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CLASSROOH RESOURCE MATERIALS / NUMBER I
THE MATHEMATICAL ASSOCIATION Of AMERICA

Classroom Resource Materials
Number 1
Proofs Without Words
Exercises in Visual Thinking
Roger B. Nelsen
Lewis and Clark College
Published and Distnbuted by
THE MATHEMATICAL ASSOCIATION OF AMERICA

©1993 by
The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 93-86388
ISBN 0-88385-700-6
Printed in the United States of America
Current Printing Oast digit):
10 987654321

Introduction
see (se) 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 (2nd Ed.)
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 end-
of-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]
Polya's "Draw a figure ..." is classic pedagogic advice, and
Einstein and Poincare'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
certain 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. I 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
Introduction ν
Geometry & Algebra 1
Trigonometry, Calculus & Analytic Geometry 27
Inequalities 47
Integer Sums 67
Sequences & Series 113
Miscellaneous 131
Sources 145
Index of Names 151

Geometry & Algebra
The Pythagorean Theorem I
The Pythagorean Theorem II
The Pythagorean Theorem III
The Pythagorean Theorem IV
The Pythagorean Theorem V
The Pythagorean Theorem VI
A Pythagorean Theorem: aa' = bb' + c-c'
The Rolling Circle Squares Itself
On Trisecting an Angle .
Trisection of an Angle in an Infinite Number of Steps
Trisection of a Line Segment ....
The Vertex Angles of a Star Sum to 180 е
Viviani's Theorem .....
A Theorem About Right Triangles
Area and the Projection Theorem of a Right Triangle
Chords and Tangents of Equal Length .
Completing the Square .....
Algebraic Areas I .
Algebraic Areas II .
Diophantus of Alexandria's "Sum of Squares" Identity
The kth n-gonal Number ....
The Volume of a Frustum of a Square Pyramid
The Volume of a Hemisphere via Cavalieri's Principle
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

Geometry & Algebra 3
The Pythagorean Theorem I
—adapted from the Chou pet suan ching
(author unknown, circa B.C. 200?)

4 Proofs without Words
The Pythagorean Theorem II
Behold!
—Bhaskara (12th century)

Geometry & Algebra 5
The Pythagorean Theorem III
—based on Euclid's proof

6
Proofs without Words
The Pythagorean Theorem IV
-Η. Ε. Dudeney (1917)

Geometry & Algebra
The Pythagorean Theorem V
a
A = 2-±ab +±c2 =±(a+b)
2 2 2
—James A. Garfield (1876)
2Qth President of the United States

8 Proofs without Words
The Pythagorean Theorem VI
c + a _ b
b ~ c-a
a + Ό = с
—Michael Hardy

Geometry & Algebra
9
A Pythagorean Theorem: a-a' = b-b' + c-c'
a(x + y) = bb' +c-c'.
—Enzo R. Gentile

10
Proofs without Words
The Rolling Circle Squares Itself
—Thomas Eisner

Geometry & Algebra Π
On Trisecting an Angle
—Rufus Isaacs

12
Proofs without Words
Trisection of an Angle in an Infinite Number
of Steps
ill i_J_
3"2"4 + 8"16
—Eric Kincanon

Geometry & Algebra
13
Trisection of a Line Segment
AF = |-AB
—Scott Coble

14
Proofs without Words
The Vertex Angles of a Star Sum to 180"
1+3
2+4/
' 1+3
—Fouad Nakhli

Geometry & Algebra
15
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.
C
A"
D
B'
В
—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
CD2 = AD-DB
—Sidney H. Kung

18
Proofs without Words
Chords and Tangents of Equal Length
If circle C| passes through the center О of circle C2, 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 = (χ + a/2)2 -(a/2)2
—Charles D. Gallant

20
Proofs without Words
Algebraic Areas I
(a + bf + (a-b)2 = 2(a2 + b2)
Ш!Ш!ШШЯШЯ55ЩШ1Ш1^
1ШМ
Ж
Ш
+
а-Ь
а-Ь Ъ
—Shirley Wakin

Geometry & Algebra
21
Algebraic Areas II
(a + b + c)2 + (a + b - c)2 + (a - b + c)2 + (a - b - c)2
= (2a)2 + (2b)2 + (2c)2
4 a + b + c ►
a-b + c
a-b-c
lb
2c
—Sam Pooley and K. Ann Drude

