle with  B =  C = 80o. Draw BD and CE, making
angles 50o and 60o with the base. Then  CED = 20o.
JRM 15 (1982-83) 150 cites Math. Quest. Educ. Times 17 (1910) 75. ??NYS
Peterhouse and Sidney Entrance Scholarship Examination. Jan 1916. ??NYS.
E. M. Langley. Note 644: A Problem. MG 11 (No. 160) (Oct 1922) 173.
Thirteen solvers, including Langley. Solutions to Note 644. MG 11 (No. 164) (May 1923)
321-323.
Gerrit Bol. Beantwoording van prijsvraag No. 17. Nieuw Archief voor Wiskunde (2) 18
(1936) 14-66. ??NYS. Coxeter (CM 3 (1977) 40) and Rigby (below) describe this.
The prize question was to completely determine the concurrent diagonals of regular
polygons. The 18-gon is the key to Langley's problem. However Bol's work was not
geometrical.
Birtwistle. Math. Puzzles & Perplexities. 1971. Find the angle, pp. 86-87. Short solution
using law of sines and other simple trigonometric relations.
Colin Tripp. Adventitious angles. MG 59 (No. 408) (Jun 1975) 98-106. Studies when
 CED can be determined and all angles are an integral number of degrees. Computer
search indicates that there are at most 53 cases.
SOURCES - page 231
CM 3 (1977) 12 gives 1939 & 1950 reappearances of the problem and a 1974 variation.
D. A. Q. [Douglas A. Quadling]. The adventitious angles problem: a progress report. MG 61
(No. 415) (Mar 1977) 55-58. Reports on a number of contributions resolving the cases
which Tripp could not prove. All the work is complicated trigonometry -- no further
cases have been demonstrated geometrically.
CM 4 (1978) 52-53 gives more references.
D. A. Q. [Douglas A. Quadling]. Last words on adventitious angles. MG 62 (No. 421) (Oct
1978) 174-183. Reviews the history, reports on geometric proofs for all cases and
various generalizations.
J[ohn]. F. Rigby. Adventitious quadrangles: a geometrical approach. MG 62 (No. 421) (Oct
1978) 183-191. Gives geometrical proofs for almost all cases. Cites Bol and a long
paper of his own to appear in Geom. Dedicata (??NYS). He drops the condition that
ABC be isosceles. His adventitious quadrangles correspond to Bol's triple intersections
of diagonals of a regular n-gon.
MS 27:3 (1994/5) 65 has two straightforward letters on the problem, which was mentioned in
ibid. 27:1 (1994/5) 7. One letter cites 1938 and 1955 appearances. P. 66 gives another
solution of the problem. See next item.
Douglas Quadling. Letter: Langley's adventitious angles. MS 27:3 (1994/5) 65-66. He was
editor of MG when Tripp's article appeared. He gives some history of the problem and
some life of Langley (d. 1933). Edward Langley was a teacher at Bedford Modern
School and the founding editor of the MG in 1894-1895. E. T. Bell was a student of
Langley's and contributed an obituary in the MG (Oct 1933) saying that Langley was the
finest expositor he ever heard -- ??NYS. Langley also had botanical interests and a
blackberry variety is named for him.
6.AA.
NETS OF POLYHEDRA
Albrecht Dürer. Underweysung der messung mit dem zirckel uň [NOTE: ň denotes an n
with an overbar.] richtscheyt, in Linien ebnen unnd gantzen corporen. Nürnberg, 1525,
revised 1538. Facsimile of the 1525 edition by Verlag Dr. Alfons Uhl, Nördlingen,
1983. German facsimile with English translation of the 1525 edition, with notes about
the 1538 edition: The Painter's Manual; trans. by Walter L. Strauss; Abaris Books,
NY, 1977. Figures 29-43 (erroneously printed 34) (pp. 316-347 in The Painter's
Manual, Dürer's 1525 ff. M-iii-v - N-v-r) show nets and pictures of the regular
polyhedra, an approximate sphere (16 sectors by 8 zones), truncated tetrahedron,
truncated cube, cubo-octahedron, truncated octahedron, rhombi-cubo-octahedron, snub
cube, great rhombi-cubo-octahedron, truncated cubo-octahedron (having a pattern of
four triangles replacing each triangle of the cubo-octahedron -- not an Archimedean
solid) and an elongated hexagonal bipyramid (not even regular faced). (See 6.AT.3 for
more details.) (Panofsky's biography of Dürer asserts that Dürer invented the concept of
a net -- this is excerpted in The World of Mathematics I 618-619.) In the revised
version of 1538, figure 43 is replaced by the icosi-dodecahedron and great rhombicubo-octahedron (figures 43 & 43a, pp. 414-419 of The Painter's Manual) to make 9 of
the Archimedean polyhedra.
Albrecht Dürer. Elementorum Geometricorum (?) -- the copy of this that I saw at the Turner
Collection, Keele, has the title page missing, but Elementorum Geometricorum is the
heading of the first text page and appears to be the book's title. This is a Latin
translation of Unterweysung der Messung .... Christianus Wechelus, Paris, 1532. This
has the same figures as the 1525 edition, but also has page numbers. Liber quartus,
fig. 29-43, pp. 145-158 shows the same material as in the 1525 edition.
Cardan. De Rerum Varietate. 1557, ??NYS = Opera Omnia, vol. III, pp. 246-247. Liber
XIII. Corpora, qua regularia diei solent, quomodo in plano formentur. Shows nets of
the regular solids, except the two halves of the dodecahedron have been separated to fit
into one column of the text.
Barbaro, Daniele. La Practica della Perspectiva. Camillo & Rutilio Borgominieri, Venice,
(1569); facsimile by Arnaldo Forni, 1980, HB. [The facsimile's TP doesn't have the
publication details, but they are given in the colophon. Various catalogues say there are
several versions with dates on the TP and colophon varying independently between
1568 and 1569. A version has both dates being 1568, so this is presumed to be the first
appearance. Another version has an undated title in an elaborate border and this
facsimile must be from that version.] Pp. 45-104 give nets and drawings of the regular
SOURCES - page 232
polyhedra and 11 of the 13 Archimedean polyhedra -- he omits the two snub solids.
E. Welper. Elementa geometrica, in usum geometriae studiosorum ex variis Authoribus
collecta. J. Reppius, Strassburg, 1620. ??NYS -- cited, with an illustration of the nets
of the octahedron, icosahedron and dodecahedron, in Lange & Springer Katalog 163 -Mathematik & Informatik, Oct 1994, item 1350 & illustration on back cover, but the
entry gives Trassburg.
Athanasius Kircher. Ars Magna, Lucis et Umbrae. Rome, 1646. ??NX. Has net of a
rhombi-cuboctahedron.
Pike. Arithmetic. 1788. Pp. 458-459. "As the figures of some of these bodies would give
but a confused idea of them, I have omitted them; but the following figures, cut out in
pasteboard, and the lines cut half through, will fold up into the several bodies." Gives
the regular polyhedra.
Dudeney. MP. 1926. Prob. 146: The cardboard box, pp. 58 & 149 (= 536, prob. 316, pp. 109
& 310). All 11 nets of a cube.
Perelman. FMP. c1935? To develop a cube, pp. 179 & 182-183. Asserts there are 10 nets
and draws them, but two "can be turned upside down and this will add two more ...."
One shape is missing. Of the two marked as reversible, one is symmetric, hence equal
to its reverse, but the other isn't.
C. Hope. The nets of the regular star-faced and star-pointed polyhedra. MG 35 (1951) 8-11.
Rather technical.
H. Steinhaus. One Hundred Problems in Elementary Mathematics. (As: Sto Zadań, PWN -Polish Scientific Publishers, Warsaw, 1958.) Pergamon Press, 1963. With a Foreword
by M. Gardner; Basic Books, NY, 1964. Problem 34: Diagrams of the cube, pp. 20 &
95-96. (Gives all 11 nets.) Gardner (pp. 5-6) refers to Dudeney and suggests the four
dimensional version of the problem should be easy.
M. Gardner. SA (Nov 1966) c= Carnival, pp. 41-54. Discusses the nets of the cube and the
Answers show all 11 of them. He asks what shapes these 11 hexominoes will form -they cannot form any rectangles. He poses the four dimensional problem; the
Addendum says he got several answers, no two agreeing.
Charles J. Cooke. Nets of the regular polyhedra. MTg 40 (Aut 1967) 48-52. Erroneously
finds 13 nets of the octahedron.
Joyce E. Harris. Nets of the regular polyhedra. MTg 41 (Winter 1967) 29. Corrects Cooke's
number to 11.
A. Sanders & D. V. Smith. Nets of the octahedron and the cube. MTg 42 (Spring 1968)
60-63. Finds 11 nets for the octahedron and shows a duality with the cube.
Peter Turney. Unfolding the tesseract. JRM 17 (1984-85) 1-16. Finds 261 nets of the
4-cube. (I don't believe this has ever been confirmed.)
Peter Light & David Singmaster. The nets of the regular polyhedra. Presented at New York
Acad. Sci. Graph Theory Day X, 213 Nov 1985. In Notes from New York Graph
Theory Day X, 23 Nov 1985; ed. by J. W. Kennedy & L. V. Quintas; New York Acad.
Sci., 1986, p. 26. Based on Light's BSc project in 1984-1984 under my supervision.
Shows there are 43,380 nets for the dodecahedron and icosahedron. I may organize
this into a paper, but several others have since verified the result.
6.AB.
SELF-RISING POLYHEDRA
H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938. (= Kalejdoskop Matematyczny.
Książnica-Atlas, Lwów and Warsaw, 1938, ??NX.) Pp. 74-75 describes the
dodecahedron and says to see the model in the pocket at the end, but makes no special
observation of the self-rising property. Described in detail with photographs in OUP,
NY, eds: 1950: pp. 161-164; 1960: pp. 209-212; 1969 (1983): pp. 196-198.
Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, p. 29, section
66: Pop-up dodecahedron.
M. Kac. Hugo Steinhaus -- a reminiscence and a tribute. AMM 81 (1974) 572-581. Material
is on pp. 580-581, with picture on p. 581.
A pop-up octahedron was used by Waddington's as an advertising insert in a trade journal at
the London Toy Fair about 1981. Pop-up cubes have also been used.
6.AC.
CONWAY'S LIFE
There is now a web page devoted to Life run by Bob Wainwright -- address is:
SOURCES - page 233
http://members.aol.com/life1ine/life/lifepage.htm [sic!].
M. Gardner. Solitaire game of "Life". SA (Oct 1970). On cellular automata,
self-reproduction, the Garden-of-Eden and the game of "Life". SA (Feb 1971).
c= Wheels, chap. 20-22. In the Oct 1970 issue, Conway offered a $50 prize for a
configuration which became infinitely large -- Bill Gosper found the glider gun a month
later. At G4G2, 1996, Bob Wainwright showed a picture of Gosper's telegram to
Gardner on 4 Nov 1970 giving the coordinates of the glider gun. I wasn't clear if
Wainwright has this or Gardner still has it.
Robert T. Wainwright, ed. (12 Longue Vue Avenue, New Rochelle, NY, 10804, USA).
Lifeline (a newsletter on Life), 11 issues, Mar 1971 -- Sep 1973. ??NYR.
John Barry. The game of Life: is it just a game? Sunday Times (London) (13 Jun 1971).
??NYS -- cited by Gardner.
Anon. The game of Life. Time (21 Jan 1974). ??NYS -- cited by Gardner.
Carter Bays. The Game of Three-dimensional Life. Dept. of Computer Science, Univ. of
South Carolina, Columbia, South Carolina, 29208, USA, 1986. 48pp.
A. K. Dewdney. The game Life acquires some successors in three dimensions. SA 256:2
(Feb 1987) 8-13. Describes Bays' work.
Bays has started a quarterly 3-D Life Newsletter, but I have only seen one (or two?) issues.
??get??
Alan Parr. It's Life -- but not as we know it. MiS 21:3 (May 1992) 12-15. Life on a
hexagonal lattice.
6.AD.
ISOPERIMETRIC PROBLEMS
There is quite a bit of classical history which I have not yet entered. Magician's Own
Book notes there is a connection between the Dido version of the problem and Cutting a card
so one can pass through it, Section 6.BA. There are several relatively modern surveys of the
subject from a mathematical viewpoint -- I will cite a few of them.
Virgil. Aeneid. -19. Book 1, lines 360-370. (p. 38 of the Penguin edition, translated by
W. F. Jackson Knight, 1956.) Dido came to a spot in Tunisia and the local chiefs
promised her as much land as she could enclose in the hide of a bull. She cut it into a
long strip and used it to cut off a peninsula and founded Carthage. This story was later
adapted to other city foundations. John Timbs; Curiosities of History; With New
Lights; David Bogue, London, 1857, devotes a section to Artifice of the thong in
founding cities, pp. 49-50, relating that in 1100, Hengist, the first Saxon King of Kent,
similarly purchased a site called Castle of the Thong and gives references to Indian,
Persian and American versions of the story as well as several other English versions.
Pappus. c290. Synagoge [Collection]. Book V, Preface, para. 1-3, on the sagacity of bees.
Greek and English in SIHGM II 588-593. A different, abridged, English version is in
HGM II 389-390.
The Friday Night Book (A Jewish Miscellany). Soncino Press, London, 1933. Mathematical
Problems in the Talmud: Arithmetical Problems, no. 2, pp. 135-136. A Roman
Emperor demanded the Jews pay him a tax of as much wheat as would cover a space
40 x 40 cubits. Rabbi Huna suggested that they request to pay in two instalments of 20
x 20 and the Emperor granted this. [The Talmud was compiled in the period -300 to
500. This source says he is one of the few mathematicians mentioned in the Talmud,
but gives no dates and he is not mentioned in the EB. From the text, the problem would
seem to be sometime in the 1-5 C.]
