BookReview2011.docx for IMAGE
Science, Recreational Mathematics and Magic.
A review of
“Before Sudoku – The World of Magic Squares”,
by Seymour S. Block and Santiago A. Tavares, 2009, Oxford.
239 pages. ISBN:
Reviewed by Peter D. Loly
Department of Physics and Astronomy, University of Manitoba.
loly@cc.umanitoba.ca; http://home.cc.umanitoba.ca/~loly/
The authors, both chemical engineers, have written a concise survey of a fascinating
family of integer number squares. Readers of this magazine will have a special
interest in the doubly affine square arrays through nth order Latin squares with
elements 1..n repeated n times with each once in every row and column so that they
have the same line sum of elements in all Rows and Columns but not necessarily for
the diagonals (RC squares). Ninth order Sudoku solutions have the additional
constraint that each appears only once in each of the nine adjoining 3-by-3
subsquares.
By contrast natural magic squares with elements 1..n*n with both Diagonals having
the same linesum as the rows and columns) are RCD squares. Latin squares and
magic squares are all doubly affine integer matrices which often exhibit more
elegant results than real number matrices, e.g. their determinant vanishes in
singular cases. Some Latin squares are RCD, but with repeated elements. Since the
widespread reappearance of Sudoku puzzles in 2005, they have been used as a pivot
to the study of magic squares, e.g. Arno van den Essen in “Magische vierkanten: van
Lo-Shu tot sudoku”, 2008, and of course the present authors, Block and Tavares,
hereafter: B&T.
The early history of magic squares dates back about two and a half millennia to
China (though there is a certain fondness for legends that suggest four millennia),
where the 3-by-3 Lo Shu [Luoshu, ...] was contemporaneous with the binary Yin-Yang
trigrams (Loly, " A Logical Way of Ordering the Trigrams and Hexagrams of the
Yijing." The Oracle - The Journal of Yijing Studies, vol. 2, no. 12, January 2002, p.2-
13), and is thought to have been passed along and extended through India and
Persia by itinerant scholars until they reached the Mediterranean area by the 9th
century. In the tenth century Islamic scholars wrote lengthy texts about magic
squares up to order nine (9-by-9), with another recorded in a Chinese text of 1275
C.E., and a Latin squares of the same order is found in a XII-XIII century manuscript
by Albuni. [Descombes, “Les Carrés Magiques”, Vuibert, 2000] From the 14th
century on magic squares attracted the attention of Manuel Moschopoulos, Agrippa
of Nettlesheim, Albrecht Dϋrer, and many others.
The past few centuries have seen a growing literature of archived works and more
recently the volume of self-published material has grown exponentially with the
advent of the World Wide Web. Now recreational mathematicians interact with the
authors of archived articles via email. So it is helpful to have this addition to a
number of recent books by other scholars.
The present work consists of a rather personal selection of topics, but is a useful
addition to recreational mathematics, with substantial sections not covered in
previous treatments. B&T Tavares include extensive discussion of magic squares and
art in chapter 10 where earlier connected line patterns have been extended
considerably by them. There is also a welcome discussion of musical renderings of
magic squares.
While published in 2009 B&T reference contributions up to 2007, but omit progress
around 2006 on Franklin bent diagonal squares which has now been covered by van
den Essen (see above). Also missing is the even earlier landmark work of Dame
Kathleen Ollerenshaw and David Brée (1997) on most-perfect pandiagonal magic
squares., as well as the epic statistical population estimates of magic squares for
orders 6 through 10 by Walter Trump [http://www.trump.de/magicsquares/howmany.html]. There is also no mention of compound or composite magic
squares [W. Chan and P. D. Loly, "Iterative Compounding of Square Matrices to Generate
Large-Order Magic Squares", Mathematics Today, 38(4), 113-118, August 2002.]
Incidentally Block and Tavares’s Figure 7.9 order 4 Franklin square is found earlier
in their own Figure 5.8 by Dϋrer in 1514. A tad before Ben’s birth!
A measure of the amateur activity concerned with magic squares may be seen from
the excellent web site of Harvey Heinz: http://www.magic-squares.net/ from which
many other sites can be reached. B&T also include some results from Harvey’s
prolific colleague John Hendricks. The latter encouraged the present reviewer in his
first venture into these waters with a non-magic pandiagonal square [P. D. Loly. "A
purely pandiagonal 4*4 square and the Myers-Briggs Type Table." J. Rec. Math., 31(1), 29-31,
2000/2001].