22
Proofs without Words
Diophantus of Alexandria's "Sum of Squares
Identity
(я2 + b2)(c2 + d2) = (ad + be)2 + (bd-ac)2
< a2 N-
2 2
α с
2 2
Λ2
bV
=φ
ι/
(ее)2
(лО2
(bef
(Μ)2
.w
Μ
ad
be
J
=Ь
abed
be
ad
||||||i
—RBN

Geometry & Algebra 23
The № и-gonal Number is
1 + (jfc- l)(n -1) + \(k - 2)(k - l)(n - 2)
—Dave Logothetti

24
Proofs without Words
The Volume of a Frustum of a Square Pyramid
[Problem 14, The Moscow Papyrus, circa 1850 B.C.]
REFERENCES
1. С. В. 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

Geometry & Algebra 25
The Volume of a Hemisphere via
Cavalieri's Principle*
2m
Vs = VP = з^яг = зяг3
*Tzu Geng, son of the most celebrated mathematician Tzu Chung Chih
in ancient China, was believed to be the first to develop the principle in
the 5th century A. D.
—Sidney H. Kung

Trigonometry, Calculus & Analytic Geometry
Sine of the Sum .....
Area and Difference Formulas .
The Law of Cosines I
The Law of Cosines II ... .
The Law of Cosines III (via Ptolemy's Theorem)
The Double-Angle Formulas
The Half-Angle Tangent Formulas
Mollweide's Equation ....
(tan0 + l)2 + (cot© + l)2 = (sec0 + csc0)2
The Substitution to Make a Rational Function of
the Sine and Cosine . . . . .
Sums of Arctangents ......
The Distance Between a Point and a Line
The Midpoint Rule is Better than the Trapezoidal Rule
for Concave Functions . . . . .
Integration by Parts ......
The Graphs of / and /-1 are Reflections about the Line у = χ
The Reflection Property of the Parabola
Area under an Arch of the Cycloid . . . .
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

Trigonometry, Calculus & Analytic Geometry 29
Sine of the Sum
sin (x + y) = sin χ cos у + cos χ sin у for χ + у < π
с = a cos у + Ь cos x
г = 1 /2 =» sin z = (с/2)/(1 /2) = с, sin x-a, sin у = b;
sin (x + у) = sin (π - (χ + у)) = sin ζ = sin χ cos у + sin y cos дт
—Sidney Η. Kung

30
Proofs without Words
Area and Difference Formulas
asmx
bsiny
bcosy
я cos*
sinOt-y) = sin χ cos у - cos χ sin у
bcosy + asiruc
я cos*
bsiny
cosOt - y) = cos χ cos у + sin χ sin у
—Sidney H. Kung

Trigonometry, Calculus & Analytic Geometry
31
The Law of Cosines I
bcos0
c2 = (b sin Θ)2 + (a-b cos Θ)2
= a2 + b2-2ab cos θ
—Timothy A. Sipka

32
Proofs without Words
The Law of Cosines II
(2a cos 0-b)b = (a - c)(a + c)
c2 =a2 + b2- lab cos θ
—Sidney H. Kung

Trigonometry, Calculus & Analytic Geometry 33
The Law Of Cosines III (via Ptolemy's Theorem)
с-с = b-b + (a + 2b cos(k-θ))α
с2 = a2 +b2 - 2abcose
—Sidney H. Kung

34
Proofs without Words
The Double-Angle Formulas
x2+y2=l
C(cos20, sin20)
2sin0
AACD ~ AABC
CD I AC = ВС I AB
sin20/2cos0 =2sin0/2
sin20 = 2sin0cos0
AD I AC = AC I AB
(1 + cos20)/2cos0 = 2cos0/2
cos20 = 2cos20-l
—RBN

Trigonometry, Calculus & Analytic Geometry
35
The Half-Angle Tangent Formulas
1 - cos0
tan 2
θ sin0 1 - cos0
1 + cose sine
—R. J. Walker

36
Proofs without Words
Mollweide's Equation
(a - b) cos 2 =
• ία~β\
с sin V о ;
α + β
a
'а-Ъ
—H. Arthur DeKleine

Trigonometry, Calculus & Analytic Geometry 37
(tan0+l)2 + (cot0+l)2 = (sec0+csc0)2
cot0 1
tan20 + 1 = sec20
cot20 + l = esc2©
(tan© + l)2 + (cot0 + l)2 = (sec0 + csc0)2
/ , Λ tan0 + 1 ч
V, also tan0 = )
cot0+l
—William Romaine

38
Proofs without Words
The Substitution to Make a Rational Function of
the Sine and Cosine
—rbn

Trigonometry, Calculus & Analytic Geometry 39
Sums of Arctangents
1 Ι π
arctan 2 + arctan 3 = 4
arctanl + arctan 2 + arctan 3 = π
—Edward M. Harris