The 5C Saxon mercenary, Hengist or Hengest, is said to have requested from Vortigern: "as
much land as can be encircled by a thong". He "then took the hide of a bull and cut it
into a single leather thong. With this thong he marked out a certain precipitous site,
which he had chosen with the greatest possible cunning." This is reported by Geoffrey
of Monmouth in the 12C and this is quoted by the editor in: The Exeter Book Riddles;
8-10C (the book was owned by Leofric, first Bishop of Exeter, who mentioned it in his
will of 1072); Translated and edited by Kevin Crossley-Holland; (As: The Exeter
Riddle Book, Folio Society, 1978, Penguin, 1979); Revised ed., Penguin, 1993; pp.
101-102.
Lucca 1754. c1330. Ff. 8r-8v, pp. 31-33. Several problems, e.g. a city 1 by 24 has
perimeter 50 while a city 8 by 8 has perimeter 32 but is 8/3 as large; stitching two
SOURCES - page 234
sacks together gives a sack 4 times as big.
Calandri. Arimethrica. 1491. F. 97v. Joining sacks which hold 9 and 16 yields a sack
which holds 49!!
Pacioli. Summa. 1494. Part II, ff. 55r-55v. Several problems, e.g. a cord of length 4
encloses 100 ducats worth, how much does a cord of length 10 enclose? Also stitching
bags together.
Buteo. Logistica. 1559. Prob. 86, pp. 298-299. If 9 pieces of wood are bundled up by 5½
feet of cord, how much cord is needed to bundle up 4 pieces? 5 pieces?
Pitiscus. Trigonometria. Revised ed., 1600, p. 223. ??NYS -- described in: Nobuo Miura;
The applications of trigonometry in Pitiscus: a preliminary essay; Historia Scientarum
30 (1986) 63-78. A square of side 4 and triangle of sides 5, 5, 3 have the same
perimeter but different areas. Presumably he was warning people not to be cheated in
this way.
J. Kepler. The Six-Cornered Snowflake, op. cit. in 6.AT.3. 1611. Pp. 6-11 (8-19). Discusses
hexagons and rhombic interfaces, but only says "the hexagon is the roomiest" (p. 11
(18-19)).
van Etten. 1624. Prob. 90 (87). Pp. 136-138 (214-218). Compares fields 6 x 6 and 9 x 3.
Compares 4 sacks of diameter 1 with 1 sack of diameter 4. Compares 2 water pipes of
diameter 1 with 1 water pipe of diameter 2.
Ozanam. 1725.
Question 1, 1725: 327. Question 3, 1778: 328; 1803: 325; 1814: 276; 1840: 141.
String twice as long contains four times as much asparagus.
Question 2, 1725: 328. If a cord of length 10 encloses 200, how much does a cord of
length 8 enclose?
Question 3, 1725: 328. Sack 5 high by 4 across versus 4 sacks 5 high by 1 across.
c= Q. 2, 1778: 328; 1803: 324; 1814: 276; 1840: 140-141, which has sack 4
high by 6 around versus two sacks 4 high by 3 around.
Question 4, 1725: 328-329. How much water does a pipe of twice the diameter
deliver?
Les Amusemens. 1749.
Prob. 211, p. 376. String twice as long contains four times as much asparagus.
Prob. 212, p. 377. Determine length of string which contains twice as much asparagus.
Prob. 223-226, pp. 386-389. Various problems involving changing shape with the
same perimeter. Notes the area can be infinitely small.
Ozanam-Montucla. 1778.
Question 1, 1778: 327; 1803: 323-324; 1814: 275-276; 1840: 140. Square versus
oblong field of the same circumference.
Prob. 35, 1778: 329-333; 1803: 326-330; 1814: 277-280; 1840: 141-143. Les
alvéoles des abeilles (On the form in which bees construct their combs).
Jackson. Rational Amusement. 1821. Geometrical Puzzles.
No. 30, pp. 30 & 90. Square field versus oblong (rectangular?) field of the same
perimeter.
No. 31, pp. 30 & 90-91. String twice as long contains four times as much asparagus.
Magician's Own Book (UK version). 1871. To cut a card for one to jump through, p. 124,
says: "The adventurer of old, who, inducing the aborigines to give him as much land as
a bull's hide would cover, and made it into one strip by which acres were enclosed, had
probably played at this game in his youth." See 6.BA.
M. Zacharias. Elementargeometrie und elementare nicht-Euklidische Geometrie in
synthetischer Behandlung. Encyklopädie der Mathematischen Wissenschaften.
Band III, Teil 1, 2te Hälfte. Teubner, Leipzig, 1914-1931. Abt. 28: Maxima und
Minima. Die isoperimetrische Aufgabe. Pp. 1118-1128. General survey, from
Zenodorus (-1C) and Pappus onward.
6.AD.1.
LARGEST PARCEL ONE CAN POST
New section. I have just added the problem of packing a fishing rod as the diagonal of a
box. Are there older examples?
Richard A. Proctor. Greatest content with parcels' post. Knowledge 3 (3 Aug 1883) 76.
Height + girth  6 ft. States that a cylinder is well known to be the best solution.
Either for a cylinder or a box, the optimum has height = 2, girth = 4, with optimum
SOURCES - page 235
volumes 2 and 8/π = 2.54... ft3.
R. F. Davis. Letter: Girth and the parcel post. Knowledge 3 (17 Aug 1883) 109-110, item
897. Independent discussion of the problem, noting that length  3½ ft is specified,
though this doesn't affect the maximum volume problem.
H. F. Letter: Parcel post problem. Knowledge 3 (24 Aug 1883) 126, item 905. Suppose
'length' means "the maximum distance in a straight line between any two points on its
surface". By this he means the diameter of the solid. Then the optimum shape is the
intersection of a right circular cylinder with a sphere, the axis of the cylinder passing
through the centre of the sphere, and this has the 'length' being the diameter of the
sphere and the maximum volume is then 2⅓ ft3.
Algernon Bray. Letter: Greatest content of a parcel which can be sent by post. Knowledge 3
(7 Sep 1883) 159, item 923. Says the problem is easily solved without calculus.
However, for the box, he says "it is plain that the bulk of half the parcel will be greatest
when [its] dimensions are equal".
Pearson. 1907. Part II, no. 20: Parcel post limitations, pp. 118 & 195. Length  3½ ft;
length + girth  6 ft. Solution is a cylinder.
M. Adams. Puzzle Book. 1939. Prob. B.86: Packing a parcel, pp. 79 & 107. Same as
Pearson, but first asks for the largest box, then the largest parcel.
Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965.
Prob. 18, pp. 27 & 89. Ship a rifle about 1½ yards long when the post office does not
permit any dimension to be more than 1 yard.
T. J. Fletcher. Doing without calculus. MG 55 (No. 391) (Feb 1971) 4-17. Example 5,
pp. 8-9. He says only that length + girth  6 ft. However, the optimal box has length
2, so the maximal length restriction is not critical.
I have looked at the current parcel post regulations and they say length  1.5m and
length + girth  3m, for which the largest box is 1 x ½ x ½, with volume 1/4 m3. The
largest cylinder has length 1 and radius 1/π with volume 1/π m3.
I have also considered the simple question of a person posting a fishing rod longer than the
maximal length by putting it diagonally in a box. The longest rod occurs at a boundary
maximum, at 3/2 x 3/4 x 0 or 3/2 x 0 x 3/4, so one can post a rod of length
35/4 = 1.677... m, which is about 12% longer than 1.5m. In this problem, the use
of a cylinder actually does worse!
6.AE.
6" HOLE THROUGH SPHERE LEAVES CONSTANT VOLUME
Hamnet Holditch. Geometrical theorem. Quarterly J. of Pure and Applied Math. 2 (1858)
??NYS, described by Broman. If a chord of a closed curve, of constant length a+b, be
divided into two parts of lengths a, b respectively, the difference between the areas of
the closed curve, and of the locus of the dividing point as the chord moves around the
curve, will be πab. When the closed curve is a circle and a = b, then this is the two
dimensional version given by Jones, below. A letter from Broman says he has found
Holditch's theorem cited in 1888, 1906, 1975 and 1976.
Richard Guy (letter of 27 Feb 1985) recalls this problem from his schooldays, which would be
late 1920s-early 1930s, and thought it should occur in calculus texts of that time, but
could not find it in Lamb or Caunt.
Samuel I. Jones. Mathematical Nuts. 1932. P. 86. ??NYS. Cited by Gardner, (SA,
Nov 1957) = 1st Book, chap. 12, prob. 7. Gardner says Jones, p. 93, also gives the two
dimensional version: If the longest line that can be drawn in an annulus is 6" long,
what is the area of the annulus?
L. Lines. Solid Geometry. Macmillan, London, 1935; Dover, 1965. P. 101, Example 8W3:
"A napkin ring is in the form of a sphere pierced by a cylindrical hole. Prove that its
volume is the same as that of a sphere with diameter equal to the length of the hole."
Solution is given, but there is no indication that it is new or recent.
L. A. Graham. Ingenious Mathematical Problems and Methods. Dover, 1959. Prob. 34: Hole
in a sphere, pp. 23 & 145-147. [The material in this book appeared in Graham's
company magazine from about 1940, but no dates are provided in the book. (??can date
be found out.)]
M. H. Greenblatt. Mathematical Entertainments, op. cit. in 6.U.2, 1965. Volume of a
modified bowling ball, pp. 104-105.
C. W. Trigg. Op. cit. in 5.Q. 1967. Quickie 217: Hole in sphere, pp. 59 & 178-179. Gives
an argument based on surface tension to see that the ring surface remains spherical as
SOURCES - page 236
the hole changes radius. Problem has a 10" hole.
Andrew Jarvis. Note 3235: A boring problem. MG 53 (No. 385) (Oct 1969) 298-299. He
calls it "a standard problem" and says it is usually solved with a triple integral (??!!).
He gives the standard proof using Cavalieri's principle.
Birtwistle. Math. Puzzles & Perplexities. 1971.
Tangential chord, pp. 71-73. 10" chord in an annulus. What is the area of the annulus?
Does traditionally and then by letting inner radius be zero.
The hole in the sphere, pp. 87-88 & 177-178. Bore a hole through a sphere so the
remaining piece has half the volume of the sphere. The radius of the hole is
approx. .61 of the radius of the sphere.
Another hole, pp. 89, 178 & 192. 6" hole cut out of sphere. What is the volume of the
remainder? Refers to the tangential chord problem.
Arne Broman. Holditch's theorem: An introductory problem. Lecture at ICM, Helsinki,
Aug 1978. Broman then sent out copies of his lecture notes and a supplementary letter
on 30 Aug 1978. He discusses Holditch's proof (see above) and more careful modern
versions of it. His letter gives some other citations.
6.AF.
WHAT COLOUR WAS THE BEAR?
A hunter goes 100 mi south, 100 mi east and 100 mi north and finds himself where he
started. He then shoots a bear -- what colour was the bear?
Square versions: Perelman; Klamkin, Breault & Schwarz; Kakinuma, Barwell &
Collins; Singmaster.
I include other polar problems here. See also 10.K for related geographical problems.
"A Lover of the Mathematics." A Mathematical Miscellany in Four Parts. 2nd ed., S. Fuller,
Dublin, 1735. The First Part is: An Essay towards the Probable Solution of the Forty
five Surprising PARADOXES, in GORDON's Geography, so the following must have
appeared in Gordon. Part I, no. 10, p. 9. "There is a particular Place of the Earth where
the Winds (tho' frequently veering round the Compas) do always blow from the North
Point."
Philip Breslaw (attrib.). Breslaw's Last Legacy; or the Magical Companion: containing all
that is Curious, Pleasing, Entertaining and Comical; selected From the most celebrated
Masters of Deception: As well with Slight of Hand, As with Mathematical Inventions.
Wherein is displayed The Mode and Manner of deceiving the Eye; as practised by those
celebrated Masters of Mirthful Deceptions. Including the various Exhibitions of those
wonderful Artists, Breslaw, Sieur, Comus, Jonas, &c. Also the Interpretation of
Dreams, Signification of Moles, Palmestry, &c. The whole forming A Book of real
Knowledge in the Art of Conjuration. (T. Moore, London, 1784, 120pp.) With an
accurate Description of the Method how to make The Air Balloon, and inject the
Inflammable Air. (2nd ed., T. Moore, London, 1784, 132pp; 5th ed., W. Lane, London,
1791, 132pp.) A New Edition, with great Additions and Improvements. (W. Lane,
London, 1795, 144pp.) Facsimile from the copy in the Byron Walker Collection, with
added Introduction, etc., Stevens Magic Emporium, Wichita, Kansas, 1997. [This was
first published in 1784, after Breslaw's death, so it is unlikely that he had anything to do
with the book. There were versions in 1784, 1791, 1792, 1793, 1794, 1795, 1800,
1806, c1809, c1810, 1811, 1824. Hall, BCB 39-43, 46-51. Toole Stott 120-131,
966-967. Heyl 35-41. This book went through many variations of subtitle and contents
-- the above is the largest version.]. I will cite the date as 1784?.
Geographical Paradoxes.
Paradox I, p. 35. Where is it noon every half hour? Answer: At the North Pole in
Summer, when the sun is due south all day long, so it is noon every moment!
Paradox II, p. 36. Where can the sun and the full moon rise at the same time in the
same direction? Answer: "Under the North Pole, the sun and the full moon, both
decreasing in south declination, may rise in the equinoxial points at the same
time; and under the North Pole, there is no other point of compass but south." I
think this means at the North Pole at the equinox.
Carlile. Collection. 1793. Prob. CXVI, p. 69. Where does the wind always blow from the
north?
Jackson. Rational Amusement. 1821. Geographical Paradoxes.
No. 7, pp. 36 & 103. Where do all winds blow from the north?
SOURCES - page 237
No. 8, pp. 36 & 110. Two places 100 miles apart, and the travelling directions are to
go 50 miles north and 50 miles south.
Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:3 (Jul 1903) 246-247. A safe catch.
Airship starts at the North Pole, goes south for seven days, then west for seven days.