Clearly the interpretation of Latin and magic squares as matrices opens up many
interesting mathematical explorations, not the least as a vehicle for learning about
eigenproperties, e.g. using MATLAB’s magic(n) function. Progress continues to be
made in scientific studies of magic squares, much of which can be reached from a
paper by the present reviewer and his colleagues: Loly, Cameron, Trump and
Schindel, "Magic square spectra", Linear Algebra and its Applications, 430 (2009)
2659-2680. That work laid the foundations for the use of singular value
decomposition as a critical step in comparing and indexing magical squares [TBA].
While Latin squares have well known design applications, the authors also discuss
some uses of magic squares in chapter ? Readers of IMAGE may be interested to
know that that George Styan and this reviewer published two comments on 4x4 and
5x5 Philatelic Latin Squares in CHANCE [vol. 23, no. 1 & 2, both pp. 57-62, 2010].
This is a book that I am keeping in my library and which I recommend as a good
starting point for newcomers to the field for the variety of magical squares
introduced, however it contains no linear significant algebra.
Contents:
Ch1
Ch2
Ch3
Ch4
Introduction
Magic Squares and Sudoku
History of Sudoku
Some Techniques for Solving Sudoku Puzzles
What Is Sudoku?
Solving the Sudoku
The Naked Double
Another Option
Ch5 History of Magic Squares
Magic Squares 4,000 Years Ago: China
India and the Middle East
Europe
America: Benjamin Franklin
Nineteenth, Twentieth, and Twenty-First Centuries
Ch6 When a Magic Square Is Not Square
Magic Triangles
Magic Rectangles
Magic Cubes
Magic Pyramids
Magic Circles
Magic Spheres
Magic Stars
Magic Pentagrams
1
3
6
11
11
11
16
17
20
20
22
25
28
30
33
33
34
38
42
44
46
47
53
Ch7
Ch8
Ch9
Ch10
Ch11
Magic Crosses
Magic Serrated Edge Figures
Magic Squares and Arithmetic
Magic Square Subtraction
Magic Square Multiplication
Magic Square Division
Addition-Multiplication Magic Square
Bimagic Square
Geometric Magic Square
Sequence of Three Magic Squares
Magic Square of Square Numbers
Power Magic Square
Exotic Magic Squares
Antimagic Squares
Reversible Magic Squares
Multiplication Reversible Magic Squares
Forward-Backward Magic Squares
Concentric (Bordered) Magic Squares
Universal Magic Squares
Annihilation Magic Squares
Palindromic Magic Squares
Concatenation Magic Squares
Alphamagic Squares
Domino Latin Squares
Magic Hexagonal Tiles
Other Magic Squares
Associated Magic Squares
Symmetrical Magic Squares
Complementary Magic Squares
Magic Square with Consecutive Pairs of Numbers
Latin Square Made Up of Magic Squares
Franklin Lightning Baseball: Three Strokes and You're Out
Magic Squares with 666 Magic Constant
Pandigital Magic Squares
Magic Square Olympics
A Magic Square to Show Off
Magic Squares in Art. Design, and Music
Art and Mathematics
Magic Squares and Art
Magic Squares and Music
The Fourth Dimension
Magic Square in Three Dimensions
Leaping into the Higher Dimensions
Zero-Dimensional Space
One-Dimensional Space
Two-Dimensional Space
56
64
71
71
72
72
74
74
78
82
83
84
88
88
91
92
94
98
99
99
100
101
105
107
107
109
109
111
112
112
113
114
116
120
121
122
125
125
130
159
168
168
169
172
173
174
Three-Dimensional Space
Four-Dimensional Space and Beyond
Final Comments on Tesseracts
Ch12 Practical Applications of Magic Squares
Magic Squares and Statistics
Error Correcting Codes
Combinatorics
The Future
Ch13 Some Puzzles for You
Magic Square Puzzles
Sudoku Puzzles
Ch14 Further Reading
Ch15 Magic Squares Terminology
Ch16 Solutions to the Puzzles
Magic Square Puzzles
Sudoku Puzzles
References
Index
175
178
186
187
187
192
194
194
196
196
201
202
204
209
209
223
225