40
Proofs without Words
The Distance Between a Point and a Line
(a, ma + c)
ma + c-b\
d
1
I ma + с - b I
Vl + tn2
—R. L. Eisenman

Trigonometry, Calculus & Analytic Geometry 41
The Midpoint Rule is Better than the
Trapezoidal Rule for Concave Functions
—Frank Burk

42
Proofs without Words
Integration by Parts
s = gib)
r=g(a)
p=№
q=fQ»
Area ^
^1
+ Area
Ш = ч*~
pr
s 1
judv + jvdu = uv
r ρ
(qf)
(p,r)
b \u b
\f(x)g'(x)dx = f(x)g(x)\b - \g(x)fXx)dx
—Richard Courant

Trigonometry, Calculus & Analytic Geometry 43
The Graphs of / and / are Reflections about
the Line у = χ
—Ayoub В. Ayoub

44
Proofs without Words
The Reflection Property of the Parabola
у
A
-M>
II
Η
к
\
\
W ν
5
0
·* / /
*/ /
/\ /
/ \ /
f \ /
\ /
t Ηρ,Ο)
QF = QD & mrm2= -1
Zl = Z2 = Z3
—Ayoub B. Ayoub

Trigonometry, Calculus & Analytic Geometry
45
Area under an Arch of the Cycloid
/ТТл
2nR
+ 2R
2R +
+ 2R
^KRTR
kR2 +
A = 3 KR2
2KR-2R
—Richard M. Beekman

Inequalities
The Arithmetic Mean—Geometric Mean Inequality I
The Arithmetic Mean—Geometric Mean Inequality II
The Arithmetic Mean—Geometric Mean Inequality III
Two Extremum Problems .....
The Harmonic Mean—Geometric Mean—Arithmetic Mean-
Root Mean Square Inequality I .
The Harmonic Mean—Geometric Mean—Arithmetic Mean-
Root Mean Square Inequality II .
The Harmonic Mean—Geometric Mean—Arithmetic Mean-
Root Mean Square Inequality III
Five Means — and Their Means
гп > л?
AB > BA for e < A < В .
The Mediant Property
Regie des Nombres Moyens
The Sum of a Positive Number and its Reciprocal
is at least Two
Aristarchus' Inequalities .
The Cauchy-Schwarz Inequality
Bernoulli's Inequality
Napier's Inequality
49
50
51
52
53
54
55
56
58
59
60
61
62
63
64
65
66

Inequalities
49
The Arithmetic Mean—Geometric Mean
Inequality I
v5F<iLfli
—Charles D. Gallant

50
Proofs without Words
The Arithmetic Mean—Geometric Mean
Inequality II
ь
a
(a + b) -(a-b) = 4ab
a + b
>Jab
—Doris Schattschneider

Inequalities
51
The Arithmetic Mean—Geometric Mean
Inequality III
—2~~ ^ ^ab, with equality if and only if a = b
—Roland H. Eddy

52
Proofs without Words
Two Extremum Problems
For a given product, the sum of two positive numbers
is minimal when the numbers are equal.
(Z[P,0) (Я0)
For a given sum, the product of two positive numbers
is maximal when the numbers are equal.
(0,S)
(S,0)
—Paolo Montuchi and Warren Page

Inequalities
53
The Harmonic Mean—Geometric Mean—
Arithmetic Mean—Root Mean Square
Inequality I
PM = a, QM = b, a > b > 0
HM < GM < AM < RM
—RBN

54
Proofs without Words
The Harmonic Mean—Geometric Mean—
Arithmetic Mean—Root Mean Square
Inequality II
-K4-
b-a
-►4-
AB=a, BC = b
AD = DC=±±b
2
BE 1AB, DE = AD
FE1 ED, FB || ED
EG = BD=^-
b + a
—Sidney H. Kung

Inequalities
55
The Harmonic Mean—Geometric Mean—
Arithmetic Mean—Root Mean Square
Inequality III
<—a —χ b ►
2 2 2
2a + 2b > (a + b)
-Га—x Jb
a2+b2 > a+b
(Vfl+Ь)2 > 4· ±ГаЛ
a+b
> Tab
a+b
-x-
a+b
1 > 4
a+b a+b
a+b
—RBN