Which way must it go to get back to its starting point? No solution given.
Pearson. 1907.
Part II, no. 21: By the compass, pp. 18 & 190. Start at North Pole and go 20 miles
southwest. What direction gets back to the Pole the quickest? Answer notes that
it is hard to go southwest from the Pole!
Part II, no. 15: Ask "Where's the north?" -- Pope, pp. 117 & 194. Start 1200 miles
from the North Pole and go 20 mph due north by the compass. How long will it
take to get to the Pole? Answer is that you never get there -- you get to the North
Magnetic Pole.
Ackermann. 1925. P. 116. Man at North Pole goes 20 miles south and 30 miles west.
How far, and in what direction, is he from the Pole?
Richard Guy (letter of 27 Feb 1985) recalls this problem (I think he is referring to the 'What
colour was the bear' version) from his schooldays in the 1920s.
H. Phillips. Week-End. 1932. Prob. 8, pp. 12 & 188. = his Playtime Omnibus, 1933,
prob. 10: Popoff, pp. 54 & 237. House with four sides facing south.
H. Phillips. The Playtime Omnibus. Faber & Faber, London, 1933. Section XVI, prob. 11:
Polar conundrum, pp. 51 & 234. Start at the North Pole, go 40 miles South, then 30
miles West. How far are you from the Pole. Answer: "Forty miles. (NOT thirty, as
one is tempted to suggest.)" Thirty appears to be a slip for fifty??
Perelman. FFF. 1934. 1957: prob. 6, pp. 14-15 & 19-20: A dirigible's flight; 1979: prob. 7,
pp. 18-19 & 25-27: A helicopter's flight. MCBF: prob. 7, pp. 18-19 & 25-26: A
helicopter's flight. Dirigible/helicopter starts at Leningrad and goes 500km N,
500km E, 500km S, 500km W. Where does it land? Cf Klamkin et seq., below.
Phillips. Brush. 1936. Prob. A.1: A stroll at the pole, pp. 1 & 73. Eskimo living at North
Pole goes 3 mi south and 4 mi east. How far is he from home?
Haldeman-Julius. 1937. No. 51: North Pole problem, pp. 8 & 23. Airplane starts at North
Pole, goes 30 miles south, then 40 miles west. How far is he from the Pole?
J. R. Evans. The Junior Week-End Book. Gollancz, London, 1939. Prob. 9, pp. 262 & 268.
House with four sides facing south.
Leopold. At Ease! 1943. A helluva question!, pp. 10 & 196. Hunter goes 10 mi south, 10
mi west, shoots a bear and drags it 10 mi back to his starting point. What colour was
the bear? Says the only geographic answer is the North Pole.
E. P. Northrop. Riddles in Mathematics. 1944. 1944: 5-6; 1945: 5-6; 1961: 15-16. He
starts with the house which faces south on all sides. Then he has a hunter that sees a
bear 100 yards east. The hunter runs 100 yards north and shoots south at the bear -what colour .... He then gives the three-sided walk version, but doesn't specify the
solution.
E. J. Moulton. A speed test question; a problem in geography. AMM 51 (1944) 216 & 220.
Discusses all solutions of the three-sided walk problem.
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 50: A fine outlook, pp. 54-55.
House facing south on all sides used by an artist painting bears!
Leeming. 1946. Chap. 3, prob. 32: What color was the bear?, pp. 33 & 160. Man walks 10
miles south, then 10 miles west, where he shoots a bear. He drags it 10 miles north
to his base. What color .... He gives only one solution.
Darwin A. Hindman. Handbook of Indoor Games & Contests. (Prentice-Hall, 1955);
Nicholas Kaye, London, 1957. Chap. 16, prob. 4: The bear hunter, pp. 256 & 261.
Hunter surprises bear. Hunter runs 200 yards north, bear runs 200 yards east, hunter
fires south at bear. What colour ....
Murray S. Klamkin, proposer; D. A. Breault & Benjamin L. Schwarz, solvers. Problem 369.
MM 32 (1958/59) 220 & 33 (1959/60) 110 & 226-228. Explorer goes 100 miles
north, then east, then south, then west, and is back at his starting point. Breault gives
only the obvious solution. Schwartz gives all solutions, but not explicitly. Cf
Perelman, 1934.
Benjamin L. Schwartz. What color was the bear?. MM 34 (1960) 1-4. ??NYS -- described
by Gardner, SA (May 1966) = Carnival, chap. 17. Considers the problem where the
hunter looks south and sees a bear 100 yards away. The bear goes 100 yards east and
the hunter shoots it by aiming due south. This gives two extra types of solution.
SOURCES - page 238
Ripley's Puzzles and Games. 1966. Pp. 18, item 5. 50 mi N, 1000 mi W, 10 mi S to return
to your starting point. Answer only gives the South Pole, ignoring the infinitely many
cases near the North Pole. Looking at this made me realise that when the sideways
distance is larger than the circumference of the parallel at that distance from the pole,
then there are other solutions that start near the pole. Here there are three solutions
where one starts at distances 109.2, 29.6 or 3.05 miles from the South Pole, circling it
1, 2 or 3 times.
Yasuo Kakinuma, proposer; Brian Barwell and Craig H. Collins, solvers. Problem 1212 -Variation of the polar bear problem. JRM 15:3 (1982-83) 222 & 16:3 (1983-84)
226-228. Square problem going one mile south, east, north, west. Barwell gets the
explicit quadratic equation, but then approximates its solutions. Collins assumes the
earth is flat near the pole.
David Singmaster. Bear hunting problems. Submitted to MM, 1986. Finds explicit solutions
for the general version of Perelman/Klamkin's problem. [In fact, I was ignorant of (or
had long forgotten) the above when I remembered and solved the problem. My thanks
to an editor (Paul Bateman ??check) for referring me to Klamkin. The Kakinuma et al
then turned up also.] Analysis of the solutions leads to some variations, including the
following.
David Singmaster. Home is the hunter. Man heads north, goes ten miles, has lunch, heads
north, goes ten miles and finds himself where he started.
Used as: Explorer's problem by Keith Devlin in his Micromaths Column; The
Guardian (18 Jun 1987) 16 & (2 Jul 1987) 16.
Used by me as one of: Spring term puzzles; South Bank Polytechnic Computer
Services Department Newsletter (Spring 1989) unpaged [p. 15].
Used by Will Shortz in his National Public Radio program 6? Jan 1991.
Used as: A walk on the wild side, Games 15:2 (No. 104) (Aug 1991) 57 & 40.
Used as: The hunting game, Focus 3 (Feb 1993) 77 & 98.
Used in my Puzzle Box column, G&P, No. 11 (Feb 1995) 19 & No. 12 (Mar
1995) 41.
Bob Stanton. The explorers. Games Magazine 17:1 (No. 113) (Feb 1993) 61 & 43. Two
explorers set out and go 500 miles in each direction. Madge goes N, W, S, E, while
Ellen goes E, S, W, N. At the end, they meet at the same point. However, this is not at
their starting point. How come? and how far are they from their starting point, and in
what direction? They are not near either pole.
Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995.
Chap. 11, prob. 9: What color was that bear? (A lesson in non-Euclidean geometry),
pp. 97 & 185-191. Camper walks south 2 km, then west 5 km, then north 2 km; how
far is he from his starting point? Solution analyses this and related problems, finding
that the distance x satisfies 0  x  7.183, noting that there are many minimal cases
near the south pole and if one is between them, one gets a local maximum, so one has to
determine one's position very carefully.
David Singmaster. Symmetry saves the solution. IN: Alfred S. Posamentier & Wolfgang
Schulz, eds.; The Art of Problem Solving: A Resource for the Mathematics Teacher;
Corwin Press, NY, 1996, pp. 273-286. Sketches the explicit solution to Klamkin's
problem as an example of the use of symmetric variables to obtain a solution.
Anonymous. Brainteaser B163 -- Shady matters. Quantum 6:3 (Jan/Feb 1996) 15 & 48. Is
there anywhere on earth where one's shadow has the same length all day long?
6.AG.
MOVING AROUND A CORNER
There are several versions of this. The simplest is moving a ladder or board around a
corner -- here the problem is two-dimensional and the ladder is thin enough to be considered
as a line. There are slight variations -- the corner can be at a T or + junction; the widths of
the corridors may differ; the angle may not be a right angle; etc. If the object being moved is
thicker -- e.g. a table -- then the problem gets harder. If one can use the third dimension, it
gets even harder.
H. E. Licks. Op. cit. in 5.A, 1917. Art. 110, p. 89. Stick going into a circular shaft in the
ceiling. Gets [h2/3 + d2/3)]3/2 for maximum length, where h is the height of the room
and d is the diameter of the shaft. "A simple way to solve a problem which has proved
a stumbling block to many."
SOURCES - page 239
Abraham. 1933. Prob. 82 -- Another ladder, pp. 37 & 45 (23 & 117). Ladder to go from one
street to another, of different widths.
E. H. Johnson, proposer; W. B. Carver, solver. Problem E436. AMM 47 (1940) 569 & 48
(1941) 271-273. Table going through a doorway. Obtains 6th order equation.
J. S. Madachy. Turning corners. RMM 5 (Oct 1961) 37, 6 (Dec 1961) 61 & 8 (Apr 1962)
56. In 5, he asks for the greatest length of board which can be moved around a corner,
assuming both corridors have the same width, that the board is thick and that vertical
movement is allowed. In 6, he gives a numerical answer for his original values and
asserts the maximal length for planar movement, with corridors of width w and plank
of thickness t, is 2 (w2 - t). In vol. 8, he says no two solutions have been the same.
L. Moser, proposer; M. Goldberg and J. Sebastian, solvers. Problem 66-11 -- Moving
furniture through a hallway. SIAM Review 8 (1966) 381-382 & 11 (1969) 75-78 &
12 (1970) 582-586. "What is the largest area region which can be moved through a
"hallway" of width one (see Fig. 1)?" The figure shows that he wants to move around a
rectangular corner joining two hallways of width one. Sebastian (1970) studies the
problem for moving an arc.
J. M. Hammersley. On the enfeeblement of mathematical skills .... Bull. Inst. Math. Appl. 4
(1968) 66-85. Appendix IV -- Problems, pp. 83-85, prob. 8, p. 84. Two corridors of
width 1 at a corner. Show the largest object one can move around it has area < 2 2
and that there is an object of area  π/2 + 2/π = 2.2074.
Partial solution by T. A. Westwell, ibid. 5 (1969) 80, with editorial comment
thereon on pp. 80-81.
T. J. Fletcher. Easy ways of going round the bend. MG 57 (No. 399) (Feb 1973) 16-22.
Gives five methods for the ladder problem with corridors of different widths.
Neal R. Wagner. The sofa problem. AMM 83 (1976) 188-189. "What is the region of largest
area which can be moved around a right-angled corner in a corridor of width one?"
Survey.
R. K. Guy. Monthly research problems, 1969-77. AMM 84 (1977) 807-815. P. 811 reports
improvements on the sofa problem.
J. S. Madachy & R. R. Rowe. Problem 242 -- Turning table. JRM 9 (1976-77) 219-221.
G. P. Henderson, proposer; M. Goldberg, solver; M. S. Klamkin, commentator. Problem
427. CM 5 (1979) 77 & 6 (1979) 31-32 & 49-50. Easily finds maximal area of a
rectangle going around a corner.
Research news: Conway's sofa problem. Mathematics Review 1:4 (Mar 1991) 5-8 & 32.
Reports on Joseph Gerver's almost complete resolution of the problem in 1990. Says
Conway asked the problem in the 1960s and that L. Moser is the first to publish it. Says
a group at a convexity conference in Copenhagen improved Hammersley's results to
2.2164. Gerver's analysis gives an object made up of 18 segments with area 2.2195.
The analysis depends on some unproven general assumptions which seem reasonable
and is certainly the unique optimum solution given those assumptions.
A. A. Huntington. More on ladders. M500 145 (Jul 1995) 2-5. Does usual problem, getting a
quartic. The finds the shortest ladder. [This turns out to be the same as the longest
ladder one can get around a corner from corridors of widths w and h, so 6.AG is
related to 6.L.]
6.AH.
TETHERED GOAT
A goat is grazing in a circular field and is tethered to a post on the edge. He can reach
half of the field. How long is the rope? There are numerous variations obtained by modifying
the shape of the field or having buildings within it. In recent years, there has been study of the
form where the goat is tethered to a point on a circular silo in a large field -- how much area
can he graze?
Upnorensis, proposer; Mr. Heath, solver. Ladies Diary, 1748-49 = T. Leybourn, II: 6-7,
quest. 302. [I have a reference to p. 41 of the Ladies' Diary.] Circular pond enclosed by
a circular railing of circumference 160 yards. Horse is tethered to a post of the railing
by a rope 160 yards long. How much area can he graze?
Dudeney. Problem 67: Two rural puzzles -- No. 67: One acre and a cow. Tit-Bits 33 (5 Feb
& 5 Mar 1898) 355 & 432. Circular field opening onto a small rectangular paddock
with cow tethered to the gate post so that she can graze over one acre. By skilful choice
of sizes, he avoids the usual transcendental equation.
SOURCES - page 240
Arc. [R. A. Archibald]. Involutes of a circle and a pasturage problem. AMM 28 (1921)
328-329. Cites Ladies Diary and it appears that it deals with a horse outside a circle.
J. Pedoe. Note 1477: An old problem. MG 24 (No. 261) (Oct 1940) 286-287. Finds the
relevant area by integrating in polar coordinates centred on the post.
A. J. Booth. Note 1561: On Note 1477. MG 25 (No. 267) (Dec 1941) 309-310. Goat
tethered to a point on the perimeter of a circle which can graze over ½, ⅓, ¼ of the
area.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968.
No. 8: "Don't fence me in", pp. 87. Equilateral triangular field of area 120. Three goats
tethered to the corners with ropes of length equal to the altitude. Consider an
area where n goats graze as contributing 1/n to each goat. What area does each
goat graze over?
