Proofs Without Words
Exercises in Visual Thinking
© 1993 by
The Mathematical Association of America (Incorporated)
Library ofCongress Catalog Card Number 93-86388
ISBN 0-88385-700-6
Printedinthe United States ofAmerica
Current Printing (last digit):
9
9
£2
7
6
Introduction
see (sé) v., saw, seen, seeing. —v.t.
5. to perceive (things) mentally; discern;
understand: to see the point of an argument.
—THE RANDOM HOUSE DICTIONARY
OF THE ENGLISH LANGUAGE (24 Ep.)
UNABRIDGED.
“Proofs without words” (PWWs) have become regular features in
the journals published by the Mathematical Association of America —
notably Mathematics Magazine and The College Mathematics Journal.
PWWs began to appear in Mathematics Magazine about 1975, and, in
an editors’ note in the January 1976 issue of the Magazine, J. Arthur
Seebach and Lynn Arthur Steen encouraged further contributions of
PWWs to the Magazine. Although originally solicited for “use as endof-article fillers,” the editors went on to ask “What could be better for
this purpose than a pleasing illustration that made an important mathematical point?”
A few years earlier Martin Gardner, in his popular “Mathematical
Games” column in the October 1973 issue of the Scientific American,
discussed PWWs as “look-see” diagrams. Gardner points out that “in
many cases a dull proof can be supplemented by a geometric analogue
so simple and beautiful that the truth of a theorem is almost seen at a
glance.” This dramatically illustrates the dictionary quote above: in
English “to see” is often “to understand.”
In the same vein, the editorial policy of The College Mathematics
Journal throughout most of the 1980s stated that, in addition to expository articles, “The Journal also invites other types of contributions,
most notably: proofs without words, mathematical poetry, quotes, ...”
(their italics). But PWWs are not recent innovations — they have a
long history. Indeed, in this volume you will find modern renditions
of proofs without words from ancient China, classical Greece, and India
of the twelfth century.
vi
Proofs without Words
Of course, “proofs without words” are not really proofs. As
Theodore Eisenberg and Tommy Dreyfus note in their paper “On the
Reluctance to Visualize in Mathematics” [in Visualization in Teaching
and Learning Mathematics, MAA Notes Number 19], some consider
such visual arguments to be of little value, and “that there is one and
only one way to communicate mathematics, and ‘proofs without
words’ are not acceptable.” But to counter this viewpoint, Eisenberg
and Dreyfus go on to give us some quotes on the subject:
[Paul] Halmos, speaking of Solomon Lefschetz (editor of
the Annals), stated: “He saw mathematics not as logic but
as pictures.” Speaking of what it takes to be a mathematician, he stated: “To be a scholar of mathematics you must
be born with ... the ability to visualize” and most teachers
try to develop this ability in their students. [George]
Pélya’s “Draw a figure ...” is classic pedagogic advice, and
Einstein and Poincaré’s views that we should use our visual intuitions are well known.
So, if “proofs without words” are not proofs, what are they? As
you will see from this collection, this question does not have a simple,
concise answer. But generally, PWWs are pictures or diagrams that
help the observer see why a particular statement may be true, and also
to see how one might begin to go about proving it true. In some an
equation or two may appear in order to guide the observer in this process. But the emphasis is clearly on providing visual clues to the observer to stimulate mathematical thought.
I should note that this collection is not intended to be complete. It
does not include all PWWs which have appeared in print, but is rather
a sample representative of the genre. In addition, as readers of the Association’s journals are well aware, new PWWs appear in print rather
frequently, and I anticipate that this will continue. Perhaps some day a
second volume of PWWs will appear!
I hope that the readers of this collection will find enjoyment in discovering or rediscovering some elegant visual demonstrations of cer-
tain mathematical ideas; that teachers will want to share many of them
with their students; and that all will find stimulation and encouragement to try to create new “proofs without words.”
Introduction
vii
Acknowledgment. 1 would like to express my appreciation and
gratitude to the many people who have played a part in the publication
of this collection: to Gerald Alexanderson and Martha Siegel, who, as
editors of Mathematics Magazine, gave me encouragement over the
years as I learned to read and write PWWs; to Doris Schattschneider,
Eugene Klotz, and Richard Guy for sharing with me their collections of
PWWs;
and finally, to all those individuals who have contributed
“proofs without words” to the mathematical literature (see the Index of
Names on pp. 151-152), without whom this collection simply would
not exist.
Note. All the drawings in this collection were redone to create a
uniform appearance. In a few instances titles were changed, and shading or symbols were added (or deleted) for clarity. Any errors resulting
from that process are entirely my responsibility.
Roger B. Nelsen
Lewis and Clark College
Portland, Oregon
Contents
fo
Sa
eee
ee OED
Oe
eee
RR
TT Vv
Geometry & Algebra a nsessssssssssssssssnsoseccceseesssstnsencreceeeseressnnsnsnereceeeseseseseee 1
Trigonometry, Calculus & Analytic Geometry ........scscssssccsssssseusseeeneees 27
MECCA
EC
a
it castes ead
xsceectecis cecccscseusescossteee 47
ne Ber:Suns a
Re
SOQMEHOCS CISCOOS as ccscocieih
a
coccinea
ara 113
Pao
as el NN, scl 131
IMIS
Ce LOUIE i
at aetna co
SOWNCCS ie ceseeassess cts cee cies occa
Index of Names
cca belcbe ocala
a 67
tsb 145
151
Geometry & Algebra
The Pythagowean Theorem Vassnseunsnecssisensereisanenoosatensusseune 3
Wives PyPinnaggporeseins Tara Tics
aerated 4
The Pythagorean Theorem TT o...ssssssssssssssssssssssssssssssesessessesseccssesssesseeee 5
The Pythagorean Theorem IV oo .ccccssssscssssssscssssssssssssnsssssessssssscsnesnnsseseese 6
The Pythagorean Theomenn Vee
hve: Pythagoneatt Theorem VU
csseoecacenecssnrvvencersnvsovnnnseeensovne 7
a aenareecnsssmsevreeonrsnceeenereeesnpervmsse 8
A Pythagorean Theorem: -0' = D-b' + 0-C" iccscssssssssssccsssccsssscssnssusessssseees 9
The Rolling Circle Squares Itself oo. .ssssssscssssssssnssessssssesssssesseseessesenes 10
On Trisecting an Amgye oo... csessssssccssssssssssssscescesesesesenssseeseecessseusunnsesseeees 11
Trisection of an Angle in an Infinite Number of Steps oo... .ocssssssssssees 12
Trisection of a Line Segment .o.cesssscsssssscssssssssssussussesessseseeeseseeeseeeseesnsene 13
Completing: the Square :ssccccsscssossscssasscsencssnccntnchitaensartshabetarcoaseasstestansehioinn 19
Alfpebraic Areas Uacc.ccscaccsdscsucsittteuctatgasetsctchctonssesnsciumcascpccscattedersnuasbttscaaes 20
Nipebr
alcAreas M0 a scccscccsssisusenctsctscoeecnscasisias
cetacean noesoennctscecsqnantatnci2tsel 21
Diophantus of Alexandria’s “Sum of Squares” Identity 1.22
The Kth n-gonal Number .........-ssssesecccceessenoeeessssnnrseceesennnneseneesnnereesesseanare 23
The Volume of a Frustum of a Square Pyramid. oo.
ssssscccsssnseseseeesee 24
The Volume of a Hemisphere via Cavalieri’s Principle 2000... 25
Geometry & Algebra
3
The Pythagorean Theorem I
—adapted from the Chou pei suan.ching
(author unknown, circa B.c. 200?)
4
mmm
Proofs without Words
een
ce
ce
>
SR
EE
AT
ES
The Pythagorean Theorem II
Behold!