56
Proofs without Words
Five Means — and Their Means
ty
cm
rms
am
gm
hm
a + b
Arithmetic: am = AM(a,b) = 2
Contraharmonic: cm = CM(a,b) = *
Geometric: gm = GM(a,b) = *fab
lab
Harmonic: hm = HM(a,b) = «
Root Mean Square: rms = RMS(fl,b) = Д/ —5-
дс = у
a + b
*y='
2 2 (fl + b)
χ + y =—
2 2 2 t2
^x +y =д + b
дс + у = д + b
/zm #m дт rms cm b

Inequalities 57
I. 0<a<b
2 . ,2 2 . ,2
a <
2яЬ гт д + Ь a+b a+b ,
< Vflb < —r— < \ /—-— < r- < b
a+b 2 V 2 я+Ь
II. hm + cm = a + b => AM(hm, cm) = am.
III. hm-am=a-b => GMQim, am) = gm.
a2 + b2
TV. am cm = —«— => GM(«»i, cm) = rws.
tt о 0 (a + b) τ,, ,„,
V. gwr + rwsz = —2— =* ^MS(^w, rws) = am.
(r
■AM
GM ν I / GM ν
~2 Л 2 ,2
i
^ 2дЬ гт a + b I a+b a + b ,
к t ;
RMS
—RBN

58
Proofs without Words
βπ > 71?
J
i
I
0.4-
1/e .
шк/к -
0.35-
0.3-
<
f
i
^**^
/
ι
К 1
V '
1 1
1 1
^\^^
^\s^ lruc
Ι ι ^^
1 1 ^
e 3 π
—Fouad Nakhli

Inequalities
59
Ав > BA iore<A<B
e<A<B => ША> wig
In Λ In В
=*AB > BA
—Charles D. Gallant

60
Proofs without Words
The Mediant Property
a_ c_ a a + с с_
b K d =* b <b + d< d
—Richard A. Gibbs

Inequalities
61
Regie des Nombres Moyens (Two Proofs)
[Nicolas Chuquet, be Triparty en la Science des Nombres, 1484]
I.
, Λ a c
a,b,c,d>0; г < τ
a
a + с
b <b + d < d
W-j < W3
тг < m2 < w3
II.
. 1
1
1 '
1
i—
JL
b
1 —
—►
t
JL
b
1
1
1
α
IIJ^JI
b+d
1
c_
d
1
—Li Changming
b+d
a_ а с с_
b < b + d + b + d < d
-RBN

62
Proofs without Words
The Sum of a Positive Number and its
Reciprocal is at least Two (Four Proofs)
I.
A
■
λ
A
i
X
i
1
X
,
.
ШШ
ШШШ$:Ш$1$$|:|$
liliJll
1
$1*
1
1111Ш1111111
$Й^ЩЙ^^ВД^^|
If llllll
ιιιιιιβιιιΐ!
+<— —
X
y = 2-x
III.
IV.
t
1
χ
u
I
r
X>\ => X + ~>2
—RBN

Inequalities
63
Aristarchus' Inequalities
0<β<α<?-
since a tana
smp β tanjS
у = tan χ
► χ
β α
sin/ϊ tanj3
sina < ——a; ~~z~a < tana
β β
sina a tana
sin)3 0 tan0
—RBN

64
Proofs without Words
The Cauchy-Schwarz Inequality
\{а,ЪУ(х,у)\ < ||<а,Ь>|| ||<x,y>||
Ы
\b\
\a\
\y\
(1я1 + \y\)(\b\ + \x\) < 2<ka\\h\ + i*llyl) + V«2+ *>2 V^ + y2
л \ax + by\ <> \a\\x\ +\b\\y\ < yja2+ b2^x2+ y2
—RBN

Inequalities
65
Bernoulli's Inequality (two proofs)
x > ο, χ φ ι, r > ι => xr -1 > r(x -1)
I. (first semester calculus)
У
= /-1
y = r(x-l)
► X
II. (second semester calculus)
► t
/-1 =
.r-1
\rtr dt > rix-\)
ι
0<x<l
1 - / = J гГ df < Kl - x)
—RBN

66
Proofs without Words
Napier's Inequality (two proofs)
1 lnb-lna 1
b>a>0 => r < —г—-— < -
о о — a a
I. (first semester calculus)
У
y = ln χ
► x
m(L ) < m(L ) < m(L )
l 2 3
II. (second semester calculus)
У
i(b-a)< p-dx<l(b-a)
—RBN

Integer Sums
Sums of Integers I .
Sums of Integers II ....
Sums of Odd Integers I
Sums of Odd Integers II
Sums of Odd Integers III .
Squares and Sums of Integers
Arithmetic Progressions with Sum Equal to the
of the Number of Terms .
Sums of Squares I .
Sums of Squares II .
Sums of Squares III ....
Sums of Squares IV ... .
Sums of Squares V ....
Alternating Sums of Squares
Sums of Squares of Fibonacci Numbers
Sums of Cubes I
Sums of Cubes II
Sums of Cubes III .
Sums of Cubes IV .
Sums of Cubes V .
Sums of Cubes VI .
Sums of Integers and Sums of Cubes
Sums of Odd Cubes are Triangular Numbers .