No. 53: Around the silo, pp. 71 & 112-113. Goat tethered to the outside of a silo of
diameter 20 by a rope of length 10π, i.e. he can just get to the other side of the
silo. How big an area can he graze? The curve is a semicircle together with two
involutes of a circle, so the solution uses some calculus.
Marshall Fraser. A tale of two goats. MM 55 (1982) 221-227. Gives examples back to 1894.
Marshall Fraser. Letter: More, old goats. MM 56 (1983) 123. Cites Arc[hibald].
Bull, 1998, below, says this problem has been discussed by the Internet newsgroup sci.math
some years previously.
Michael E. Hoffman. The bull and the silo: An application of curvature. AMM 105:1 (Jan
1998) ??NYS -- cited by Bull. Bull is tethered by a rope of length L to a circular silo
of radius R. If L  πR, then the grazeable area is L3/3R + πL2/2. This paper
considers the problem for general shapes.
John Bull. The bull and the silo. M500 163 (Aug 1998) 1-3. Improves Hoffman's solution
for the circular silo by avoiding polar coordinates and using a more appropriate
variable, namely the angle between the taut rope and the axis of symmetry.
Keith Drever. Solution 186.5 -- Horse. M550 188 (Oct 2002) 12. A horse is tethered to a
point on the perimeter of a circular field of radius 1. He can graze over all but 1/π of
the area. How long is the rope? This turns out to make the problem almost trivial -- the
rope is 2 long and the angle subtended at the tether is π/2.
6.AI.
TRICK JOINTS
S&B, pp. 146-147, show several types.
These are often made in two contrasting woods and appear to be physically impossible.
They will come apart if one moves them in the right direction. A few have extra
complications. The simplest version is a square cylinder with dovetail joints on each face -called common square version below. There are also cases where one thinks it should come
apart, but the wood has been bent or forced and no longer comes apart -- see also 6.W.5.
See Bogesen in 6.W.2 for a possible early example.
Johannes Cornelus Wilhelmus Pauwels. UK Patent 15,307 -- Improved Means of Joining or
Fastening Pieces of Wood or other Material together, Applicable also as a Toy.
Applied: 9 Nov 1887; complete specification: 9 Aug 1888; accepted: 26 Oct 1888.
2pp + 1p diagrams. It says Pauwels is a civil engineer of The Hague. Common square
version.
Tom Tit, vol. 2. 1892. Assemblage paradoxal, pp. 231-232. = K, no. 155: The paradoxical
coupling, pp. 353-354. Common square version with instructions for making it by
cutting the corners off a larger square.
Emery Leverett Williams. The double dovetail and blind mortise. SA (25 Apr 1896) 267.
The first is a trick T-joint.
T. Moore. A puzzle joint and how to make it. The Woodworker 1:8 (May 1902) 172. S&B,
p. 147, say this is the earliest reference to the common square version -- but see
Pauwels, above. "... the foregoing joint will doubtless be well-known to our
professional readers. There are probably many amateur woodworkers to whom it will
be a novelty."
Hasluck, Paul N. The Handyman's Book. Cassell, 1903; facsimile by Senate (Tiger Books),
Twickenham, London, 1998. Pp. 220-223 shows various joints. Dovetail halved joint
with two bevels, p. 222 & figs. 703-705 of pp. 221-222. "... of but little practical value,
but interesting as a puzzle joint."
SOURCES - page 241
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Shows the common square version
"given to me some ten years ago, but I cannot say who first invented it." He previously
published it in a newspaper. ??look in Weekly Dispatch.
Samuel Hicks. Kinks for Handy Men: The dovetail puzzle. Hobbies 31 (No. 790) (3 Dec
1910) 248-249. Usual square dovetail, but he suggests to glue it together!
Dudeney. AM. 1917. Prob. 424: The dovetailed block, pp. 145 & 249. Shows the common
square version -- "... given to me some years ago, but I cannot say who first invented it."
He previously published it in a newspaper. ??as above
Anon. Woodwork Joints, 1918, op. cit. in 6.W.1. A curious dovetail joint, pp. 193, 195.
Common square version. Dovetail puzzle joint, pp. 194-195. A singly mortised
T-joint, with an unmortised second piece.
E. M. Wyatt. Woodwork puzzles. Industrial-Arts Magazine 12 (1923) 326-327. Doubly
dovetailed tongue and mortise T-joint called 'The double (?) dovetail'.
Sherman M. Turrill. A double dovetail joint. Industrial-Arts Magazine 13 (1924) 282-283. A
double dovetail right angle joint, but it leaves sloping gaps on the inside which are filled
with blocks.
Collins. Book of Puzzles. 1927. Pp. 134-135: The dovetail puzzle. Common square version.
E. M. Wyatt. Puzzles in Wood, 1928, op. cit. in 5.H.1.
The double (?) dovetail, pp. 44-45. Doubly dovetailed tongue and mortise T-joint.
The "impossible" dovetail joint, p. 46. Common square version.
Double-lock dovetail joint, pp. 47-49. Less acceptable tricks for a corner joint.
Two-way fanned half-lap joint, pp. 49-50. Corner joint.
A. B. Cutler. Industrial Arts and Vocational Education (Jan 1930). ??NYS. Wyatt, below,
cites this for a triple dovetail, but I could not not find it in vols. 1-40.
R. M. Abraham. Prob. 225 -- Dovetail Puzzle. Winter Nights Entertainments. Constable,
London, 1932, p. 131. (= Easy-to-do Entertainments and Diversions with coins, cards,
string, paper and matches; Dover, 1961, p. 225.) Common square version.
Abraham. 1933. Prob. 304 -- Hexagon dovetail; Prob. 306 -- The triangular dovetail,
pp. 142-143 (100 & 102).
Bernard E. Jones, ed. The Practical Woodworker. Waverley Book Co., London, nd [1940s?].
Vol. 1: Lap and secret dovetail joints, pp. 281-287. This covers various secret joints -i.e. ones with concealed laps or dovetails. Pp. 286-287 has a subsection: Puzzle
dovetail joints. Common square version is shown as fig. 28. A pentagonal analogue is
shown as fig. 29, but it uses splitting and regluing to produce a result which cannot be
taken apart.
E. M. Wyatt. Wonders in Wood. Bruce Publishing Co., Milwaukee, 1946.
Double-double dovetail joint, pp. 26-27. Requires some bending.
Triple dovetail puzzle, pp. 28-29. Uses curved piece with gravity lock.
S&B, p. 146, reproduces the above Wyatt and shows a 1948 example.
W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Dovetail deceptions, p. 64. Common
square version and a tapered T joint.
Allan Boardman. Up and Down Double Dovetail. Shown on p. 147 of S&B. Square version
with alternate dovetails in opposite directions. This is impossible!
I have a set of examples which belonged to Tom O'Beirne. There is a common square version
and a similar hexagonal version. There is an equilateral triangle version which requires
a twist. There is a right triangle version which has to be moved along a space diagonal!
[One can adapt the twisting method to n-gons!]
Dick Schnacke (Mountain Craft Shop, American Ridge Road, Route 1, New Martinsville,
West Virginia, 26155, USA) makes a variant of the common square version which has
two dovetails on each face. I bought one in 1994.
6.AJ. GEOMETRIC ILLUSIONS
There are a great many illusions. This will only give some general studies and some
specific sources, though the sources of many illusions are unknown.
An exhibition by Al Seckel says there are impossible geometric patterns in a mosaic floor in
the Roman villa at Fishbourne, c75, but it is not clear if this was intentional.
Anonymous 15C French illustrator of Giovanni Boccaccio, De Claris Mulieribus, MS Royal
16 Gv in the British Library. F. 54v: Collecting cocoons and weaving silk. ??NYS -reproduced in: The Medieval Woman An Illuminated Book of Postcards,
SOURCES - page 242
HarperCollins, 1991. This shows a loom(?) frame with uprights at each corner and the
crosspieces joining the tops of the end uprights as though front and rear are reversed
compared to the ground.
Seckel, 2002a, below, p. 25 (= 2002b, p. 175), says Leonardo da Vinci created the first
anamorphic picture, c1500.
Giuseppe Arcimboldo (1537-1593). One of his paintings shows a bowl of vegetables, but
when turned over, it is a portrait. Seckel, 2000, below, fig. 109, pp. 120 & 122
(= 2002b, fig, 107, pp. 118 & 120), noting that this is the first known invertible picture,
but see next entry.
Topsy turvy coin, mid 16C. Seckel, 2002a, fig. 65, p. 80 (omitted in 2002b), shows an
example which shows the Pope, but turns around to show the Devil. Inscription around
edge reads: CORVI MALUM OVUM MALII.
Robert Smith. A Compleat System of Opticks in Four Books. Cambridge, 1738. He includes
a picture of a distant windmill for which one cannot tell whether the sails are in front or
behind the mill, apparently the first publication of this visual ambiguity. ??NYS -- cited
by: Nicholas J. Wade; Visual Allusions Pictures of Perception; Lawrence Erlbaum
Associates, Hove, East Sussex, 1990, pp. 17 & 25, with a similar picture.
L. A. Necker. LXI. Observations on some remarkable optical phœnomena seen in
Switzerland; and on an optical phœnomenon which occurs on viewing a figure of a
crystal or geometrical solid. Phil. Mag. (3) 1:5 (Nov 1832) 329-337. This is a letter
from Necker, written on 24 May 1832. On pp. 336-337, Necker describes the visual
reversing figure known as the Necker cube which he discovered in drawing rhomboid
crystals. This is also quoted in Ernst; The Eye Beguiled, pp. 23-24]. Richard L.
Gregory [Mind in Science; Weidenfeld and Nicolson, London, 1981, pp. 385 & 594]
and Ernst say that this was the first ambiguous figure to be described.
See Thompson, 1882, in 6.AJ.2, for illusions caused by rotations.
F. C. Müller-Lyer. Optische Urtheilstusehungen. Arch. Physiol. Suppl. 2 (1889) 263-270.
Cited by Gregory in The Intelligent Eye. Many versions of the illusion. But cf below.
Wehman. New Book of 200 Puzzles. 1908. The cube puzzle, p. 37. A 'baby blocks' pattern
of cubes, which appears to show six cubes piled in a corner one way and seven cubes
the other way. I don't recall seeing this kind of puzzle in earlier sources, though this
pattern of rhombuses is common on cathedral floors dating back to the Byzantine era or
earlier.
James Fraser. British Journal of Psychology (Jan 1908). Introduces his 'The Unit of Direction
Illusion' in many forms. ??NYS -- cited in his popular article in Strand Mag., see
below. Seckel, 2000, below, has several versions. On p. 44, note to p. 9 (= 2002b, p.
44, note to p. 9), he says Fraser created a series of these illusions in 1906.
H. E. Carter. A clever illusion. Curiosities section, Strand Mag. 378 (No. 219) (Mar 1909)
359. An example of Fraser's illusion with no indication of its source.
James Fraser. A new illusion. What is its scientific explanation? Strand Mag. 38 (No. 224)
(Aug 1909) 218-221. Refers to the Mar issue and says he introduced the illusion in the
above article and that the editors have asked him for a popular article on it. 16
illustrations of various forms of his illusion.
Lietzmann, Walther & Trier, Viggo. Wo steckt der Fehler? 3rd ed., Teubner, 1923. [The
Vorwort says that Trier was coauthor of the 1st ed, 1913, and contributed most of the
Schülerfehler (students' mistakes). He died in 1916 and Lietzmann extended the work
in a 2nd ed of 1917 and split it into Trugschlüsse and this 3rd ed. There was a 4th ed.,
1937. See Lietzmann for a later version combining both parts.] II. Täuschungen der
Anschauung, pp. 7-13.
Lietzmann, Walther. Wo steckt der Fehler? 3rd ed., Teubner, Stuttgart, (1950), 1953.
(Strens/Guy has 3rd ed., 1963.) (See: Lietzmann & Trier. There are 2nd ed, 1952??;
5th ed, 1969; 6th ed, 1972. Math. Gaz. 54 (1970) 182 says the 5th ed appears to be
unchanged from the 3rd ed.) II. Täuschungen der Anschauung, pp. 15-25. A
considerable extension of the 1923 ed.
Williams. Home Entertainments. 1914. Colour discs for the gramophone, pp. 207-212.
Discusses several effects produced by spirals and eccentric circles on discs when
rotated.
Gerald H. Fisher. The Frameworks for Perceptual Localization. Report of MOD Research
Project70/GEN/9617, Department of Psychology, University of Newcastle upon Tyne,
1968. Good collection of examples, with perhaps the best set of impossible figures.
Pp. 42-47 -- reversible perspectives.
SOURCES - page 243
Pp. 56-65 -- impossible and ambiguous figures.
Appendix 6, p.190 -- 18 reversible figures.
Appendix 7, pp. 191-192 -- 12 reversible silhouettes.
Appendix 8, p. 193 -- 12 impossible figures.
Appendix 14, pp. 202-203 -- 72 geometrical illusions.
Harvey Long. "It's All In How You Look At It". Harvey Long & Associates, Seattle, 1972.
48pp collection of examples with a few references.
Bruno Ernst [pseud. of J. A. F. Rijk]. (Avonturen met Onmogelijke Figuren; Aramith
Uitgevers, Holland, 1985.) Translated as: Adventures with Impossible Figures.
Tarquin, Norfolk, 1986. Describes tribar and many variations of it, impossible
staircase, two-pronged trident. Pp. 76-77 reproduces an Annunciation of 14C in the
Grote Kerk, Breda, with an impossible perspective. P. 78 reproduces Print XIV of
Giovanni Battista Piranesi's "Carceri de Invenzione", 1745, with an impossible 4-bar.
Diego Uribe. Catalogo de impossibilidades. Cacumen (Madrid) 4 (No. 37) (Feb 1986) 9-13.
Good summary of impossible figures. 15 references to recent work.