—Bhaskara (12 century)
Geometry & Algebra
=a
The Pythagorean Theorem III
—based on Euclid’s proof
6
RR
Proofs without Words
SR, ES
SR
A
SAR
ha
PS
FC
A
SE LSFE
The Pythagorean Theorem IV
—H. E. Dudeney (1917)
Geometry & Algebra
_ 7
The Pythagorean Theorem V
A=2: Fab +i¢ =1a+by
Cf =a" +h"
—James A. Garfield (1876)
20% President of the United States
8
a
a
Proofs without
eS Words
SN a
The Pythagorean Theorem VI
f£+a__b
b>
Oca
+h
=
—Michael Hardy
Geometry & Algebra
9
A Pythagorean Theorem: a-a’ = b-b’ + c-c’
—Enzo R. Gentile
10
a
ag
Proofs without Words
The Rolling Circle Squares Itself
\
—Thomas Elsner
Geometry & Algebra
11
On Trisecting an Angle
—Rufus Isaacs
12
a
Proofs without Words
Trisection of an Angle in an Infinite Number
of Steps
—Eric Kincanon
Trisection of a Line Segment
Cooled
De
—Scott Coble
4
Proofs without Words
The Vertex Angles of a Star Sum to 180°
—Fouad Nakhli
Geometry & Algebra
5
Viviani’s Theorem
The perpendiculars to the sides from a point on the boundary
or within an equilateral triangle add up to the height
of the triangle.
—Samuel Wolf
16
Proofs without Words
A Theorem About Right Triangles
The internal bisector of the right angle of a right triangle
bisects the square on the hypotenuse.
—Roland H. Eddy
Geometry
& Algebra
17
Area and the Projection Theorem
of a Right Triangle
—Sidney H. Kung
18
Proofs without Words
Chords and Tangents of Equal Length
If circle C, passes through the center O of circle C,, the length
of the common chord PQ is equal to the tangent segment PR.
—Roland H. Eddy
Geometry & Algebra
__ 19
Completing the Square
x +ax = (x+a/2)?-(@/2)?
—Charles D. Gallant
Proofs without Words
20
Algebraic Areas I
(a+b)?
+(a—b)* = 2a +b’)
—Shirley Wakin
ey
N
NS
See
a
BSS
\\
—Sam Pooley and K. Ann Drude
Proofs without Words
22
N
N
N
Diophantus of Alexandria’s “Sum of Squares”
Identity
(a2 + b?)\(c2 + 2) = (ad + bc)* + (bd - aac)?
—RBN
Geometry & Algebra
23.
The kth n-gonal Number is
1+ (k-1)(n-1) +5 (k-2k-)(n-2)
—Dave Logothetti
24
eS
Proofs without Words
The Volume of a Frustum of a Square Pyramid
[Problem 14, The Moscow Papyrus, circa 1850 B.c.]
h
h
1
VP) = 54 VP) = bag")
= fa ab+P)
REFERENCES
1. C. B. Boyer, A History of Mathematics, John Wiley & Sons, New
York, 1968, pp. 20-22.
2. R. J. Gillings, Mathematics in the Time of the Pharaohs, The MIT
Press, Cambridge, 1972, pp. 187-193.
—RBN
4
Geometry & Algebra
The Volume of a Hemisphere via
Cavalieri’s Principle*
¢— 2n(r-h) +
|ee
eee
a
ng Chih
*Tzu Geng, son of the most celebrated mathematician Tzu Chu
ciple in
in ancient China, was believed to be the first to develop the prin
the 5th century A. D.
—Sidney H. Kung
Trigonometry, Calculus & Analytic Geometry
SINE) OF THO SU ies esseseesecccoss
Se csraeeiet 29
Area and Difference Formulas .....essssssssssssssssscssnssssssusessessnessenessesesese 30
TUT RW OE CORT
SB CR
AN
sc aecsctee
isin ancatnbemai
anetnaieicn
estes 31
cnpa
tan
sca
serindeg aan
arcana
amie 32
The Law of Cosines III (via Ptolemy’s Theorem) 00.0... ssssccccccscssssnsseseess 33
The Double-Angle Formulas ............csccsssssssssssssssscssssscssesssssssssssesssesssssessssessee 34
The Half-Angle Tangent Formulas ............esssssossssssssssecssessecesesssssssssssssseeeeee’ 35
Moll weide’s Equation ...........cccsssssscssssssccssssssscsssssssssssssssssssssssssssessssssssssssnsnenees 36
(tan + 1)? + (cotO +1)? = (SeCO + CSCO)e eesnesrtnernetnetnetneteeereee 37
The Substitution to Make a Rational Function of
the Sime and Coste o.......cccscconsossssssssssssseccsecesescsssessnsnnssseseseseeeeeeeses 38
Sums of Arctangemts _.sesessnonssnsssnsoocssssvssssosssctecsacccescassccencsesecssssercssesten 39
The Distance Between a Point and a Lime oo sesansosccccsseeesetsnteneeeere 40
The Midpoint Rule is Better than the Trapezoidal Rule
for Concave FUMCtions o.....ccsssssssssssssssssssceseseccecccssnsnnsussseseseeeseseees 41
Inte eration Dy Panes ccisecerssrdi tenement aerate
reali tcre
d b |42
The Graphs of f and f~! are Reflections about the Line y = x
The Reflection Property of the Parabola ...............scsssssssssssssssssssssssssssssssssssend 44
Area under an Arch of the Cycloidd oo... scsssssssssssecscssssecssssesssssssssessnesenses 45
29
Trigonometry, Calculus & Analytic Geometry
Sine of the Sum
sin(x+y) = sinxcosy+cosxsiny
for x+y<7
c=acosy+bcosx
r=1/2>sinz=(c/2)/(1/2)
=c, sinx=a, siny=b;
sin (x+y) =sin (W-(x+)) =sinz=sin x cos y + sin y cos x
—Sidney H. Kung
Area and Difference Formulas
x-y
sin(x-y) = sin x cos y—cosx
sin y
b
acosx
bsiny
cos(x—y) = cosxcosy+sinxsiny
—Sidney H. Kung
Trigonometry, Calculus & Analytic Geometry
31
The Law of Cosines I
2 = (bsin 6) + (a—b cos 6)"
=a’ +b" —2ab cos 6
—Timothy A. Sipka
32
easin
ess
Proofs without Words
faseence es
aicmecaennicans aviacneneine
The Law of Cosines II
(2a cos @-b)b = (a-c(a+c)
ce =? +b ~ 2b cos @
—Sidney H. Kung
Trigonometry, Calculus & Analytic Geometry
33
The Law Of Cosines III (via Ptolemy’s Theorem)
cc = bb+(a+2b cos(z- 6))-a
2 = a* +b* -2ab-cosd
—Sidney H. Kung
Proofs without Words
4
The Double-Angle Formulas
C(cos26, sin26)
AACD ~ AABC
‘CD/ AC = BC/ AB
‘AD
/ AC = AC/ AB
sin26/2cos@ = 2sin6/2
(1 + cos26)/2cos@ = 2cos6/2
sin2@ = 2sin@cos@
cos20 = 2cos2@-1
—RBN
Trigonometry, Calculus & Analytic Geometry
35
The Half-Angle Tangent Formulas
6
sin®
tan>= T+ cos6 =
1-cosé
sind
—R. J. Walker
36
Proofs without
—————
0
0S
Without Words
WOT:
Mollweide’s Equation
(a-b) cos ¥= csin (=)
—H. Arthur DeKleine
Trigonometry, Calculus & Analytic Geometry
37
(tan@ + 1)? + (cot + 1)? = (sec6+ csc6)*
tan26+1 = sec2@
cot2@+1 = csc20
(tan + 1)2 + (cot@+1)? = (sec@ + csc6)2
(iiss —s
daa
coté+1
—William Romaine
38
Proofs without Words
The Substitution to Make a Rational Function of
the Sine and Cosine
7
=)
a
CI
22
1
1
=>
1427
a
a
1
=>
z2
1+22
a
1-2?