68
Proofs without Words
Sums of Fourth Powers ....
fcth Powers as Sums of Consecutive Odd Numbers
Sums of Triangular Numbers I
Sums of Triangular Numbers II
Sums of Triangular Numbers III
Sums of Oblong Numbers I
Sums of Oblong Numbers II
Sums of Oblong Numbers III
Sums of Pentagonal Numbers
On Squares of Positive Integers
Consecutive Sums of Consecutive Integers
Count the Dots ....
Identities for Triangular Numbers
A Triangular Identity
Every Hexagonal Number is a Triangular Number
One Domino = Two Squares: Concentric Squares
Sums of Consecutive Powers of Nine are
Sums of Consecutive Integers
Sums of Hex Numbers Are Cubes
Every Cube is the Sum of Consecutive Odd Numbers
The Cube as an Arithmetic Sum
108
109
110
111

Integer Sums
69
Sums of Integers I
^^ ^Ш ^^
1 + 2 + · · · + η = 2«(и + 1)
—'The ancient Greeks"
(as cited by Martin Gardner)

70
Proofs without Words
Sums of Integers II
1 + 2 + ·
η2 η
+ η =~2" + 2
—Ian Richards

Integer Sums
71
Sums of Odd Integers I
φ φ φ
m mm
1 + 3 + 5 + · · · + (2л -1) = и2
—Nicomachus of Gerasa (circa a. d. 100)

72
Proofs without Words
Sums of Odd Integers II
1+3 + ··· + (2η-1) =ψ·ηγ
η

Integer Sums
73
Sums of Odd Integers III
2и-1
Δ + 3·Δ + · · · + (2и - 1)·Δ = А = w ·Δ
η
ί = 1
—Jeno Lehel

74
Proofs without Words
Squares and Sums of Integers
I.
1 + 2 + · ·· + (и -1) + и + (и -1) + - + 2 + 1 = η2
—"The ancient Greeks'"
(as cited by Martin Gardner)

Integer Sums
II.
75
+
1 + 3 + 1 = l2 + 22
'f&Ji ' ffiT.
^ ^
+
1 + 3 + 5 + 3+1 = 22 + 32
ШУ/ ЧЕаУ/^ \laiv^ \|В
^^ 4^
Чйй/ Xss
1 + 3 + 5 + 7 + 5 + 3 + 1 = 32 + 42
1 + 3 + .·. + (2л-1) + (2и+1) + (2л-1) + - + 3 + 1 = и2 + (л+1)2
—Нее Sik Kim

76
Proofs without Words
Arithmetic Progressions with Sum Equal to
the Square of the Number of Terms
3n-2 2
Σ k= (2и-1); n= 1,2,3,
k = n
D
рмщ
:♦¥>
И
ί Π
:?·δ·ίέ·έ·:·δ·ί>&·ίίίίίδι
*> 4ii4c+i#ii>
та'ята'Т'^я'Я'тта'Л
чщя4|ЩП^|^Щя
рптчччТфчч!1!'!1!1!11!1!?^
и = 4
4 + 5 + 6 + 7 + 8 + 9 + 10 = 7:
WHWHraj^UA1.1.1.1.1.1!' ·"·**· ·*>+ί+ί+·+·:||
:^ЖЖ^*&ЩЯ—
•^^л^чЧ^тчтата!??!!???!?1!^
IIP
-t-K-M-M-M-M-ft-
·##4&**ΛΛ * * ♦ *
—James О. Chilaka

Integer Sums
77
Sums of Squares I
|2 , o2
V + 2* + --> + nz = зИ(и + 1)(я + 2)
n + l
—Man-Keung Siu

78
Proofs without Words
Sums of Squares II
ι2 , o2
3(Г + 2Z + · · · +nz) = {In + 1)(1 + 2 + · · · + и)
IN
31
ш
Шм
1
к
^
1
1
1
WWt
1
^
^
^
%%i
1
i
1
^
£
7УУ/
ϋ
1
1
2и + 1
+
+
+
—Martin Gardner and Dan Kalman
(independently)