Bruno Ernst. Escher's impossible figure prints in a new context. In: H. S. M. Coxeter, et al.,
eds.; M. C. Escher -- Art and Science; North-Holland (Elsevier), Amsterdam, 1986,
pp. 124-134. Pp. 128-129 discusses the Breda Annunciation, saying it is 15C and
quoting a 1912 comment by an art historian on it. There is a colour reproduction on
p. 394. P. 130 shows and discusses briefly Bruegel's "The Magpie on the Gallows",
1568. Pp. 130-131 discusses and illustrates the Piranesi.
Bruno Ernst. (Het Begoochelde Oog, 1986?.) Translated by Karen Williams as: The Eye
Beguiled. Benedikt Taschen Verlag, Köln, 1992. Much expanded version of his
previous book, with numerous new pictures and models by new artists in the field.
Chapter 6: Origins and history, pp. 68-93, discusses and quotes almost everything
known. P. 68 shows a miniature of the Madonna and Child from the Pericope of Henry
II, compiled by 1025, now in the Bayersche Staatsbibliothek, Munich, which is similar
in form to the Breda Annunciation (stated to be 15C). (However, Seckel, 1997, below,
reproduces it as 2 and says it is c1250.) P. 69 notes that Escher invented the
impossible cube used in his Belvedere. P. 82 is a colour reproduction of Duchamp's
1916-1917 'Apolinère Enameled' - see 6.AJ.2. Pp. 83-84 shows and discusses Piranesi.
Pp. 84-85 show and discuss Hogarth's 'False Perspective' of 1754. Reproduction and
brief mention of Brueghel (= Bruegel) on p. 85. Discussion of the Breda Annunciation
on pp. 85-86. Pp. 87-88 show and discuss a 14C Byzantine Annunciation in the
National Museum, Ochrid. Pp. 88-89 show and discuss Scott Kim's impossible fourdimensional tribar.
J. Richard Block & Harold E. Yuker. Can You Believe Your Eyes? Brunner/Mazel, NY,
1992. Excellent survey of the field of illusions, classified into 17 major types -- e.g.
ambiguous figures, unstable figures, ..., two eyes are better than one. They give as
much information as they can about the origins. They give detailed sources for the
following -- originals ??NYS. These are also available as two decks of playing cards.
W. E. Hill. My wife and my mother-in-law. Puck, (6 Nov 1915) 11. [However, Julian
Rothenstein & Mel Gooding; The Paradox Box; Redstone Press, London, 1993; include
a reproduction of a German visiting card of 1888 with a version of this illusion. The
English caption by James Dalgety is: My Wife and my Mother-in-law. Cf Seckel,
1997, below.] Ernst, just above, cites Hill and says he was a cartoonist, but gives no
source. Long, above, asserts it was designed by E. G. Boring, an American
psychologist.
G. H. Fisher. Mother, father and daughter. Amer. J. Psychology 81 (1968) 274-277.
G. Kanisza. Subjective contours. SA 234:4 (Apr 1976) 48-52. (Kanisza triangles.)
Al Seckel, 1997. Illusions in Art. Two decks of playing cards in case with notes. Deck 1 -Classics. Works from Roman times to the middle of the 20th Century. Deck 2 -Contemporary. Works from the second half of the 20th Century. Y&B Associates,
Hempstead, NY, 1997. This gives further details on some of the classic illusions -some of this is entered above and in 6.AU and some is given below.
10: Rabbit/Duck. Devised by Joseph (but notes say Robert) Jastrow, c1900. Seckel,
2000, below, p. 159 (= 2002b, p. 156), says Joseph Jastrow, c1900.
10: My Wife and My Mother-in-Law, anonymous, 1888. However, in an exhibition,
Seckel's text implies the 1888 German card doesn't have a title and the title first
occurs on an 1890 US card. Seckel, 2000, below, p. 122 (= 2002b, p. 120), says
Boring took it from a popular 19C puzzle trading card.
SOURCES - page 244
Al Seckel, 2000. The Art of Optical Illusions. Carlton, 2000. 144 well reproduced illusions
with brief notes. All figures except 69-70 are included in Seckel, 2002b.
J. Richard Block. Seeing Double Over 200 Mind-Bending Illusions. Routledge, 2002.
Update of Block & Yuker, 1992.
Edgar Rubin. Rubin's Vase. 1921. This is the illusion where there appears to be a
vase, but the outsides appear to be two face profiles. [Pp. 8-11.] But Seckel,
2000, above, p. 122 (= 2002b, p. 120), says Rubin's inspiration was a 19C puzzle
card.
My wife and my mother-in-law. P. 17 says Hill's version may derive from a late 1880s
advertising postcard for Phenyo-Caffein (Worcester, Massachusetts), labelled
'My Girl & Her Mother', reproduced on p. 17.
P. 18 has G. H. Fisher's 1968 triple image, labelled 'Mother, Father and Daughter-inLaw'.
P. 44 says that Rabbit/Duck was devised by Joseph Jastrow in 1888.
Al Seckel, 2002a. More Optical Illusions. Carlton, 2002. 137 well reproduced illusions with
brief notes, different than in Seckel, 2000, above. All figures except 65-66, 86-87, 9595, 137 are included in Seckel, 2002b, but with different figure and page numbers.
Al Seckel, 2002b. The Fantastic World of Optical Illusions. Carlton, 2002. This is
essentially a combination of Seckel, 2000, and Seckel, 2002a, both listed above. The
Introduction is revised. Figures 69-70 of the first book and 65-66, 86-87, 94-95, 137 of
the second book are omitted. The remaining figures are then numbered consecutively.
The page of Further Reading in the first book is put at the end of this combined book.
Here I make some notes about origins of other illusions, but I have fewer details on these.
The Müller-Lyer Illusion -- <-> vs >---< was proposed by Zollner in 1859 and described by
Johannes Peter Müller (1801-1858) & Lyer in 1889. This seems to be a confusion, as
the 1889 article is by F. C. Müller-Lyer, cf above. Lietzmann & Trier, p. 7, date it as
1887.
The Bisection Illusion -- with a vertical segment bisecting a horizontal segment, but above it - was described by Albert Oppel (1831-1865) and Wilhelm Wundt (1832-1920) in
1865.
Zollner's Illusion -- parallel lines crossed by short lines at 45o, alternately in opposite
directions -- was noticed by Johann K. F. Zollner (1834-1882) on a piece of fabric with
a similar design.
Hering's Illusion -- with parallel lines crossed by numerous lines through a point between the
lines -- was invented by Ewals Hering (1834-1918) in 1860.
6.AJ.1
TWO PRONGED TRIDENT
I have invented this name as it is more descriptive than any I have seen. The object or a
version of it is variously called: Devil's Fork; Three Stick Clevis; Widgit; Blivit; Impossible
Columnade; Trichometric Indicator Support; Triple Encabulator for Tuned Manifold; Hole
Location Gage; Poiyut; Triple-Pronged Fork with only Two Branches; Old Roman
Pitchfork.
Oscar Reutersvård. Letters quoted in Ernst, 1992, pp. 69-70, says he developed an equivalent
type of object, which he calls impossible meanders, in the 1930s.
R. L. Gregory says this is due to a MIT draftsman (= draughtsman) about 1950??
California Technical Industries. Advertisement. Aviation Week and Space Technology 80:12
(23 Mar 1964) 5. Standard form. (I wrote them but my letter was returned 'insufficient
address'.)
Hole location gage. Analog Science Fact • Science Fiction 73:4 (Jun 1964) 27. Classic Two
pronged trident, with some measurements given. Editorial note says the item was 'sent
anonymously for some reason' and offers the contributor $10 or a two year subscription
if he identifies himself. (Thanks to Peter McMullen for the Analog items, but he
doesn't recall the contributor ever being named.)
Edward G. Robles, Jr. Letter (Brass Tacks column). Analog Science Fact • Science Fiction
74:4 (Dec 1964) 4. Says the Jun 1964 object is a "three-hole two slot BLIVIT" and was
developed at JPL (Jet Propulsion Laboratory, Pasadena) and published in their Goddard
News. He provides a six-hole five-slot BLIVIT, but as the Editor comments, it 'lacks
SOURCES - page 245
the classic simple elegance of the Original.' However, a letter of inquiry to JPL resulted
in an email revealing that Goddard News is not their publication, but comes from the
Goddard Space Flight Center. I have had a response from Goddard, ??NYR.
D. H. Schuster. A new ambiguous figure: a three-stick clevis. Amer. J. Psychol. 77 (1964)
673. Cites Calif. Tech. Ind. ad. [Ernst, 1992, pp. 80-81 reproduces this article.]
Mad Magazine. No. 93 (Mar 1965). (I don't have a copy of this -- has anyone got one for
sale?) Cover by Norman Poiyut (?) shows the figure and it is called a poiyut. Miniature
reproduction in: Maria Reidelbach; Completely Mad -- A History of the Comic Book
and Magazine; Little, Brown & Co., Boston, 1991, p. 82. Shows a standard version. Al
Seckel says they thought it was an original idea and they apologised in the next issue -to all of the following! I now have the relevant issue, No. 95 (Jun 1965) and p. 2 has 15
letters citing earlier appearances in Engineering Digest, The Airman (official journal of
the U.S. Airforce), Analog, Astounding Science Fact -- Science Fiction (Jun 1964, see
above), The Red Rag (engineering journal at the University of British Columbia),
Society of Automotive Engineers Journal (designed by by Gregory Flynn Jr. of General
Motors as Triple Encabulator Tuned Manifold), Popular Mechanics, Popular Science
(Jul 1964), Road & Track (Jun 1964). Other letters say it was circulating at: the
Engineering Graphics Lab of the University of Minnesota at Duluth; the Nevada Test
Site; Eastman Kodak (used to check resolution); Industrial Camera Co. of Oakland
California (on their letterhead). Two letters give an impossible crate and an impossible
rectangular frame (sort of a Penrose rectangle).
Sergio Aragones. A Mad look at winter sports. Mad Magazine (?? 1965); reprinted in: Mad
Power; Signet, NY, 1970, pp. 120-129. P. 124 shows a standard version.
Bob Clark, illustrator. A Mad look at signs of the times. Loc. cit. under Aragones,
pp. 167-188. P. 186 shows standard version.
Reveille (a UK weekly magazine) (10 Jun 1965). ??NYS -- cited by Briggs, below -- standard
version.
Don Mackey. Optical illusion. Skywriter (magazine of North American Aviation) (18 Feb
1966). ??NYS -- cited by Conrad G. Mueller et al.; Light and Vision; Time-Life Books
Pocket Edition, Time-Life International, Netherlands, 1969, pp. 171 & 190. Standard
version with nuts on the ends.
Heinz Von Foerster. From stimulus to symbol: The economy of biological computation. IN:
Sign Image Symbol; ed. Gyorgy Kepes; Studio Vista, London, 1966, pp. 42-60. On
p. 55, he shows the "Triple-pronged fork with only two branches" and on p. 54, he notes
that although each portion is correct, it is impossible overall, but he gives no indication
of its history or that it is at all new.
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Pp. 17-18
shows the unnamed trident in a version from Adcock & Shipley (Sales) Ltd., machine
tool makers in Leicester. Cites Reveille, above. Standard versions.
Harold Baldwin. Building better blivets. The Worm Runner's Digest 9:2 (1967) 104-106.
Discusses relation between numbers of slots and of prongs. Draws a three slot version
and 2 and 4 way versions.
Charlie Rice. Challenge! Op. cit. in 5.C. 1968. P. 10 shows a six prong, four slot version,
called the "Old Roman Pitchfork".
Roger Hayward. Blivets; research and development. The Worm Runner's Digest 10 (Dec
1968) 89-92. Several fine developments, including two interlaced frames and his
monumental version. Cites Baldwin.
M. Gardner. SA (May 1970) = Circus, pp. 3-15. Says this became known in 1964 and cites
Mad & Hayward, but not Schuster.
D. Uribe, op. cit. above, gives several variations.
6.AJ.2.
TRIBAR AND IMPOSSIBLE STAIRCASE
Silvanus P. Thompson. Optical illusions of motion. Brain 3 (1882) 289-298. Hexagon of
non-overlapping circles.
Thomas Foster. Illusions of motion and strobic circles. Knowledge 1 (17 Mar 1882) 421423. Says Thompson exhibited these illusions at the British Association meeting in
1877.
Pearson. 1907. Part II, no. 3: Whirling wheels, p. 3. Gives Thompson's form, but the wheels
are overlapping, which makes it look a bit like an ancestor of the tribar.
Marcel Duchamp (1887-1968). Apolinère Enameled. A 'rectified readymade' of 1916-1917
SOURCES - page 246
which turned a bedframe in an advertisement for Sapolin Enamel into an impossible
figure somewhat like a Penrose Triangle and a square version thereof. A version is in
the Philadelphia Museum of Art and is reproduced and discussed in Ernst; The Eye
Beguiled, p. 82. (Duchamp's 'readymades' were frequently reproduced by himself and
others, so there may be other versions of this.)
Oscar Reutersvård. Omöjliga Figure [Impossible Figures -- In Swedish]. Edited by Paul
Gabriel. Doxa, Lund, (1982); 2nd ed., 1984. This seems to be the first publication of
his work, but he has been exhibiting since about 1960 and some of the exhibitions seem
to have had catalogues. P. 9 shows and discusses his Opus 1 from 1934, which is an
impossible tribar made from cubes. (Reproduced in Ernst, 1992, p. 69 as a drawing
signed and dated 1934. Ernst quotes Reutersvård's correspondence which describes his
invention of the form while doodling in Latin class as a schoolboy. A school friend
who knew of his work showed him the Penroses' article in 1958 -- at that time he had
drawn about 100 impossible objects -- by 1986, he had extended this to some 2500!)
He has numerous variations on the tribar and the two-pronged trident. An exhibition by
Al Seckel says Reutersvård had produced some impossible staircases, e.g. 'Visualized
Impossible Bach Scale', in 1936-1937, but didn't go far with it until returning to the idea
in 1953.