9
Z= tanzz
:
2z
=> sind =z
I+q
z2
22
1-22
and cos0 =——
= 73
—RBN
Trigonometry, Calculus & Analytic Geometr
39
Sums of Arctangents
arctan1 +arctan2 +arctan3 = 7
—Edward M. Harris
The Distance Between a Point and a Line
(a,ma +c)
ma+c-bl
—R. L. Eisenman
Trigonometry, Calculus & Analytic Geometry
41
The Midpoint Rule is Better than the
Trapezoidal Rule for Concave Functions
—Frank Burk
roofs withou'
;
b
p
(pr)
b
Jfode'@dx = fladgta)| = Jstay '@dx
Trigonome'
, Calculus & Analytic Geometr
The Graphs of f and f-! are Reflections about
the Line y x
—Ayoub B. Ayoub
Eb
nr
The Reflection Property of the Parabola
QF=QD
&
mm=-1
=>
41=22=23
—Ayoub B. Ayoub
Trigonometry, Calculus & Analytic Geometr
45
Area under an Arch of the Cycloid
=> A=32R
—Richard M. Beekman
Inequalities
The Arithmetic Mean—Geometric Mean Inequality Tow.n....ssssscccsoesee: 49
The Arithmetic Mean—Geometric Mean Inequality Too. .ssssssssssee 50
The Arithmetic Mean—Geometric Mean Inequality TID ou...
sss 51
Two Extremmum Probbers ooo.o.nnaaaasssssssssssssssssesesssesessssnsssnsntnsscsnntneneneneeeeeeeee 52
The Harmonic Mean—Geometric Mean—Arithmetic Mean—
Root Mean Square Inequality Voo........cssscsssssscccconnssccccnnsesconseseceaneees 53
The Harmonic Mean—Geometric Mean—Arithmetic Mean—
Root Mean Square Inequality Wo...sssssscscececcceceessnsnmueesseeeeseeeee 54
The Harmonic Mean—Geometric Mean—Arithmetic Mean—
Root Mean Square Inequality Td ow ...ssssssssssscccsneecccennssseconneesesones 55
Five Means — and Their Means .............sesessssssscscesecsseeceecseceecesseseeseceseeesesee 56
BEE
IE ic sssccteoecpnuachncubeacuubietewensesaslnas
saat cxsstsrccotaescssocetsttnats casita datescati 58
ABS BAOR 6 SA Bice
dota tide
la tinned 59
The Mediant Property .oo.......csssssssccccconsssseececcscnsssseessccennunesteceecsennesssencennsnseseeee 60
Regle des Nombres MOoyenss ............ssssssssssssssssssssssssssesssssnunsnsssseeeseeseennenenssed 61
The Sum of a Positive Number and its Reciprocal
1S At LEASE TWO erasers
vcs inarancsnu slants
62
Aristarchus’ Inequalities ..0......ssssssccssssssececccesssnseeecccennsseeseceecsnsseeseceensneeessed 63
The Cauchy-Schwarz Inequality ...ssssssssssssssesssseessenseessensnssessncersteeneed 64
Bernoulli's Inequality’ 2. :asa...cssuicsscessacesscscuccscocrmstsectecssacesscssecenvesnsnccaouusesesed 65
Naapier’s Intequia ity cscs cacaicsscsoosscesscansseonceveecstcsscccntosassumnsssesienemnacnarneid 66
Inequalities
49
The Arithmetic Mean—Geometric
Mean
Inequality I
—Charles D. Gallant
Proofs without Words
50
The Arithmetic Mean—Geometric Mean
Inequality I
elk
Go wai
ait s hb
—Doris Schattschneider
Inequalities
51
The Arithmetic Mean—Geometric Mean
Inequality III
oa 2 Jab , with equality ifand only ifa=b
—
—
Jp ————>
—Roland H. Eddy
Two Extremum Problems
For a given product, the sum of two positive numbers
is minimal when the numbers are equal.
(2/P,0)
(S,0)
For a given sum, the product of two positive numbers
is maximal when the numbers are equal.
NOs
Qs
DX
RRS KT
VME
AY,
—Paolo Montuchi and Warren Page
53
Inequalities
The Harmonic Mean—Geometric Mean—
Arithmetic Mean—Root Mean Square
Inequality I
PM = a, QM = b, a>b>0
HM
< GM
< AM < RM
an < ab < ee < Vie +b2)/2
—RBN
The Harmonic
Mean—Geometric
Mean—
Arithmetic Mean—Root Mean Square
Inequality II
AB =a, BC=b
+
<<
AD=pc =2t8
2
BE.LAB, DE=AD
FELED, FB||ED
= pp 2-4
EG=BD=">
—Sidney H. Kung
Inequalities
55
The Harmonic Mean—Geometric
Mean—
Arithmetic Mean—Root Mean Square
Inequality III
ab>0=> V2 + )/2 ai, ab > atb
Zab
<—a —><—
b —>
2a + wv 2 (at wy
g+P,
2
arb
2
Wa+by > lato
tbs Sab
—RBN
_
Five Means — and Their Means
Arithmetic:
am = AM(a,b) = att
Contraharmonic:
a? +07
cm = CM(,b) = a+b
Geometric: gm = GM(a,b) = Vab
Harmonic:
hm = HM(a,b) = a
a
2
Root Mean Square: rms = RMS(a,b) =
~ rE
a
rms|--~----- ree
a+ re
ampon sono n:
2
'
a+
(a+b)
i
os ee '
hm foc sess 7 cece fusqece
z + ¥ ok
+ paasd
‘
ee
er
ed
ee
ee
a
hm gm
@m
rms
cm
b
Ine ualities
I.
57
0<a<b=
<
2c
Mab < 4
2
2.
+2
2.
arth
>
4,2
a
ae
a+b
hm +cm=a+b
= AM(hm, cm) =am.
Ill.
hmam=ab
=
IV.
2
am-cm = er
GM(hm, am) = gm.
=
GM(am, cm)
= rms.
2
gm? + rms? - oe => RMS(gm, rms)
=am.
GM
i;
2ab
ree
ee
GM
a+b
a+b
2
woo.
2.
<
ose,
42
a+b
a oan
2
4
—RBN
58
Proofs without Words
>
—Fouad Nakhii
Inequalities
59
AB> BA foresA<B
es<A<B
>
ma>mp
nA _InB
ra
= AB > BA
—Charles D. Gallant
60
Proofs without Words
The Mediant Property
—Richard A. Gibbs
Inequalities
61
Regle des Nombres Moyens (Two Proofs)
[Nicolas Chuquet, Le Triparty en la Science des Nombres, 1484]
a,b, c,d > 0; , a > 5 <b
< 7
I.
Mm,
< M3 => m, < Mm, < my
—Li Changming
Il.
>
+sclh
A
TE) > + ini
+
> + by
A
Alo
—RBN
Proofs without Words
62
The Sum of a Positive Number and its
Reciprocal is at least Two (Four Proofs)
I. ¥
x21>
1
x+ 722
—RBN
Inequalities
63
Aristarchus’ Inequalities
0<B<a<Z = Me, & .ane
oan
2 ~~ sinB B tanB
sina
a
tana
Ie acer gyt,SS psi Sear
sinB
B
tanB
—RBN
64
Proofs without Words
The Cauchy-Schwarz Inequality
Ka,b)<x,y)l < Ka,b)H IKx,y)Il
lA
lol
lal
iy
(al + Lyte + be) < 2Galibl + Sally) + Ya2 BYP oy?
lax + byl < lallxl +lbllyl < Va? + Bx? +y
—RBN
Inequalities
65
Bernoulli’s Inequality (two proofs)
x>0,x41,7r>1
=> x7-1>r(x-1)
I. (first semester calculus)
y
II. (second semester calculus)
0<x<1
x
v-1= [rt lat > re-1)
1
1
1-x = fr
at< n1-9
x
—RBN
Proofs without Words
66
Napier’s Inequality (two proofs)
0
1 Inb-lIna
1
boa>0>5<" fq
<a
I. (first semester calculus)
m(L,) < m(L,) < m(L,)
II. (Second semester calculus)
b
1
1
pe-a) < fgax < *o-0)
a
—RBN
Integer Sums
Sums’ Of Urnbegers Tasscssssscssesceceseocnasonnnsconessnocnenentncoveecenenecccenst 69
SCO)
2
2|
a
oo ae a
eae een
ee 70
Sums of Odd Integers Docsssssscssscsssssssesesssssssesssssssessssssssssssssssssssssssssssnee 71
SUMSIOF Odd Integers
ores
Sums of Odd Integers TI
oe i
rte
et
ag 72
ae cssscssssssssssssusesssseseececcsssnssunessssseeseceensnsnesses 73
Squares and Sums of Integers... .....ssessssssssssssssssssssssssssvsvevesnsessscsessesseseeeesees 74.