Integer Sums 79
Sums of Squares III
3(12 + 22 + · · · +и2) = ψι(η + 1)(2и + 1)
η η и η η и-1 ... 2 1 1 2 ... и-l η
η-\ η-1 ... и-l η η-1 ... 2 2 3 ... η
2 2 и и-1 и-1 и
1 ни
2и+1 2и+1 ... 2и+1 2и+1
2и+1 2и+1 ... 2и+1
2и+1 2и+1
2и+1
—^Sidney Η. Kung

80
Proofs without Words
Sums of Squares IV
(, 2
П \ n-\
Σ* -2Σ
1
r
ι
■ 1
1
1
3
1 1
(1+2 + 3). (4)
i
I
(1 + 2 + 3 + 4) · (5)
5
—James O. Chilaka

Integer Sums
81
Sums of Squares V
η η
Σ Σι = Σ<
—Pi-Chun Chuang

82
Proofs without Words
Alternating Sums of Squares
I.
+
II.
1 (-i)<+1 k2 = (-i)n+V„ - (-i)"+1 Sb±»
-Dave Logothetti
Лили.'.! """"""
ll.lll.lll.lll
• :§': . >
ιΙι'μΪιι'μΙ
·.·>.·:·.···: хй:·:·:·*
Ш*
^.(„.«^...^.„-ν = Σ (-i)W = ^fi)
fc = 0 2
—Steven L. Snover

Integer Sums
83
Sums of Squares of Fibonacci Numbers
ΓΤΤΤ ! I 1 ! I I ! I ! I
F1=F2 = 1; Fn+2 = Fn+l + Fn => Fj + F2 + ... + Fn = FnFn+l
—Alfred Brousseau

84
Proofs without Words
Sums of Cubes I
ι3 , o3 · o3
V + Τ + У + · · · + rf = (1 + 2 + 3 + · · · + ny
—Solomon W. Golomb

Integer Sums
85
Sums of Cubes II
l3 + 23 + 33 + · · · + n3 = (1 + 2 + 3 + · · · + n)2
—J. Barry Love

86
Proofs without Words
Sums of Cubes III
l3 + 23 + 33 + · · · + я3 = (1 + 2 + 3 + · · · + n)2
—Alan L. Fry

Integer Sums
87
Sums of Cubes IV
ι3 , ο3 , o3
V + Τ + У + · · · + пд = -finin + 1)У
—Antonella Cupillari and Warren Lushbaugh
(independently)

88
Proofs without Words
Sum of Cubes V
2 .2
tn = 1 + 2 + · · · + η => tn - tn_i = и
((тагжгт'Л'.м.'.ур.'лш
У
Zu
ЯД
π
IT ι'ι ийЕ
/»Ш!^
Λ
•^:'·:'·:·:·^·:'·:'·'.·^ί
tn = (1 + 2 + · · · + η)2 = Ι3 + 23 + 33 + · · · + η2
—RBN

Integer Sums
89
Sum of Cubes VI
+
+
+
+
1
2
3
η
2
4
6
In
3 · ·
6 < <
9 - ■
Зи . ·
η I
- 2и
• Ъп I
. и* I
+ Г~2 4|
+ 1 3 6
Гз~
6
9
+ .
+ [..я 2wЗи · · <
и
.2я
Зи
*
«2
Σ ι + 2 Σ
ί=1 ί=1
= Σ i + 2 Σ i + · · · + « Σ i = ΚΙ)2 + 2(2)2 + · · · + и(и)2
и 2
■ (ί,Ο
и
ι=1
.3
—Farhood Pouryoussefi

90
Proofs without Words
Sums of Integers and Sums of Cubes
1 + 2 + · · · + η = ψ{η + 1)
V + 23 +
n3= (jn(n + l)J
1
2
η
их η
n(n +1)
—Georg Schrage

Integer Sums
91
Sums of Odd Cubes are Triangular Numbers
1 =
3
3 =
53.
= D
= з(з2) -.
= 5(52) -■
<—►
1+2(3)
= 111111+ +1111 и
ζ
A
< 2 + 3(5)
(2и - 1)" = (2n - l)(2w - 1)" =
ζ
< (n -1) + я(2и -1) ►
3 _■
'Л
/
Ζ
(2«-l)
ι3 , q3 , c3
Γ5 + 3·* + b* + - + (2n-l)3 = 1 + 2 + 3 + .·· + (2и2-1) = n2(2n2-l)
(и -1) + n(2n -1)
—Monte J. Zerger