Oscar Reutersvård. Swedish postage stamps for 25, 50, 75 kr. 1982, based on his patterns
from the 1930s. The 25 kr. has the tribar pattern of cubes which he first drew in 1934.
(Also the 60 kr.??)
L. S. & R. Penrose. Impossible objects: A special type of visual illusion. British Journal of
Psychology 49 (1958) 31-33. Presents tribar and staircase. Photo of model staircase,
which Lionel Penrose had made in 1955. [Ernst, 1992, pp. 71-73, quotes conversation
with Penrose about his invention of the Tribar and reproduces this article. Penrose, like
the rest of us, only learned about Reutersvård's work in the 1980s.]
Anon.(?) Don't believe it. Daily Telegraph (24 Mar 1958) ?? (clipping found in an old book).
"Three pages of the latest issue of the British Journal of Psychology are devoted to
"Impossible Objects."" Shows both the tribar and the staircase.
M. C. Escher. Lithograph: Belvedere. 1958.
L. S. & R. Penrose. Christmas Puzzles. New Scientist (25 Dec 1958) 1580-1581 & 1597.
Prob. 2: Staircase for lazy people.
M. C. Escher. Lithograph: Ascending and Descending. 1960.
M. C. Escher. Lithograph: Waterfall. 1961.
Oscar Reutersvård, in 1961, produced a triangular version of the impossible staircase, called
'Triangular Fortress without Highest Level'.
Joseph Kuykendall. Letter. Mad Magazine 95 (Jun 1965) 2. An impossible frame, a kind of
Penrose rectangle.
S. W. Draper. The Penrose triangle and a family of related figures. Perception 7 (1978)
283-296. ??NYS -- cited and reproduced in Block, 2002, p. 48. A Penrose rectangle.
Uribe, op. cit. above, gives several variations, including a perspective tribar and Draper's
rectangle.
Jan van de Craats. Das unmögliche Escher-puzzle. (Taken from: De onmogelijke Escherpuzzle; Pythagoras (Amsterdam) (1988).) Alpha 6 (or: Mathematik Lehren / Heft 55 -??) (1992) 12-13. Two Penrose tribars made into an impossible 5-piece burr.
6.AJ.3.
CAFÉ WALL ILLUSION
This is the illusion seen in alternatingly coloured staggered brickwork where the lines of
bricks distinctly seem tilted. I suspect it must be apparent in brickwork going back to Roman
times.
The illusion is apparent in the polychrome brick work on the side wall inside Keble College
Chapel, Oxford, by William Butterfield, completed in 1876 [thanks to Deborah
Singmaster for observing this].
Lietzmann & Trier, op. cit. at 6.AJ, 1923. Pp. 12-13 has a striking version of this, described
as a 'Flechtbogen der Kleinen'. I can't quite translate this -- Flecht is something
interwoven but Bogen could be a ribbon or an arch or a bower, etc. They say it is
reproduced from an original by Elsner. See Lietzmann, 1953.
Ogden's Optical Illusions. Cigarette card of 1927. No. 5. Original ??NYS -- reproduced in:
Julian Rothenstein & Mel Gooding; The Paradox Box; Redstone Press, London, 1993
SOURCES - page 247
AND in their: The Playful Eye; Redstone Press, London, 1999, p. 56. Vertical version
of this illusion.
B. K. Gentil. Die optische Täuschung von Fraser. Zeitschr. f. math. u. naturw. Unterr. 66
(1935) 170 ff. ??NYS -- cited by Lietzmann.
Nelson F. Beeler & Franklyn M. Branley. Experiments in Optical Illusion. Ill. by Fred H.
Lyon. Crowell, 1951, p. 42, fig. 39, is a good example of the illusion.
Lietzmann, op. cit. at 6.AJ, 1953. P. 23 is the same as above, but adds a citation to Gentil,
listed above.
Leonard de Vries. The Third Book of Experiments. © 1965, probably for a Dutch edition.
Translated by Joost van de Woestijne. John Murray, 1965; Carousel, 1974. Illusion 10,
pp. 58-59, has a clear picture and a brief discussion.
Richard L. Gregory & Priscilla Heard. Border locking and the café wall illusion. Perception
8 (1979) 365-380. ??NYS -- described by Walker, below. [I have photos of the actual
café wall in Bristol.]
Jearl Walker. The Amateur Scientist: The café-wall illusion, in which rows of tiles tilt that
should not tilt at all. SA 259:5 (Nov 1988) 100-103. Good summary and illustrations.
6.AJ.4.
STEREOGRAMS
New section, due to reading Glass's assertion as to the inventor, who is different than
other names that I have seen.
Don Glass, ed. How Can You Tell if a Spider is Dead? and More Moments of Science.
Indiana Univ Press, Bloomington, Indiana, 1996. Now you see it, now you don't,
pp. 131-132. Asserts that Christopher Tyler, of the Smith-Kettlewell Eye Research
Institute, San Francisco, is the inventor of stereograms.
6.AJ.5.
IMPOSSIBLE CRATE.
This is like a Necker Cube where all the edges are drawn as wooden slats in an
impossible configuration.
Escher. Man with Cuboid, which is essentially a detail from Belvedere, both 1958, are
apparently the first examples of this impossible object.
Chuck Mathias. Letter Mad Magazine 95 (Jun 1965) 2. Gives an impossible crate.
Jerry Andrus developed his actual model in 1981 and it appeared on the cover of Omni in
1981. But Al Seckel's exhibition says the first physical example was The Feemish
Crate, due to C. F. Cochran.
Seckel, 2002a, figs. 27 A&B, pp. 36-37 (= 2002b, figs. 169 A&B, pp. 186-187), shows and
discusses Andrus' crate from two viewpoints.
6.AK.
POLYGONAL PATH COVERING N x N LATTICE OF POINTS,
QUEEN'S TOURS, ETC.
For magic circuits, see 7.N.4.
3x3 problem: Loyd (1907), Pearson, Anon., Bullivant, Goldston, Loyd (1914), Blyth,
Abraham, Hedges, Evans, Doubleday - 1, Piggins & Eley
4x4 problem: King, Abraham, M. Adams, Evans, Depew, Meyer, Ripley's,
Queen's tours: Loyd (1867, 1897, 1914), Loyd Jr.
Bishop's tours: Dudeney (1932), Doubleday - 2, Obermair
Rook's tours: Loyd (1878), Proctor, Loyd (1897), Bullivant, Loyd (1914), Filipiak,
Hartswick, Barwell, Gardner, Peters, Obermair
Other versions: Prout, Doubleday - 1
Trick solutions: Fixx, Adams, Piggins, Piggins & Eley
Thanks to Heinrich Hemme for pointing out Fixx, which led to adding most of the
material on trick solutions.
Loyd. ??Le Sphinx (Mar 1867 -- but the Supplement to Sam Loyd and His Chess Problems
corrects this to 15 Nov 1866). = Chess Strategy, Elizabeth, NJ, 1878, no. or p. 336(??).
= A. C. White; Sam Loyd and His Chess Problems; 1913, op. cit. in 1; no. 40, pp.
42-43. Queen's circuit on 8 x 8 in 14 segments. (I.e. closed circuit, not leaving
SOURCES - page 248
board, using queen's moves.) No. 41 & 42 of White give other solutions. White quotes
Loyd from Chess Strategy, which indicates that Loyd invented this problem. Tit-Bits
No. 31 & SLAHP: Touring the chessboard, pp. 19 & 89, give No. 41.
Loyd. Chess Strategy, 1878, op. cit. above, no. or p. 337 (??) (= White, 1913, op. cit. above,
no. 43, pp. 42-43.) Rook's circuit on 8 x 8 in 16 segments. (I.e. closed circuit, not
leaving board, using rook's moves, and without crossings.)
Richard A. Proctor. Gossip column. Knowledge 10 (Dec 1886) 43 & (Feb 1887) 92. 6 x 6
array of cells. Prisoner in one corner can exit from the opposite corner if he passes
"once, and once only, through all the 36 cells." "... take the prisoner into either of the
cells adjoining his own, and back into his own, .... This puzzle is rather a sell, ...."
Letter and response [in Gossip column, Knowledge 10 (Mar 1887) 115-116] about the
impossibility of any normal solution.
Loyd. Problem 15: The gaoler's problem. Tit-Bits 31 (23 Jan & 13 Feb 1897) 307 & 363.
Rook's circuit on 8 x 8 in 16 segments, but beginning and ending on a central square.
Cf The postman's puzzle in the Cyclopedia, 1914.
Loyd. Problem 16: The captive maiden. Tit-Bits 31 (30 Jan & 20 Feb 1897) 325 & 381.
Rook's tour in minimal number of moves from a corner to the diagonally opposite
corner, entering each cell once. Because of parity, this is technically impossible, so the
first two moves are into an adjacent cell and then back to the first cell, so that the first
cell has now been entered.
Loyd. Problem 20: Hearts and darts. Tit-Bits 31 (20 Feb, 13 & 20 Mar 1897) 381, 437,
455. Queen's tour on 8 x 8, starting in a corner, permitting crossings, but with no
segment going through a square where the path turns. Solution in 14 segments. This
is No. 41 in White -- see the first Loyd entry above.
Ball. MRE, 4th ed., 1905, p. 197. At the end of his section on knight's tours, he states that
there are many similar problems for other kinds of pieces.
Loyd. In G. G. Bain, op. cit. in 1, 1907. He gives the 3 x 3 lattice in four lines as the
Columbus Egg Puzzle.
Pearson. 1907. Part I, no. 36: A charming puzzle, pp. 36 & 152-153. 3 x 3 lattice in 4
lines.
Loyd. Sam Loyd's Puzzle Magazine (Apr 1908) -- ??NYS, reproduced in: A. C. White; Sam
Loyd and His Chess Problems; 1913, op. cit. in 1; no. 56, p. 52. = Problem 26: A brace
of puzzles -- No. 26: A study in naval warfare; Tit-Bits 31 (27 Mar 1897) 475 & 32
(24 Apr 1897) 59. = Cyclopedia, 1914, Going into action, pp. 189 & 364. = MPSL1,
prob. 46, pp. 44 & 138. = SLAHP: Bombs to drop, pp. 86 & 119. Circuit on 8 x 8 in
14 segments, but with two lines of slope 2. In White, p. 43, Loyd says an ordinary
queen's tour can be started "from any of the squares except the twenty which can be
represented by d1, d3 and d4." This problem starts at d1. However I think White must
have mistakenly put down twenty for twelve??
Anon. Prob. 67. Hobbies 31 (No. 782) (8 Oct 1910) 39 & (No. 785) (29 Oct 1910) 94.
3 x 3 lattice in 4 lines "brought under my notice some time back".
C. H. Bullivant. Home Fun, 1910, op. cit. in 5.S. Part VI, Chap. IV.
No. 1: The travelling draught-man, pp. 515 & 520. Rook's circuit on 8 x 8 in 16
segments, different than Loyd's.
No. 3: Joining the rings. 3 x 3 in 4 segments.
Will Goldston. More Tricks and Puzzles without Mechanical Apparatus. The Magician Ltd.,
London, nd [1910?]. (BMC lists Routledge & Dutton eds. of 1910.) (There is a 2nd
ed., published by Will Goldston, nd [1919].) The nine-dot puzzle, pp. 127-128 (pp.
90-91 in 2nd ed.).
Loyd. Cyclopedia, 1914, pp. 301 & 380. = MPSL2, prob. 133 -- Solve Christopher's egg
tricks, pp. 93 & 163 (with comment by Gardner). c= SLAHP: Milkman's route, pp. 34
& 96. 3 x 3 case.
Loyd. Cyclopedia, 1914, pp. 293 & 379. Queen's circuit on 7 x 7 in 12 segments.
Loyd. The postman's puzzle. Cyclopedia, 1914, pp. 298 & 379. Rook's circuit on 8 x 8
array of points, with one point a bit out of line, starting and ending at a central square,
in 16 segments. P. 379 also shows another 8 x 8 circuit, but with a slope 2 line. See
also pp. 21 & 341 and SLAHP, pp. 85 & 118, for two more examples.
Loyd. Switchboard problem. Cyclopedia, 1914, pp. 255 & 373. (c= MPSL2, prob. 145,
pp. 102 & 167.) Rook's tour with minimum turning.
Blyth. Match-Stick Magic. 1921. Four-way game, pp. 77-78. 3 x 3 in 4 segments.
King. Best 100. 1927. No. 16, pp. 12 & 43. 4 x 4 in 6 segments, not closed, but easily can
SOURCES - page 249
be closed.
Loyd Jr. SLAHP. 1928. Dropping the mail, pp. 67 & 111. 4 x 4 queen's tour in 6
segments.
Collins. Book of Puzzles. 1927. The star group puzzle, pp. 95-96. 3 x 3 in 4 segments.
Dudeney. PCP. 1932. Prob. 264: The fly's tour, pp. 82 & 169. = 536, prob. 422, pp. 159 &
368. Bishop's path, with repeated cells, going from corner to corner in 17 segments.
Abraham. 1933. Probs. 101, 102, 103, pp. 49 & 66 (30 & 118). 3 x 3, 4 x 4 and 6 x 6
cases.
The Bile Beans Puzzle Book. 1933. No. 4: The puzzled milkman. 3 x 3 array in four lines.
Sid G. Hedges. More Indoor and Community Games. Methuen, London, 1937. Nine spot,
p. 110. 3 x 3. "Of course it can be done, but it is not easy." No solution given.
M. Adams. Puzzle Book. 1939. Prob. C.64: Six strokes, pp. 140 & 178. 4 x 4 array in 6
segments which form a closed path, though the closure was not asked for.
J. R. Evans. The Junior Week-End Book. Op. cit. in 6.AF. 1939. Probs. 30 & 31, pp. 264 &
270. 3 x 3 & 4 x 4 cases in 4 & 6 segments, neither closed nor staying within the
array.