Arithmetic Progressions with Sum Equal to the Square
Of the Number Of Terms ..0.........ccsssssssssssssssssssssssssssssnsnsnsssseceeseseseeesees 76
Sums of Squares 1 ...........ecccceseeweesseeseseveeeeeesnenenininiestsnsnseeesntniedecesesesensesssssess 77
Sums of Squares We... .eeesssssssssssssnsnsnsccceeseseenesensunansseeceseneeessessnnanssonssseeseee 78
SUITS OF SUNS TT aa
i
acta
hitters 79
Sums of Squares IV ....
Sums of Squares V oo
81
Alternating Sums of Squares .o.eescsssssssssessssseseseeecceccensnnnuseneessesseseeeenanad 82
Sums of Squares of Fibonacci Numbe?s ..essssssssscccseessssssssneroeeeceeesee 83
SUTIN OF USS
SUNS Of CUO
i
aos aa seep aceesetssgedeeol 84
gsi
Sageocaeeathaeapueclaiti
aco eschea
taiespcases 85
SUMS OF Cubes TU occas
sey cece
latecencetthecteaeca teases
eal 86
Sums OF Cubes DV oon. sess cat eeccscttcs
acesttatancasena
cteansee87
cceangins
ics
ISUTIS Of CUBS! V cess
te
tale cela
ea escreceeceeet itera ecaele88
SUMS OF Ces VM scatte
a cla ners
geetansel atdnasesernneselebed 89
Sums of Integers and Sums of Cubes oo ssessessesssrsroseesessceceseneeneee 90
Sums of Odd Cubes are Triangular Numbers oo...
cssssssssssssccesssseeeseeee 91
68
Proofs without Words
Sums of Fourth Powers o.....cscsssssscsssccssesessensseceeceeseemenceeceesensessnnmnsnenscenseen 92
kth Powers as Sums of Consecutive Odd Numbers... ..csssscsssssnesessnee 93
Sums of Triangular Numbers [oosssssssssccccccccscececececeeesenssssssssssensnunsnsssee 94
Sums of Triangular Numbers Wy oi sscsssssssssssssscccccsssssnsssseseecccccensnseseeseeesees 95
Sums of Triangular Numbers IID .........ssssssssssssssssscsccecceccecececcecrererennnssssereres 96
Sums of Oblong Numbers 1ou... .sssscscssssssssssssseceseccsnsneseeseeececenssnsseseeeeeeeeeesee 97
Sums of Oblong Numbers 0 ou... ssssccsssescccccsnseecesounsesecsennsesessessnnesseceesseses
Sums of Oblong Numbers Too...
e.sssseees
Sums of Pentagonal Numbers
On Squares of Positive Integers
i. sssssssssscssseecsessesssssesessseeessesseesenes 101
Consecutive Sums of Consecutive Integers .........cssssccsssssesssssssssesssesesessee 102
UATE ETO
aa
cscs cctaieseassctep
bancsabebbecntenbmccnninicecnsnes 103
Identities for Triangular Numbers ..............ssssssssscssssssessssonsssecessorssseeeeeeceee 104
A, Tritt alar Tentty ssc
casita acted stich cwacasosn tos tabescasansentin 105
Every Hexagonal Number is a Triangular Number ............csssseeeee 106
One Domino = Two Squares: Concentric Squares oo sssssssssssssseeseees 107
Sums of Consecutive Powers of Nine are
Sums of Consecutive Integers eessssssessccceesesseseseesnnecesesnnesee 108
Sums of Hex Numbers Are Cubes oo esssssssssssssecessneeeeennesennnneesnnsee 109
Every Cube is the Sum of Consecutive Odd Numbers oo.
The Cube as an Arithmetic Sum
sssssssssss 110
Integer Sums
69
Sums of Integers I
1
14+2+---+n=5n(n+1)
—"The ancient Greeks”
(as cited by Martin Gardner)
70
Proofs without
Words
SOOT
WIE “FVORGS:
Sums of Integers II
—lan Richards
Integer Sums
71
Sums of Odd Integers I
1434+54---+(2n-1)
=
—Nicomachus of Gerasa (circa A.D. 100)
Proofs without Words
72
Sums of Odd Integers II
1+34---+(2n-1) = FOn? =r
Integer Sums
73
Sums of Odd Integers III
A+3At-+-+(2n-1)A =A=rA
¥ ci-1) mir
i=1
—Jend Lehel
74,
_————_—_—_—_—_——__—————_—_—
Proofs without Words
SS
Squares and Sums of Integers
13241=2"
14+2+34+241
= 3?
14+24+34+44+34+24+1=4
14+24+-4+(7-1)+n4+(n-1)4~4241 = 22
—“The ancient Greeks”
{as cited by Martin Gardner)
Integer Sums
75
Il.
=
©
+
14341=P+2
143454+74+54+341=37+4
1434-4 (2-1) + (2n+1) + (Qn-1) 44341
= 12+ (n+l?
—Hee Sik Kim
Proofs without Words
76
Arithmetic Progressions with Sum Equal to
the Square of the Number of Terms
3n-2
2
k = (2n-1) ; n=1,2,3,-~.
k=n
n=4
445464+74+84+94+10=7
—James O. Chilaka
Integer Sums
Sums of Squares I
P+2+.. +9
1
1
=3n (n+1I)(n+ oy
—Man-Keung Siu
78
Proofs without Words
Sums of Squares II
3(17 +2? +--+ 4?) = (n+ 1) +24-++n)
IUSiS
“+n
1+2+--
WA.
ss
—_—
Ban
2n+1
—Martin Gardner and Dan Kalman
(independently)
Integer Sums
ee,
Sums of Squares III
3(17 +27 +---4n’) = Fn(n + 1)(2n +1)
n
n
n-1n-1
non
n
n-1
2
n-1
n
n-1
2
#1
+
2
1
3
n
n
2nt+1
---
2n+1
2n+1 9 2nt1
-:-
2n+1
2n+1
«nln
n-1 n
n-1
n
2n+1
2
+
n
2
1
2n+1
2n+1
2n+1
—Sidney H. Kung
Proofs without Words
80
Sums of Squares IV
nN»
n
k=1
k=1
‘
n-1)[
k
k=1]
\i=1
VE - (34) -2 >, [S:i}een
(1+2+3+4)-(5)
—James O. Chilaka
Integer Sums
81
Sums of Squares V
—Pi-Chun Chuang
Proofs without Words
82
Alternating Sums of Squares
>.cy
KR 2 nr,
_ i
moe)
—Dave Logothetti
II.
PD
teeny = ¥
Cann”
=
MAD
=0
—Steven L. Snover
Integer Sums
83
Sums of Squares of Fibonacci Numbers
Fy = Fo = 1; Fuso =
2
n+1 +F,, => SEU
2
Oey
=
FF nei
—Alfred Brousseau
Proofs without Words
84
Sums of Cubes I
P+ 2433 4---¢0
= (1424+34---4n)P
Co
LZ
=_i
—Solomon W. Golomb
Integer Sums
Sums of Cubes II
P+23+37+.. -+0 = (1424+34-- +n?
—J. Barry Love
Proofs without Words
Sums of Cubes III
3423432 4---¢0 = (14+24+34---+0)
—Alan L. Fry
Integer Sums
87
Sums of Cubes IV
1
+343? +e tn P oain(n+ DP
aKa
—Antonella Cupillari and Warren Lushbaugh
{irdependently)
88
Proofs without Words
Sum of Cubes V
LOL
LMA ISH
%
tany=
i
4
(14+2+---¢n? = 9423433 4-.-43
—RBN
89
Integer Sums
Sum of Cubes VI
us
=
.