92
Proofs without Words
Sums of Fourth Powers
1 = 1 V = i /
k = 2\ i = 1
4 2 2
1 2 1
П 9
Σ·'
i = l
—I 1 1
1 f 1
)Vi 4:;:1
h
--1—ι—ι—ι—ι—i-T—ι—
2, 2 24
3 (1 + 2 )
IK
—1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
—ι—ι—ι—r-i—ι—ι—ι—ι—ι—^—ι—г-1—ι—
2 2 2 2Ч J
4 (1 + 2 + 3 ) 1
1РДДшШР1ш8Ш11Р811
η 9
Σ<
i = \
—Elizabeth M. Markham

Integer Sums
93
№ Powers as Sums of Consecutive Odd
Numbers
nk = (и*-1 - η + 1) + (пк~г - η + 3) + - + (и*-1 - и + 2и - 1);
к = 2,3, ···.
??P!v!?!v!v!\vl^^
¥^f
4<ч
·:·:|.ρ:·:·
■itiliMi^iliiUiiiiilkiliMitfiiili
mm
mm
Ш
$
ι
#
iii'inj
1> ■■ ■■ ■■|«(':vi':«t- «J
η k l-n
*■*
η
—N. Gopalakrishnan Nair

94
Proofs without Words
Sums of Triangular Numbers I
Tn = 1 + 2 + · · · + и => Τ1 + Τ2 + ··· + Τη =
n(n +1)(и + 2)
w+2
Г1+Т2 +
3(Τ1 + Τ2 + ··· + Τ„) = (я+2)Тп
(и+ 2) и(и + 1) и(и+1)(и+2)
+ Г =
—Monte J. Zerger

Integer Sums
95
Sums of Triangular Numbers II
η 1
k = \
I
—RBN

96
Proofs without Words
Sums of Triangular Numbers III
Tk = 1 +2 + · · + A: => 3 Σ Tk = 5«(« + D(« + 2)
A: = 1
1 1 я
12 2 1 и-1 и-1
12 3 3 2 1 я-2 и-2 w-2
12 ··· я-l n-\ n-2 ··· 1 2 2 ··· 2
12 ··· и-1 и п и-1 ··· 2 1 1 1 ··· 1 1
и+2
n+2 и+2
и+2 и+2 и+2
и+2 и+2 ··· и+2
и+2 и+2 ··· и+2 и+2
3(Тг + Т2 + ... + Тп) = VOi + 2)

Integer Sums
97
Sums of Oblong Numbers I
(1χ2) + (2χ3) + (3χ4) + ··· + (η-1)η =
(и - 1)л(л + 1)
(i)
1 1
1 1
(ii)
и + 1
w-1
я+l n-1
и + 1
—Т. С Wu

98
Proofs without Words
Sums of Oblong Numbers II
3(1-2 + 2-3 + 3-4 + · · · + n(n + 1)) = n(n + l)(n + 2)
+ 4
—Sidney H. Kung

Integer Sums
99
Sums of Oblong Numbers III
1г~з
(1х2) + (2хЗ) + --- + (и-1)хи = ι[ηό-η\
3(1 χ 2)
* Γ 1 /\ · ·/ \
23-2
3(1 χ 2)
/—7
-"/I
m
3(1 χ 2)
3(2 χ 3)
/
/ У
τ
v\
/
V
/
3(2 χ 3)
уЗ
p=/7|
тщ
• ■ 1
1 · J
33-3
3(3 χ 4)
3
4 -4
r I I
>
i L-'
5V
—Ali R. Amir-Moez

100
Proofs without Words
Sums of Pentagonal Numbers
1-2 2-5 3-8 n(3n-l) n2(n +1)
—William A. Miller

Integer Sums
101
On Squares of Positive Integers
Tn = 1 + 2 + --- + И
(2n + l)2 = 8Tn + 1
(2n)2 = 8Тп_г + 4и
—Edwin G. Landauer

102
Proofs without Words
Consecutive Sums of Consecutive Integers
η + η + 1
1 + 2 = 3
4+5+6=7+8
9 + 10 + 11 + 12 = 13 + 14 + 15
16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24
η2 + (и2 + 1) + · · · + (η2 + η) = (η2 + η + 1) + · · · + (η2 + 2η)
—RBN

Integer Sums
103
Count the Dots
я и-1
it = l k = i
η 2η
it=l Jt = w + 1
—Warren Page

104
Proofs without Words
Identities for Triangular Numbers
Tn = 1 + 2 + - + и
© © ©
© © ©
©
© ©
© © ©
© © © ©
© © © © ©
©©©©©©
зг + τ = τ
" и-1 2и
ЗТ + Τ = Г
" и+1 2и+1
© ©
© © ©
© © © ©
© © © © ©
б б 9 б О ®
© © ©
© ©
Τ + 6Г + Г = (2и + 1)
и-1 " и+1