Depew. Cokesbury Game Book. 1939. Drawing, p. 220. 4 x 4 in 6 segments, not closed,
not staying within the array.
Meyer. Big Fun Book. 1940. Right on the dot, pp. 99 & 732. 4 x 4 in 6 segments.
A. S. Filipiak. Mathematical Puzzles, 1942, op. cit. in 5.H.1, pp. 50-51. Same as Bullivant,
but opens the circuit to make a 15 segment path.
M. S. Klamkin, proposer and solver; John L. Selfridge, further solver. Problem E1123 -Polygonal path covering a square lattice. AMM 61 (1954) 423 & 62 (1955) 124 &
443. Shows N x N can be done in 2N-2 segments. Selfridge shows this is minimal.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. Joining the stars, pp.
41 & 129. 5 x 5 array of points. Using a line of four segments, pass through 17
points. This is a bit like the 3 x 3 problem in that one must go outside the array.
R. E. Miller & J. L. Selfridge. Maximal paths on rectangular boards. IBM J. Research and
Development 4:5 (Nov 1960) 479-486. They study rook's paths where a cell is deemed
visited if the rook changes direction there. They find maximal such paths in all cases.
Ripley's Puzzles and Games. 1966. Pp. 72-73, item 2. 4 x 4 cases with closed solution
symmetric both horizontally and vertically.
F. Gregory Hartswick. In: H. A. Ripley & F. Gregory Hartswick, Detectograms and Other
Puzzles, Scholastic Book Services, NY, 1969. Prob. 4, pp. 42-43 & 82. Asks for 8 x 8
rook's circuit with minimal turning and having a turn at a central cell. Solution gives
two such with 16 segments and asserts there are no others.
Doubleday - 1. 1969. Prob. 60: Test case, pp. 75 & 167. = Doubleday - 4, pp. 83-84. Two
3 x 3 arrays joined at a corner, looking like the Fore and Aft board (cf 5.R.3), to be
covered in a minimum number of segments. He does it in seven segments by joining
two 3 x 3 solutions.
Brian R. Barwell. Arrows and circuits. JRM 2 (1969) 196-204. Introduces idea of maximal
length rook's tours. Shows the maximal length on a 4 x 4 board is 38 and finds there
are 3 solutions. Considers also the 1 x n board.
Solomon W. Golomb & John L. Selfridge. Unicursal polygonal paths and other graphs on
point lattices. Pi Mu Epsilon J. 5 (1970) 107-117. Surveys problem. Generalizes
Selfridge's 1955 proof to M x N for which MIN(2M, M+N-2) segments occur in a
minimal circuit.
Doubleday - 2. 1971. Path finder, pp. 95-96. Bishop's corner to corner path, same as
Dudeney, 1932.
James F. Fixx. More Games for the Superintelligent. (Doubleday, 1972); Muller, (1977),
1981. 6. Variation on a variation, pp. 31 & 87. Trick solution in three lines, assuming
points of finite size.
M. Gardner. SA (May 1973) c= Knotted, chap. 6. Prob. 1: Find rook's tours of maximum
length on the 4 x 4 board. Cites Barwell. Knotted also cites Peters, below.
Edward N. Peters. Rooks roaming round regular rectangles. JRM 6 (1973) 169-173. Finds
maximum length on 1 x N board is N2/2 for N even; (N-1)2/2 + N-1 for N odd,
and believes he has counted such tours. He finds tours on the N x N board whose
length is a formula that reduces to 4 BC(N+1, 3) - 2[(N-1)/2]. I am a bit unsure if he
has shown that this is maximal.
James L. Adams. Conceptual Blockbusting. Freeman, 1974, pp. 16-22. 3rd ed., (A-W,
1986), Penguin, 1987, pp. 24-33. Trick solution of 3 x 3 case in three lines, assuming
SOURCES - page 250
points of finite size, which he says was submitted anonymously when he and Bob
McKim used the puzzle on an ad for a talk on problem-solving at Stanford. Also
describes a version using paperfolding to get all nine points into a line. The material is
considerably expanded in the 3rd ed. and adds several new versions. From the
references in Piggins and Eley, it seems that these all appeared in the 2nd ed of 1979 -??NYS.
Cut out the 3 x 1 parts and tape them into a straight line.
Take the paper and roll it to a cylinder and then draw a slanting line on the
cylinder which goes through all nine, largish, points.
Cut out bits with each point on and skewer the lot with a pencil.
Place the paper on the earth and draw a line around the earth to go through all
nine points. One has to assume the points have some size.
Wodge the paper, with large dots, into a ball and stick a pencil through it. Open
up to see if you have won -- if not, try again!
Use a very fat line, i.e. as thick as the spacing between the edges of the array.
David J. Piggins. Pathological solutions to a popular puzzle. JRM 8:2 (1975-76) 128-129.
Gives two trick solutions.
Three parallel lines, since they meet at infinity.
Put the figure on the earth and use a slanting line around the earth. This works in
the limit, but otherwise requires points of finite size, a detail that he doesn't mention.
No references for these versions.
David J. Piggins & Arthur D. Eley. Minimal path length for covering polygonal lattices: A
review. JRM 14:4 (1981-82) 279-283. Mostly devoted to various trick solutions of the
3 x 3 case. They cite Piggins' solution with three parallel lines. They say that Gardner
sent them the trick solution in 1973 and then cite Adams, 1979. They give solutions
using points of different sizes, getting both three and two segment solutions and
mention a two segment version that depends on the direction of view. They then give
the solution on a sphere, citing Adams, 1979, and Piggins. They give several further
versions using paper folding, including putting the surface onto a twisted triangular
prism joined at the ends to make the surfaces into a Möbius strip -- Zeeman calls this a
umbilical bracelet or a Möbius bar.
Obermair. Op. cit. in 5.Z.1. 1984.
Prob. 19, pp. 23 & 50. Bishop's path on 8 x 8 in 17 segments, as in Dudeney, PCP,
1932.
Prob. 41, p. 72. Rook's path with maximal number of segments, which is 57. [For the
2 x 2, 3 x 3, 4 x 4 boards, I get the maximum numbers are 3, 6, 13.]
Nob Yoshigahara. Puzzlart. Tokyo, 1992. Section: The wisdom of Solomon, pp. 40-47,
abridged from an article by Solomon W. Golomb in Johns Hopkins Magazine (Oct
1984). Classic 3 x 3 problem. For the 4 x 4 case: 1) find four closed paths; 2), says
there are about 30 solutions and gives 19 beyond the previous 4. Find the unique
5-segment closed path on the 3 x 4. Gives 3 solutions on 5 x 5. 10-segment solution
on 6 x 6 which stays on the board. Loyd's 1867? Queen's circuit. Queen's circuit on
7 x 7, attributed to Dudeney, though my earliest entry is Loyd, 1914 -- ??CHECK.
6.AL.
STEINER-LEHMUS THEOREM
This has such an extensive history that I will give only a few items.
C. L. Lehmus first posed the problem to Jacob Steiner in 1840.
Rougevin published the first proof in 1842. ??NYS.
Jacob Steiner. Elementare Lösung einer Aufgabe über das ebene und sphärische Dreieck.
J. reine angew. Math. 28 (1844) 375-379 & Tafel III. Says Lehmus sent it to him in
1840 asking for a purely geometric proof. Here he gives proofs for the plane and the
sphere and also considers external bisectors.
Theodor Lange. Nachtrag zu dem Aufsatze in Thl. XIII, Nr. XXXIII. Archiv der Math. und
Physik 15 (1850) 221-226. Discusses the problem and gives a solution by Steiner and
two by C. L. Lehmus. Steiner also considers the external bisectors.
N. J. Chignell. Note 1031: A difficult converse. MG 16 (No. 219) (Jul 1932) 200-202. [The
author's name is omitted in the article but appears on the cover.] 'Three fairly simple
proofs', due to: M. J. Newell; J. Travers, improving J. H. Doughty, based on material
in Lady's and Gentleman's Diary (1859) 87-88 & (1860) 84-86; Wm. Mason, found by
SOURCES - page 251
Doughty, in Lady's and Gentleman's Diary (1860) 86.
H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Section 1.5, ex. 4, p. 16. An easy
proof is posed as a problem with adequate hints in four lines.
M. Gardner. SA (Apr 1961) = New MD, chap. 17. Review of Coxeter's book, saying his brief
proof came as a pleasant shock.
G. Gilbert & D. MacDonnell. The Steiner-Lehmus theorem. AMM 70 (1963) 79-80. This is
the best of the proofs sent to Gardner in response to his review of Coxeter. A later
source says this turned out to be identical to Lehmus' original proof!
Léo Sauvé. The Steiner-Lehmus theorem. CM 2:2 (Feb 1976) 19-24. Discusses history and
gives 22 references, some of which refer to 60 proofs.
Charles W. Trigg. A bibliography of the Steiner-Lehmus theorem. CM 2:9 (Nov 1976)
191-193. 36 references beyond Sauvé's.
David C. Kay. Nearly the last comment on the Steiner-Lehmus theorem. CM 3:6 (1977)
148-149. Observes that a version of the proof works in all three classical geometries at
once and gives its history.
6.AM.
MORLEY'S THEOREM
This also has an extensive history and I give only a few items.
T. Delahaye and H. Lez. Problem no. 1655 (Morley's triangle). Mathesis (3) 8 (1908)
138-139. ??NYS.
E. J. Ebden, proposer; M. Satyanarayana, solver. Problem no. 16381 (Morley's theorem).
The Educational Times (NS) 61 (1 Feb 1908) 81 & (1 Jul 1908) 307-308 = Math.
Quest. and Solutions from "The Educational Times" (NS) 15 (1909) 23. Asks for
various related triangles formed using interior and exterior trisectors to be shown
equilateral. Solution is essentially trigonometric. No mention of Morley.
Frank Morley. On the intersections of the trisectors of the angles of a triangle. (From a letter
directed to Prof. T. Hayashi.) J. Math. Assoc. of Japan for Secondary Education 6
(Dec 1924) 260-262. (= CM 3:10 (Dec 1977) 273-275.
Frank Morley. Letter to Gino Loria. 22 Aug 1934. Reproduced in: Gino Loria; Triangles
équilatéraux dérivés d'un triangle quelconque. MG 23 (No. 256) (Oct 1939) 364-372.
Morley says he discovered the theorem in c1904 and cites the letter to Hayashi. Loria
mentions other early work and gives several generalizations.
H. F. Baker. Note 1476: A theorem due to Professor F. Morley. MG 24 (No. 261) (Oct
1940) 284-286. Easy proof and reference to other proofs. He cites a related result of
Steiner.
Anonymous [R. P.] Morley's trisector theorem. Eureka 16 (Oct 1953) 6-7. Short proof,
working backward from the equilateral triangle.
Dan Pedoe. Notes on Morley's proof of his theorem on angle trisectors. CM 3:10 (Dec 1977)
276-279. "... very tentative ... first steps towards the elucidation of his work."
C. O. Oakley & Charles W. Trigg. A list of references to the Morley theorem. CM 3:10
(Dec 1977) 281-290 & 4 (1978) 132. 169 items.
André Viricel (with Jacques Bouteloup). Le Théorème de Morley. L'Association pour le
Développement de la Culture Scientifique, Amiens, 1993. [This publisher or this book
was apparently taken over by Blanchard as Blanchard was selling copies with his label
pasted over the previous publisher's name in Dec 1994.] A substantial book (180pp) on
all aspects of the theorem. The bibliography is extremely cryptic, but says it is abridged
from Mathesis (1949) 175 ??NYS. The most recent item cited is 1970.
6.AN.
VOLUME OF THE INTERSECTION OF TWO CYLINDERS
Archimedes. The Method: Preface, 2. In: T. L. Heath; The Works of Archimedes, with a
supplement "The Method of Archimedes"; (originally two works, CUP, 1897 & 1912)
= Dover, 1953. Supplement, p. 12, states the result. The proof is lost, but pp. 48-51
gives a reconstruction of the proof by Zeuthen.
Liu Hui. Jiu Zhang Suan Chu Zhu (Commentary on the Nine Chapters of the Mathematical
Art). 263. ??NYS -- described in Li & Du, pp. 73-74 & 85. He shows that the ratio of
the volume of the sphere to the volume of Archimedes' solid, called mou he fang gai
(two square umbrellas), is π/4, but he cannot determine either volume.
Zu Geng. c500. Lost, but described in: Li Chunfeng; annotation to Jiu Zhang (= Chiu Chang
SOURCES - page 252
Suan Ching) made c656. ??NYS. Described on pp. 86-87 of: Wu Wenchun; The
out-in complementary principle; IN: Ancient China's Technology and Science;
compiled by the Institute of the History of Natural Sciences, Chinese Academy of
Sciences; Foreign Languages Press, Beijing, 1983, pp. 66-89. [This is a revision and
translation of parts of: Achievements in Science and Technology in Ancient China [in
Chinese]; China Youth Publishing House, Beijing(?), 1978.]
He considers the shape, called fanggai, within the natural circumscribed cube and
shows that, in each octant, the part of the cube outside the fanggai has cross section of
area h2 at distance h from the centre. This is equivalent to a tetrahedron, whose
volume had been determined by Liu, so the excluded volume is ⅓ of the cube.
Li & Du, pp. 85-87, and say the result may have been found c480 by Zu Geng's
father, Zu Chongzhi.
Lam Lay-Yong & Shen Kangsheng. The Chinese concept of Cavalieri's Principle and its
applications. HM 12 (1985) 219-228. Discusses the work of Liu and Zu.
Shiraishi Chōchū. Shamei Sampu. 1826. ??NYS -- described in Smith & Mikami, pp. 233236. "Find the volume cut from a cylinder by another cylinder that intersects is
orthogonally and touches a point on the surface". I'm not quite sure what the last phrase
indicates. The book gives a number of similar problems of finding volumes of
intersections.