--+nDi
yYit+= DYit+2
i=1
i=]
n
=(Di
(2)
2
2
2
= 1(1)° + 2(2) +--++
n(n)
2
i=1
n
-y?
Ef
—Farhood Pouryoussefi
Proofs without Words
90
Sums of Integers and Sums of Cubes
1
14+2+---+n=pn(n+1)
P+2P4---40=
1
(gm +)
—Georg Schrage
Integer Sums
9
Sums of Odd Cubes are Triangular Numbers
3
leo
= ag) = Fs Es =
ooo
26)
1
(2n-1) = Qn-1DQn-1) =
<—
(n-1
+nQn)1) ——>
—
(n-1) +n(2n-1) —
3
és
P4394 5% +--+ (2n-1)? = 1424344 (21) = 1220-1)
—Monte J. Zerger
92
Proofs without Words
Sums of Fourth Powers
Ne
qleres
—Elizabeth M. Markham
Integer Sums
93
kth Powers as Sums of Consecutive Odd
Numbers
nk = (nk n+ 1) 4 (nk =n +3) 4-4 (nn
+20-2);
k= 2, 3, +».
za
—N. Gopalakrishnan Nair
94
Proofs without Words
Sums of Triangular Numbers I
Ty = 1+24---+n
= Ty 4+To+--++Ty - Man 2)
3(T, + Ty ++++4+T,) = (n+ 2)-T,,
Ty +Tyte+-4Ty
= B42) n+ 1) _ nie +
+2
—Monte J. Zerger
95
Integer Sums
Sums of Triangu lar Numbers II
at
» Te “6 n(n + 1)(n +2)
k 1
Ty = 14+24+---+k>
titi IIR
n
—RBN
9%
Proofs without Words
Sums of Triangular Numbers III
n
Tk =142+---4k>
1
a), T = 5 n(n + 1)(n + 2)
k=1
1
1
12
2
#1
123
3
2
n
n-1 n-1
1
n-2 n-2 n-2
es
er)
12
+
n-1
n-1n-2
+++
1
2
2
see
2
12
+
nln
on m1
+
21
1
1
wee
1 a
+
.
+
n+2
n+2 n+2
n+2 n+2 n+2
n+2 n+2
ces
Nt2
n+2 n+2
soe
nt+2 n+2
3(T; + To +...
+ Ty) = T,,* (a +2)
Integer Sums
97
Sums of Oblong Numbers I
(1x2)
+ (2x3) +(3x4)+---+(n-In =
(n—1)n(n + 1)
3
@)
Gi)
n+1
n+1
—T.C.Wu
Proofs without Words
98
Sums of Oblong Numbers II
3(1-2+234+3-4+---+n(n+1)) = n(nt+1)(n+2)
—Sidney H. Kung
Integer Sums
99
Sums of Oblong Numbers III
(1x2) + (2x3) +---+(n-1)xn = an? —n]
fiz iz
3(1 x 2)
=
oe
fia
3(1x2)
+
3(2 x3)
3(1x2)
+
3(2 x3)
3(3 x 4)
=
A
—Ali R. Amir-Moéz
100
Proofs without Words
Sums of Pentagonal Numbers
—William A. Miller
Integer Sums
101
On Squares of Positive Integers
Ty = 1t2te-+n
=>
(2n)?? = 8T,,4+4n
—Edwin G. Landauer
102
ne
Proofs without Words
Consecutive Sums of Consecutive Integers
14+2=3
44+5+6=7+8
9+10+11+12=13+14+15
16 +17 +18 +19 +20=21+22+23+24
n+ (+1) +--+
tn) =
+nt1) +--+ + (12+20)
—RBN
Integer Sums
103
Count the Dots
0/08 006
6 ©
© 018
© © ©
© ©
es
6 ©
©0006 60
08/06
© 06066
© e180 00660
©©0|(000000
eeeeeeeeoe0o
©0006 0\000000
©0000 0l000000
n
n-1
Yik+
k=l
Sk
kel
P.
2
k=n
Sy
Si
—Warren Page
104
Proofs without Words
Identities for Triangular Numbers
Ty = 14+2+~+7n
=
© © © © O/0/0
© © © O/0/© © © ©/0 ©
© © © © O|© © © © ©/O
T
+6T,, + Fa = (2n+1)
2
© 6 © O
Integer Sums
“
A Triangular Identity
Ty = 1+24+-4+n = (2k+1/Ty + Th = Tre rnek
—RBN
Proofs without Words
106
Every Hexagonal Number is a
Triangular Number
Hy = 1+5 +--+(4n-3)
T, = 142 +--+
=> Hy =3Ty4 + Ty = To_4 = nn-I)
(:)
NE
ey eyey a)
2O000COCE
©O0000000
©OG0
©0060
9©000
59
cl oT
eger Sums
One Domino = Two Squares:
ntric Squares
Uj,
YY =
bond
Ze
—
yi
WL
WA
Wy
ULLA_
14+42482412-2+162=9
n
142,
4k=Qn+1)
k=1
—Shirley A. Wakin
Proofs without Words
108
Sums of Consecutive Powers of Nine are Sums
of Consecutive Integers
0
14+94---4+9% = 14+24+34+---4+(14+34+---43%)
—RBN
Integer Sums
109
aS
Sums of Hex Numbers Are Cubes
Q
ecese
ORE
GXKS
EE
ORO
TPR
CO
ho=1
yo
h,=7
h, = 19
h,
= 37
110
Proofs without Words
Every Cube is the Sum of Consecutive Odd
Numbers
3 = 7+9+4+11
= [n(n—1)+1]4+- + [n(n+1)- 1]
—RBN
Integer Sums
111
The Cube as an Arithmetic Sum
n—1
n= ¥ 2in+1)+1
i=0
—Robert Bronson and
Christopher Brueningsen
Sequences & Series
On a Property of the Sequence of Odd Integers (Galileo, 1615) .............. 115
A Monotone Sequence Bounded by @.........cssssssssemcssscsssssssseesesssesssestsneeee 116
A Recusively Defined Sequence £09 © .........ssssssssssssssvsssssssssssesssseseseseseseeeee 117
Geometric Sums
a ciccoonsesencieresntoesnsteesectonibesensbentnontiobctbtbtcbuoninsososbanibenieeee 118
The Series of Reciprocals of Triangular Numbers .o...u......sosssscsscsssssseee 127
The Alternating Harmonic Series .............ssssssssssssssssssssssssssceceseesesseseseeceenees 128
sin(2n +1) = sinO + 2sinO Dj 1 COS2O nnn
129
An Arctangent Identity and Series _........sssscsccsssssnssnesseseseseeessssssunsessses 130
Sequences & Series
115
On a Property of the Sequence of Odd Integers
(Galileo, 1615)
1 _ 1s8
12326
3°54+7~>74+9+l1
-"°
14+34...+(2n-1)
(2n + 1) + (2n+ 3) +...+ (4n—-1)
ol
REFERENCE
S. Drake, Galileo Studies, The University of Michigan Press, Ann Ar-
bor, 1970, pp. 218-219.
—RBN
A Monotone Sequence Bounded by e
Vn21,
1+ _2 yl ee
(1+2)" Z (
n+1
y=In(1 +x)
1 my < m2 <1
n2=
ID
ing)
ee Se
ine
1
n+1
yn < (+o) m+]
=> (1+5)"
—RBN
Sequences & Series
117
A Recusively Defined Sequence for e
—Thomas P. Dence
Geometric Sums
r+r1-7) + r—-12 +
=1
—Warren Page
luences & Series
119
Geometric Series I
» ais 17
atartar-t+ar+-++ _ar
1/r
1-r
—J. H. Webb
Proofs without Words
120
Geometric Series II
APQR = ATSP
wltrtre..
1
= ers
—Benjamin G. Klein and Irl C. Bivens
121
Sequences & Series
Geometric Series III
waghe
(1-r)*
+ r1
—9)2
re + AQ
=
r? tee
(1-7)?
(1-1)? + 2r(1-7r)
2 SLL
1-r
+r
—Sunday A. Ajose
122
Proofs without Words
Geometric Series IV
She
27 a
Tl
=r
—Elizabeth M. Markham
Sequences & Series
123
Gabriel’s Staircase
Yak
k
r
= 'fi-=
ye for 0<r<1
—Stuart G. Swain
Proofs without Words
124
Differentiated Geometric Series
1 +26) +3() +44
=4
125
ences & Series
14+ 27+ 37444...
I+
(+), 0 <r<i
r(1-1)
—RBN
126
gh
Proofs without Words
be
12° 2.3.7 °° *n(n+l) = n+1
n/(n +1)
(n-1)/n
—Roman W. Wong
Sequences & Series
127
The Series of Reciprocals of Triangular
Numbers
—RBN
Proofs without Words
128
The Alternating Harmonic Series
1G4—-
n
2 )-51~
2 No" + 2k-1 0 2" 42k7
4 2k-1
a= fF = (1-3) (Ed)
42k’
ee
=
1,2,-+
n=1,2,---.
ee
I-3+3
—Mark Finkelstein
Sequences & Series
129
n
sin(2n +1)@ = sinO+2sin@ >, cos2ké
k=1
—J. Chris Fisher and E. L. Koh
Proofs without Words
130
r
r
reee
e
nn
An Arctangent Identity and Series
(n2+1n?+n ay n+1)
>
(n2+n+1,n°+n-+n)
arctan
hf(1,n)
———
arctan +=—
arctian n + arctan
ene
l = arctan(n
+ 1)
clan ena] 7 atctan(n +1) — arctann
Sain
n2+n+ 1
= lim
MIN-90 ATC
tan (N
tN
nek
=F
Miscellaneous
A 2x 2 Determinant is the Area of a Parallelogram ooo.
escscsosseeeeee 133
Area of the Parallelogram Determined by Vectors (a,b) and (c,d).........134
The Characteristic Polynomials of AB and BA are Equal ooo... 135
The Gaussian Quadrature as the Area of Either Trapezoid... 136
Inductive Construction of an Infinite Chessboard with
Maximal Placement of Nonattacking Queens .200.ooou...sssssssssesooee 137
Combinatorial Identities
138
3
yr 0 (3/)= 8" + 2(-1)", by Inclusion-Exclusion
in Pascal’s Triangle
The Existence of Infinitely Many Primitive Pythagorean Triples ........ 140
Pythagorean Triples via Double Angle Formulas
The Problem of the CalisSOms .0uu........sssssssscssscsesssssesssnssessennsecconssesconseeesense 142
RECUISION ii28 ssseeeieee er
n
Tel
k=1
ee
NT
acti
a SO
tO
SORE Shea ecise triads 143
eee earieaieresrammeeanans 144
Miscellaneous
Ol
133
A 2x 2 Determinant is the Area
of a Parallelogram
ab
cd
-et-e
>| Y-1Z7]
—Solomon W. Golomb
134
Proofs without Words
Area of the Parallelogram Determined by
b
og |= £@d-be)
Vectors (a,b) and (c,d) = + |
—Yihnan David Gau
Miscellaneous
135
The Characteristic Polynomials of
AB and BA are Equal
-A"|AB- Al] =
n
(4=~
a0
0
<pa-atl=|(54°
AW
“l(a a)lo
A
I\f-I
0
é |
n
Ail
n
al-(C (6 a)l-[4 soo
—Sidney H. Kung
Proofs without Words
136
The Gaussian Quadrature as the Area of
Either Trapezoid
Sb — a)(f@) + f(D) = 5(b - a)(h(a) + h))
A
| <i
—Mike Akerman
Miscellaneous
137
Inductive Construction of an Infinite
Chessboard with Maximal Placement of
Nonattacking Queens
REFERENCES
1. Dean S. Clark and Oved Shisha, Invulnerable Queens on an Infinite
Chessboard, Annals of the New York Academy of Sciences, The
Third International Conference on Combinatorial Mathematics,
1989, 133-139.
. M. Kraitchik, La Mathématique des Jeux ou Récréations Mathématiques, Imprimerie Stevens Fréres, Bruxelles, 1930, 349-353.
—Dean S. Clark and Oved Shisha
138
Proofs without Words
Combinatorial Identities
SRERSSESEL
(1)= Horm ="5
Ny
_ 1(,2
y= "S i
"2 )= (2) +"
EESEERRRRE:
—James O. Chilaka
Miscellaneous
139
n
3>y (3;= 8" + 2(-1)", by Inclusion-Exclusion in
j=0
Pascal’s Triangle
—Dean S. Clark
i
iE
The Existence of Infinitely Many Primitive
Pythagorean Triples
C10 © G0 OOO OGCO00O
e}9 OOOOH OOGOOO
C9 GOOOCHOOOCO8O
COG OOOOCOCOCOCO
COG OOGCOOOOOCOO
G0 © O0OGO08OCO8O8
O}9 OGCOCOGCOOSCOHOOO
OO O©OOSCOOCOOOOO
OOOO OOOH OCOOO
OS OOOCOSOOSOCO
GOS GOOOOOOOOCOO
0GC O00
O|6 © © OOOO
©GGOOOooooooceode
i—
———
n= 2k+1 = P+n2
= k+12 & &k+1=1
—Charles Vanden Eynden
Miscellaneous
141
a
Pythagorean Triples via Double Angle
Formulas
cos@=
ciel
m
m+
mone
x
sin29 =
2mn
a
*
m+
cos29 = & =
m+n
2mn
x
a
m-
—David Houston
Proofs without Words
142
The Problem of the Calissons
A calisson is a French sweet that looks like two equilateral triangles
meeting along an edge. Calissons could come in a box shaped like a
regular hexagon, and their packing would suggest an interesting
combinatorial problem. Suppose
a box with side of length n is
filled with sweets of sides of
length 1. The short diagonal of
each calisson in the box is parallel
to a pair of sides of the box.
We refer to these three possibilities by saying that a calisson
admits three distinct orientations.
THEOREM: In any packing, the number of calissons with a given orientation is one-third of the total number of calissons in the box.
PROOF:
—Guy David and Carlos Tomei
Miscellaneous
143
Recursion
A3 = 24241
= Ag = 2Ag34+1
Ag = 2Ag+1
Ay = 3 & A, = 2A,4+1 © A, = 2(21)-1 = 2-1
—Shirley Wakin
144
Proofs without Words
—Edward T. H. Wang
Sources
age
source
Geometry & Algebra
3
Howard Eves, Great Moments in Mathematics (Before 1650),
The Mathematical Association of America, Washington, 1980,
pp. 27-28.
4
Howard Eves, Great Moments in Mathematics (Before 1650),
The Mathematical Association of America, Washington, 1980,
pp. 29-32.
5
Howard Eves, Great Moments in Mathematics (Before 1650),
The Mathematical Association of America, Washington, 1980,
pp. 31, 33.
6
Howard Eves, Great Moments in Mathematics (Before 1650),
The Mathematical Association of America, Washington, 1980,
pp- 29-30.
7
Howard Eves, Great Moments in Mathematics (Before 1650),
The Mathematical Association of America, Washington, 1980,
pp. 34-36.
8
9
College Mathematics Journal, vol. 17, no. 5 (Nov. 1986), p. 422.
College Mathematics Journal, vol. 20, no. 1 (Jan. 1989), p. 58.
10
11
12
13
14
15
Mathematics Magazine, vol. 50, no. 3 (May 1977), p. 162.
Mathematics Magazine, vol. 48, no. 4 (Sept.-Oct. 1975), p. 198.
College Mathematics Journal, vol. 21, no. 5 (Nov. 1990), p. 393.