P. R. Rider, proposer; N. B. Moore, solver. Problem 3587. AMM 40 (1933) 52 (??NX) &
612. Gives the standard proof by cross sections, then considers the case of unequal
cylinders where the solution involves complete elliptic integrals of the first and second
kinds. References to solution and similar problem in textbooks.
Leo Moser, solver; J. M. Butchart, extender. MM 25 (May 1952) 290 & 26 (Sep 1952) 54.
??NX. Reproduced in Trigg, op. cit. in 5.Q: Quickie 15, pp. 6 & 82-83. Moser gives
the classic proof that V = 16r3/3. Butchart points out that this also shows that the shape
has surface area 16r2.
6.AO.
CONFIGURATION PROBLEMS
NOTATION: (a, b, c) denotes the configuration of a points in b rows of c each.
The index below covers articles other than the surveys of Burr et al. and Gardner.
(
(
(
(
(
(
5, 2, 3):
Sylvester
6, 3, 3):
Mittenzwey
7, 6, 3):
Criton
9, 8, 3):
Sylvester; Carroll; Criton
9, 9, 3):
Carroll; Bridges; Criton
9, 10, 3):
Jackson; Family Friend; Parlour Pastime; Magician's Own Book;
The Sociable; Book of 500 Puzzles; Charades etc.; Boy's Own Conjuring Book;
Hanky Panky; Carroll; Crompton; Berkeley & Rowland; Hoffmann;
Dudeney (1908); Wehman; Williams; Loyd Jr; Blyth; Rudin; Young World;
Brooke; Putnam; Criton
(10, 5, 4):
The Sociable; Book of 500 Puzzles; Carroll; Hoffmann;
Dudeney (1908); Wehman; Williams; Dudeney (1917); Blyth; King; Rudin; Young
World; Hutchings & Blake; Putnam
(10, 10, 3):
Sylvester
(11, 11, 3):
The Sociable; Book of 500 Puzzles; Wehman
(11, 12, 3):
Hoffmann; Williams; Young World
(11, 13, 3):
Prout
(11, 16, 3):
Wilkinson -- in Dudeney (1908 & 1917); Macmillan
(12, 4, 5) -- Trick version of a hollow 3 x 3 square with doubled corners, as in 7.Q:
Family Friend (1858); Secret Out; Illustrated Boy's Own Treasury;
(12, 6, 4):
Endless Amusement II; The Sociable; Book of 500 Puzzles;
Boy's Own Book; Cassell's; Hoffmann; Wehman; Rudin; Criton
(12, 7, 4) -- Trick version of a 3 x 3 square with doubled diagonal: Secret Out;
Hoffmann (1876); Mittenzwey; Hoffmann (1893), no. 8
(12, 7, 4):
Dudeney (1917); Putnam
(12, 19, 3):
Macmillan
(13, 9, 4):
Criton
(13, 12, 3):
Criton
SOURCES - page 253
(13, 18, 3):
Sylvester
(13, 22, 3):
Criton
(15, 15, 3):
Jackson
(15, 16, 3):
The Sociable; Book of 500 Puzzles; H. D. Northrop; Wehman
(15, 23, 3):
Jackson
(15, 26, 3):
Woolhouse
(16, 10, 4):
The Sociable; Book of 500 Puzzles; Hoffmann; Wehman
(16, 12, 4):
Criton
(16, 15, 4):
Dudeney (1899, 1902, 1908); Brooke; Putnam; Criton
(17, 24, 3):
Jackson
(17, 28, 3):
Endless Amusement II; Pearson
(17, 32, 3):
Sylvester
(17, 7, 5):
Ripley's
(18, 18, 4):
Macmillan
(19, 19, 4):
Criton
(19, 9, 5):
Endless Amusement II; The Sociable; Book of 500 Puzzles; Proctor;
Hoffmann; Clark; Wehman; Ripley; Rudin; Putnam; Criton
(19, 10, 5):
Proctor
(20, 12, 5):
trick method: Doubleday - 3
(20, 18, 4):
Loyd Jr
(20, 21, 4):
Criton
(21, 9, 5):
Magician's Own Book; Book of 500 Puzzles; Boy's Own Conjuring Book;
Blyth; Depew
(21, 10, 5):
Mittenzwey
(21, 11, 5):
Putnam
(21, 12, 5):
Dudeney (1917); Criton
(21, 30, 3):
Secret Out; Hoffmann
(21, 50, 3):
Sylvester
(22, 15, 5):
Macmillan
(22, 20, 4):
Dudeney (1899)
(22, 21, 4):
Dudeney (1917); Putnam
(24, 28, 3):
Jackson; Parlour Pastime
(24, 28, 4):
Jackson; Héraud; Benson; Macmillan
(24, 28, 5):
Jackson
(25, 12, 5):
Endless Amusement II; Young Man's Book; Proctor; Criton
(25, 18, 5):
Bridges
(25, 30, 4):
Macmillan
(25, 72, 3):
Sylvester
(26, 21, 5):
Macmillan
(27, 9, 6):
The Sociable; Book of 500 Puzzles; Hoffmann; Wehman
(27, 10, 6):
The Sociable; Book of 500 Puzzles; Wehman
(27, 15, 5):
Jackson
(29, 98, 3):
Sylvester
(30, 12, 7):
Criton
(30, 22, 5):
Criton
(30, 26, 5):
Macmillan
(31, 6, 6) -- with 7 circles of 6: The Sociable; Book of 500 Puzzles;
Magician's Own Book (UK version); Wehman
(31, 15, 5):
Proctor
(36, 55, 4):
Macmillan
(37, 18, 5):
Proctor
(37, 20, 5):
The Sociable; Book of 500 Puzzles; Illustrated Boy's Own Treasury;
Hanky Panky; Wehman
(49, 16, 7):
Criton
Trick versions -- with doubled counters: Family Friend (1858); Secret Out;
Illustrated Boy's Own Treasury; Hoffmann (1876); Mittenzwey;
Hoffmann (1893), nos. 8 & 9; Pearson; Home Book ....; Doubleday - 3. These could
also be considered as in 7.Q.2 or 7.Q.
A different type of configuration problem is considered by Shepherd, 1947.
SOURCES - page 254
Jackson. Rational Amusement. 1821. Trees Planted in Rows, nos. 1-10, pp. 33-34 & 99-100
and plate IV, figs. 1-9. [Brooke and others say this is the earliest statement of such
problems.]
1. (9, 10, 3). Quoted in Burr, below.
"Your aid I want, nine trees to plant
In rows just half a score;
And let there be in each row three.
Solve this: I ask no more."
2. (n, n, 3), He does the case n = 15.
3. (15, 23, 3).
4. (17, 24, 3).
5. (24, 24, 3) with a pond in the middle.
6. (24, 28, 4).
7. (27, 15, 5)
8. (25, 28, c) with c = 3, 4, 5.
9. (90, 10, 10) with equal spacing -- decagon with 10 trees on each side.
10. Leads to drawing square lattice in perspective with two vanishing points, so the
diagonals of the resulting parallelograms are perpendicular.
Endless Amusement II. 1826?
Prob. 13, p. 197. (19, 9, 5). = New Sphinx, c1840, p. 135.
Prob. 14, p. 197. (12, 6, 4). = New Sphinx, c1840, p. 135.
Prob. 26, p. 202. (25, 12, 5). Answer is a 5 x 5 square array.
Ingenious artists, how may I dispose
Of five-and-twenty trees, in just twelve rows;
That every row five lofty trees may grace,
Explain the scheme -- the trees completely place.
Prob. 35, p. 212. (17, 28, 3). [This is the problem that is replaced in the 1837 ed.]
Young Man's Book. 1839. P. 239. Identical to Endless Amusement II.
Crambrook. 1843. P. 5, no. 15: The Puzzle of the Steward and his Trees. This may be a
configuration problem -- ??
Boy's Own Book. 1843 (Paris): 438 & 442, no. 15: "Is it possible to place twelve pieces of
money in six rows, so as to have four in each row?" I. e. (12, 6, 5). = Boy's Treasury,
1844, pp. 426 & 429, no. 13. = de Savigny, 1846, pp. 355 & 358, no. 11.
Family Friend 1 (1849) 148 & 177. Family Pastime -- Practical Puzzles -- 1. The puzzle of
the stars. (9, 10, 3).
Friends of the Family Friend, pray show
How you nine stars would so bestow
Ten rows to form -- in each row three -Tell me, ye wits, how this can be?
Robina.
Answer has
Good-tempered Friends! here nine stars see:
Ten rows there are, in each row three!
W. S. B. Woolhouse. Problem 39. The Mathematician 1 (1855) 272. Solution: ibid. 2
(1856) 278-280. ??NYS -- cited in Burr, et al., below, who say he does (15, 26, 3).
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868.
Mechanical puzzles.
No. 1, p. 176 (1868: 187). (9, 10, 3).
Ingenious artist pray disclose,
How I nine trees can so dispose,
That these ten rows shall formed be,
And every row consist of three?
No. 12, p. 182 (1868: 192-193). (24, 28, 3), but with a central pond breaking 4 rows of
6 into 8 rows of 3.
Magician's Own Book. 1857.
Prob. 33: The puzzle of the stars, pp. 277 & 300. (9, 10, 3),
Friends one and all, I pray you show
How you nine stars would so bestow,
Ten rows to form -- in each row three -Tell me, ye wits, how this can be?
Prob. 41: The tree puzzle, pp. 279 & 301. (21, 9, 5), unequally spaced on each row.
SOURCES - page 255
Identical to Book of 500 Puzzles, prob. 41.
The Sociable. 1858. = Book of 500 Puzzles, 1859, with same problem numbers, but page
numbers decreased by 282.
Prob. 3: The practicable orchard, pp. 286 & 302. (16, 10, 4).
Prob. 8: The florist's puzzle, pp. 289 & 303-304. (31, 6, 6) with 7 circles of 6.
Prob. 9: The farmer's puzzle, pp. 289 & 304. (11, 11, 3).
Prob. 12: The geometrical orchard, p. 291 & 306. (27, 9, 6).
Prob. 17: The apple-tree puzzle, pp. 292 & 308. (10, 5, 4).
Prob. 22: The peach orchard puzzle, pp. 294 & 309. (27, 10, 6).
Prob. 26: The gardener's puzzle, pp. 295 & 311. (12, 6, 4) two ways.
Prob. 27: The circle puzzle, pp. 295 & 311. (37, 20, 5) equally spaced along each row.
Prob. 29: The tree puzzle, pp. 296 & 312. (15, 16, 3) with some bigger rows. Solution
is a 3 x 4 array with three extra trees halfway between the points of the middle
line of four.
Prob. 32: The tulip puzzle, pp. 296 & 314. (19, 9, 5).
Prob. 36: The plum tree puzzle, pp. 297 & 315. (9, 10, 3).
Family Friend (Dec 1858) 359. Practical puzzles -- 2. "Make a square with twelve counters,
having five on each side." (12, 4, 5). I haven't got the answer, but presumably it is the
trick version of a hollow square with doubled corners, as in 7.Q. See Secret Out, 1859
& Illustrated Boy's Own Treasury, 1860.
Book of 500 Puzzles. 1859. Prob. 3, 9, 12, 17, 22, 26, 27, 29, 32, 36 are identical to those in
The Sociable, with page numbers decreased by 282.
Prob. 33: The puzzle of the stars, pp. 91 & 114. (9, 10, 3), identical to Magician's Own
Book, prob. 33.
Prob. 41: The tree puzzle, pp. 93 & 115. (21, 9, 5), identical to Magician's Own Book,
prob. 41. See Illustrated Boy's Own Treasury.
The Secret Out. 1859.
To place twelve Cards in such a manner that you can count Four in every direction,
p. 90. (12, 7, 4) trick of a 3 x 3 array with doubling along a diagonal. 'Every
direction' must refer to just the rows and columns, but one diagonal also works.
The magical arrangement, pp. 381-382 = The square of counters, (UK) p. 9. (12, 4, 5)
-- trick version. Same as Family Friend & Illustrated Boy's Own Treasury,
prob. 13.
The Sphynx, pp. 385-386. (21, 30, 3). = Hoffmann, no. 15.
Charades, Enigmas, and Riddles. 1860: prob. 13, pp. 58 & 61; 1862: prob. 13, pp. 133 &
139; 1865: prob. 557, pp. 105 & 152. (9, 10, 3). (The 1862 and 1865 have slightly
different typography.)
Sir Isaac Newton's Puzzle (versified).
Ingenious Artist, pray disclose
How I, nine Trees may so dispose,
That just Ten Rows shall planted be,
And every Row contain just Three.
Boy's Own Conjuring Book. 1860.
Prob. 40: The tree puzzle, pp. 242 & 266. (21, 9, 5), identical to Magician's Own
Book, prob. 41.
Prob. 42: The puzzle of the stars, pp. 243 & 267. (9, 10, 3), identical to Magician's
Own Book, prob. 33, with commas omitted.
Illustrated Boy's Own Treasury. 1860.
Prob. 2, pp. 395 & 436. (37, 20, 5), equally spaced on each row, identical to The
Sociable, prob. 27.
Prob. 13, pp. 397 & 438. "Make a square with twelve counters, having five on each
side." (12, 4, 5). Trick version of a hollow square with doubled corners.
Presumably identical to Family Friend, 1858. Same as Secret Out.
J. J. Sylvester. Problem 2473. Math. Quest. from the Educ. Times 8 (1867) 106-107. ??NYS
-- Burr, et al. say he gives (10, 10, 3), (81, 800, 3) and (a, (a-1)2/8, 3).
Magician's Own Book (UK version). 1871. The solution to The florist's puzzle (The
Sociable, prob. 8) is given at the bottom of p. 284, apparently to fill out the page as
there is no relevant text anywhere.
Hanky Panky. 1872.
To place nine cards in ten rows of three each, p. 291. I.e. (9, 10, 3).
Diagram with no text, p. 128. (37, 20, 5), equal