Mathematics Magazine, vol. 67, no. 5 (Dec. 1994), p. 354.
College Mathematics Journal, vol. 17, no. 4 (Sept. 1986), p. 338.
Mathematics Magazine, vol. 62, no. 3 (June 1989), p. 190.
16
17
College Mathematics Journal, vol. 22, no. 5 (Nov. 1991), p. 420.
Mathematics Magazine, vol. 66, no. 1 (Feb. 1993), p. 13.
18
Mathematics Magazine, vol. 65, no. 5 (Dec. 1992), p. 356.
19
Mathematics Magazine, vol. 56, no. 2 (March 1983), p. 110.
20
21
22
23
24
25
Mathematics Magazine, vol. 57, no. 4 (Sept. 1984), p. 231.
Mathematics Magazine, vol. 64, no. 2 (April 1991), p. 138.
Mathematics Magazine, vol. 66, no. 3 (June 1993), p. 180.
Mathematics Magazine, vol. 64, no. 2 (April 1991), p. 114.
Mathematics Magazine, vol. 68, no. 2 (April 1995), p. 109.
Mathematics Magazine, vol. 67, no. 4 (Oct. 1994), p. 302.
page
source
Trigonometry, Calculus & Analytic Geometry
29
Mathematics Magazine, vol. 64, no. 2 (April 1991), p. 97.
30
31
32
33.
34
35
Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 317.
Mathematics Magazine, vol. 61, no. 4 (Oct. 1988), p. 259.
Mathematics Magazine, vol.63, no. 5 (Dec. 1990), p. 342.
Mathematics Magazine, vol. 64, no. 2 (April 1992), p. 103.
College Mathematics Journal, vol. 20, no. 1 (Jan. 1989), p. 51.
American Mathematical Monthly, vol. 49, no. 5 (May 1942),
36
37
p. 325.
Mathematics Magazine, vol. 61, no. 5 (Dec. 1988), p. 281.
Mathematics Magazine, vol. 61, no. 2 (April 1988), p. 113.
38
Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 267.
39
40
41
42
College Mathematics Journal, vol. 18, no. 2 (March 1987), p. 141.
Mathematics Magazine, vol. 42, no. 1 (Jan.-Feb. 1969), pp. 40-41.
College Mathematics Journal, vol. 16, no. 1 (Jan. 1985), p. 56.
Richard Courant, Differential and Integral Calculus, p. 219,
copyright © 1937, Interscience. Reprinted by permission of John
Wiley & Sons, Inc.
43
College Mathematics Journal, vol. 18, no. 1 (Jan. 1987), p. 52.
44
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45
Mathematics Magazine, vol. 66, no. 1 (Feb. 1993), p- 39.
Inequalities
49
50
51
52
53
54
55
56
58
59
60
61
Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 98.
Mathematics Magazine, vol. 59, no. 1 (Feb. 1986), p. 11.
College Mathematics Journal, vol. 16, no. 3 (June 1985), p. 208.
College Mathematics Journal, vol. 19, no. 4 (Sept. 1988), p. 347.
Mathematics Magazine, vol. 60, no. 3 (June 1987), p. 158.
College Mathematics Journal, vol. 21, no. 3 (May 1990), p. 227.
College Mathematics Journal, vol. 20, no. 3 (May 1989), p. 231.
Mathematics Magazine, vol. 69, no. 1 (Feb. 1996), p. 64.
Mathematics Magazine, vol. 60, no. 3 (June 1987), p. 165.
Mathematics Magazine, vol. 64, no. 1 (Feb. 1991), p. 31.
Mathematics Magazine, vol. 63, no. 3 (June 1990), p- 172.
I. Reprinted by permission from Mathematics Teacher, vol. 81,
No. 1 (Jan. 1988), p. 63, author Li Changming, copyright © 1988 by
the National Council of Teachers cf Mathematics, Inc.
Sources
a
147
age
source
Inequalities (continued)
61
62
63
64
65
66
II. Mathematics Magazine, vol. 67, no. 1 (Feb. 1994), p. 34.
Mathematics Magazine, vol. 67, no. 5 (Dec. 1994), p. 374.
Mathematics Magazine, vol. 66, no. 1 (Feb. 1993), p. 65.
Mathematics Magazine, vol. 67, no. 1 (Feb. 1994), p. 20.
Mathematics Magazine, in press.
College Mathematics Journal, vol. 24, no. 2 (March 1993). p. 165.
Integer Sums
69
70
71
Scientific American, vol. 229, no. 4 (Oct. 1973), p. 114.
Mathematics Magazine, vol. 57, no. 2 (March 1984), p. 104.
Reprinted by permission from Historical Topics for the
Mathematics Classroom, p. 54, author Bernard H. Gundlach,
copyright © 1969 by the National Council of Teachers of
Mathematics, Inc.
73
Mathematics Magazine, vol. 64, no. 2 (April 1991), p. 103.
74 ~~Scientific American, vol 229, no. 4 (Oct. 1973), p. 115.
75 Mathematics Magazine, vol. 66, no. 3 (June 1993), p. 166.
76 Mathematics Magazine, vol. 59, no. 2 (April 1986), p. 92.
77 ~—Mathematics Magazine, vol. 57, no. 2 (March 1984), p. 92.
78 — Scientific American, vol. 229, no. 4 (Oct. 1973), p. 115.
College Mathematics Journal, vol. 22, no. 2 (March 1991), p. 124.
79 College Mathematics Journal, vol. 20, no. 3 (May 1989), p. 205.
80
Mathematics Magazine, vol. 56, no. 2 (March 1983), p. 90.
81
College Mathematics Journal, vol. 20, no. 2 (March 1989), p. 123.
82
I.
Mathematics Magazine, vol. 60, no. 5 (Dec. 1987), p. 291.
Il. Mathematics Magazine, vol. 65, no. 2 (April 1992), p. 90.
83
M. Bicknell & V. E. Hoggatt, Jr. (eds.), A Primer for the Fibonacci
Numbers, The Fibonacci Association, San Jose, 1972, p. 147.
84
85
86
87
Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 199.
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88
Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 178.
89
90
Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 323.
Mathematics Magazine, voi. 65, no. 3 (June 1992), p. 185.
Proofs without Words
148
page
source
Integer Sums (continued)
Mathematics Magazine, vol. 68, no. 5 (Dec. 1995), p. 371.
Mathematics Magazine, vol. 65, no. 1 (Feb. 1992), p. 55.
Mathematics Magazine, vol. 66, no. 5 (Dec. 1993), p. 329.
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College Mathematics Journal, vol. 23, no. 5 (Nov: 1992), p. 417.
Richard K. Guy, written communication.
Mathematics Magazine, vol. 62, no. 1 (Feb. 1989), p. 27.
Mathematics Magazine, vol. 62, no. 2 (April 1989), p. 96.
College Mathematics Journal, vol. 18, no. 4 (Sept. 1987), p. 318.
Mathematics Magazine, vol. 66, no. 5 (Dec. 1993), p. 325.
Mathematics Magazine, vol. 58, no. 4 (Sept. 1985), pp. 203, 236.
Mathematics Magazine, vol. 63, no. 1 (Feb. 1990), p. 25.
Mathematics Magazine, vol. 55, no. 2 (March 1982), p. 97.
Richard K. Guy, written communication.
Mathematics Magazine, vol. 67, no. 4 (Oct. 1994), p. 293.
Richard K. Guy, written communication.
Mathematics Magazine, vol. 60, no. 5 (Dec. 1987), p. 327.
Mathematics Magazine, vol. 63, no. 4 (Oct. 1990), p. 225.
American Mathematical Monthly, vol. 95, no. 8 (Oct. 1988),
pp. 701, 709.
Mathematics Magazine, vol. 66, no. 5 (Dec. 1993), p. 316.
Mathematics Magazine, vol. 63, no. 5 (Dec. 1990), p. 349.
Sequences & Series
115
116
117
118
119
120
121
122
123
124
126
127
Mathematics Magazine, vol. 68, no. 1 (Feb. 1995), p. 41.
Mathematics Magazine, vol. 67, no. 5 (Dec. 1994), p. 379.
Mathematics Magazine, vol. 66, no. 3 (June 1993), p. 179.
Mathematics Magazine, vol. 54, no. 4 (Sept. 1981), p. 201.
Mathematics Magazine, vol. 60, no. 3 June 1987), p. 177.
Mathematics Magazine, vol. 61, no. 4 (Oct. 1988), p. 219.
Mathematics Magazine, vol. 67, no. 3 (June 1994), p. 230.
Mathematics Magazine, vol. 66, no. 4 (Oct. 1993), p. 242.
Mathematics Magazine, vol. 67, no. 3 une 1994), p. 209.
Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), pp. 332-333.
Mathematics Magazine, vol. 65, no. 5 (Dec. 1992), p. 338.
Mathematics Magazire, vol 64, ro. 3 (June 1991), p. 167.
Sources
age
149
source
Sequences & Series (continued)
128
American Mathematical Monthly, vol. 94, no. 6 (June-July 1988),
129.
pp. 541-42.
Mathematics Magazine, vol. 65, no. 2 (April 1992), p. 136.
130
Mathematics Magazine, vol. 64, no. 4 (Oct. 1991), p. 241.
Miscellaneous
133
Mathematics Magazine, vol. 58, no. 2 (March 1985), p. 107.
134
135
136
Mathematics Magazine, vol. 64, no. 5 (Dec. 1991), p. 339.
Mathematics Magazine, vol. 61, no. 5 (Dec. 1988), p. 294.
Mathematics Magazine, vol. 60, no. 2 (April 1987), p. 89.
137
Mathematics Magazine, vol. 61, no. 2 (April 1988), p. 98.
138
Mathematics Magazine, vol. 52, no. 4 (Sept. 1979), p. 206.
139
Mathematics Magazine, vol. 63, no. 1 (Feb. 1990), p. 29.
140
Elementary Number Theory, Random House, New York, 1987,
p. 227.
Mathematics Magazine, vol. 67, no. 3 (June 1994), p. 187.
American Mathematical Monthly, vol. 96, no. 5 (May 1989),
pp. 429-30.
141
142
143.
144
Mathematics Magazine, vol. 62, no. 3 (June 1989), p. 172.
College Mathematics Journal, vol. 20, no. 2 (March 1989), p. 152.
Index of Names
Ajose,Sunday A.
121
Akerman, Mike
136
Fibonacci (Leonardo of Pisa)
Finkelstein, Mark
128
Fisher, J. Chris
129
Fry, AlanL.
86
Amir-Moéz, AliR.
99
Aristarchus of Samos
63
Ayoub, Ayoub B.
43, 44
Galileo Galilei
115
Gallant, Charles D.
19, 49, 59
Beekman, Richard M.
45
Bernoulli, Jacques
65
Bhaskara 4
Bicknell, M.
147
Bivens, Irl1C.
120
Bronson, Robert
Brousseau, Alfred
Gardner, Martin
69, 74, 78
Garfield, James A.
111
83
111
41
Cauchy, Augustin-Louis
64
Hardy, Michael
Chuang, Pi-Chun
Houston, David
Chuquet, Nicolas
61
Clark, Dean S. 137,139
Coble, Scott
13
Courant, Richard
42
Cupillari, Antonella
147
87
142
141
11
Klein, Benjamin G.
Diophantus of Alexandria
Drude,K.Ann
Dudeney,H.E.
21
6
Eddy, RolandH.
Eisenman, R.L.
Elsner, Thomas
Euclid
5
16,18, 51
40
10
145
39
Kalman, Dan
78
Kim, Hee Sik
75
Kincanon, Eric
12
Koh, E.L.
DeKleine,H. Arthur
36
Dence, Thomas P.
117
Eves, Howard
Hoggatt, V.E.
Isaacs, Rufus
84, 133
147
8
Harris, Edward M.
81
134
Gibbs, Richard A. 60
Golomb, Solomon W.
Gundlach, Bernard H.
Guy, Richard
148
Cavalieri, Bonaventura
25
Chilaka, JamesO.
76, 80, 138
David,Guy
7
Gau, Yihnar David
Gentile,EnzoR.
9
Brueningsen, Christopher
Burk, Frank
83
129
Kung, Sidney H.
22
120
17, 25, 29, 30,
32, 33, 54, 79, 98, 135
Landauer, EdwinG.
Lehel, Jend 73
LiChangming
101
61
Logothetti, Dave
23,82
Love,J. Barry 85
Lushbaugh, Warren
87
Proofs without Words
152
Markham, Elizabeth M.
92, 122
Miller, William A.
100
Mollweide, Karl
36
Montucci, Paolo
52
Nair, N. Gopalakrishnan
93
Nakhli, Fouad
14,58
Napier, John
66
Nicomachus of Gerasa
71
64
Schwarz, Hermann A.
137
Shisha, Oved
31
Sipka, Timothy A.
Siu, Man-Keung
77
82
Snover, Steven L.
123
Swain, StuartG.
Page, Warren
52, 103, 118
Pascal, Blaise
139
Vanden Eynden, Charles
Viviani, Vincenzo
15
Pooley,Sam
21
Pouryoussefi, Farhood
Wolf, Samuel
37
Schattschneider, Doris
Schrage,Georg
Walker, R.J.
35
119
Webb,J.H.
70
Romaine, William
142
140
20, 107, 143
Wakin, Shirley
144
Wang, Edward T.H.
89
Ptolemy, Claudius
33
Pythagoras
3-9, 140, 141
,
RichardsJan
Tomei, Carlos
15
Wong, Roman W.
Wu,T.C.
97
= 126
Zerger, Monte J.
91,94
50
90
Technical Note
The manuscript for this book was edited and printed
using Microsoft® Word 5.1 on an Apple® Macintosh™ JIfx computer.
The graphics were produced
in Claris™ MacDraw® Pro 1.5v1. The text is set in
the Palatino font, with special characters in the
Symbol font. Many of the displayed equations were
produced with MacXqn™ 3.0 from Software for
Recognition Technologies. The manuscript was
printed on an Apple® Laserwriter® II NTX.
PROOFS
WITHOUT
WORDS
EXERCISES
IN VISUAL
THINKING
Roger Nelsen received his Ph.D.
in
Mathematics
from
Duke
University. Since 1969 he has
taught at Lewis and Clark College,
in Portland Oregon, where he is a
professor of mathematics. He is
currently an Associate Editor of
the “Problems and Solutions”
section of the College Mathematics
Journal. hisSpesany and research papers in mathematics have
appeared in the American Mathematical Monthly, Mathematics
Magazine, The College Mathematics Journal, Nature, Journal of.
Applied Probability, Communications in Statistics, Probability Theory
the Journal of Nonparametric Statistics. His main research
interests are in probability and mathematical statistics.
Just what are “proofs without words?” First of all, most
mathematicians would agree that they certainly are not
“proofs” in the formal sense. Indeed, the question does not
have a simple answer. But, as you will see in this book, proofs
without words are generally pictures or diagrams that help the
reader see why a particular mathematical statement is true, and
also to see how one could begin to go about proving ittrue.
While in some proofs without words an equation or two may
appear to help guide that process, the emphasis is clearly on
providing visual clues to stimulate mathematical thought. Proofs
without words bear witness to the observation that often in
the English language to see means to understand, asin “to see
the point of anargument.”
Proofs without words have a long history. In this collection you
will find modern renditions of proofs without words from
ancient China, classical Greece, twelfth-century India—even one
based on a published proof by a former President of the United
States! However, most of the proofs are relatively more recent
Creations, and many are taken from the pages of MAA journals.
The proofs in this collection are arranged by topic into six
chapters: Geometry
and Algebra; Trigonometry, Calculus and
Analytic Geometry; Inequalities; Integer Sums; Sequences and
Series, and Miscellaneous. Teachers will find that many of the
proofs without words in this collection are well suited for
classroom discussion and for helping students to think visually
in mathematics.