Integer Sums
105
A Triangular Identity
49T + 6 = Τ
n 7n+3
Tn = l + 2 + ... + n => (2k + l)% + Tk = T{2k + 1)n + k
—RBN

106
Proofs without Words
Every Hexagonal Number is a
Triangular Number
нп = 1+5 +-+(4n-3)
Tn = 1+2 +-+n
=> Hn = 3V! + Tw = T^ = n(2n-l)
φ
^^
с
φ
Η
®@®€)
5-9

Integer Sums
107
One Domino = Two Squares:
Concentric Squares
1 + 4-2+ 8-2+ 12-2 +16-2 = 92
η
1 + 2 £ 4k = (2и + 1)
Jt = l
-Shirley A. Wakin

108
Proofs without Words
Sums of Consecutive Powers of Nine are Sums
of Consecutive Integers
D
Ψ
щт<
Шш
mm
и
ш
■
■
ί%
'&
7/t
%
ш
i
D
,-J
J
π
m
Ш
1+9 + ···+9" = 1 + 2 + 3 + ··· +(1 + 3 + ··· +3й)
—RBN

Integer Sums
109
Sums of Hex Numbers Are Cubes
о
й = 1 h = 7 h = 19
1 2 3
<^>
К = «3-("-D3
·. h+h + · · -h„ = ΐΐ
12 я

по
Proofs without Words
Every Cube is the Sum of Consecutive Odd
Numbers
3
1 =1
3
2 = 3 + 5
3 = 7 + 9+11
(и+D-l
(и+1)-3
n(w-l)+l
tf = [n(n- !)+!]+ ·· + [и(и+ 1)- 1]
—RBN

The Cube as an Arithmetic Sum
i=0
5 = 1 + 13 + 25 + 37 + 49
—Robert Bronson and
Christopher Brueningsen

Sequences & Series
On a Property of the Sequence of Odd Integers (Galileo, 1615)
A Monotone Sequence Bounded by e .
A Recusively Defined Sequence for e .
Geometric Sums .......
Geometric Series I .
Geometric Series II ......
Geometric Series III ......
Geometric Series IV ..... .
Gabriel's Staircase .......
Differentiated Geometric Series .....
1 1 1 _ _n_
1-2 + 2-3 + '" +и(и+1) ~ и+1 '
The Series of Reciprocals of Triangular Numbers
The Alternating Harmonic Series ....
sin(2n + 1)0 = sin0 + 2sin0Ejt= ι cos2fc0
An Arctangent Identity and Series ....

Sequences & Series
115
On a Property of the Sequence of Odd Integers
(Galileo, 1615)
l
3
1 + 3 1 + 3 + 5
5 + 7 " 7 + 9 + 11
2w-l
_·
\@_
•
©
©
1
©
з|о © ©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
_©]
4и-1
#·
[©_
2и
!©
©
2w
+ 3
©
©
©
+ ι|©
© ©
© ©
© ©
© ©
©
©
©
©
©
©
©
©
©
©
©
©
Ο Ο Ο Ο Ο Ο Ο
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
_©)
1 + 3 + ... + (2и - 1) Ι
(2η + 1) + (2и+ 3) + ... + (4и - 1) " 3
REFERENCE
S. Drake, Galileo Studies, The University of Michigan Press, Ann
Arbor, 1970, pp. 218-219.
—RBN

116
Proofs without Words
A Monotone Sequence Bounded by г
ν^Μΐ+^ίΐ+^Γ1 <e.
w>l=»m1<m2<l
η л+1
—RBN

Sequences & Series
117
A Recusively Defined Sequence for e
e Jtj
Xn
x0> 1 & xn+1 = щр => hmxn = e
—Thomas P. Dence

118
Proofs without Words
Geometric Sums
1/2
1/2*
•
•
• 1
77\
m2
111
2+4+8+
= 1
r(l-r) r(l-r)
r
r(l-r)2
• 1
• 1
3
r(l-r)
r(l-r)
ή\-τΐ
r(l-r)
r(l-r)
r + r(l - r) + r(l - r)2 + ···=!
—Warren Page

Sequences & Series
119
Geometric Series I
oo
Σ «* - ib
и = 0
1
r
\-r
or
a + ar + ar2 + ar* +
ar
Mr \-r
—J. H. Webb

120 Proofs without Words
Geometric Series II
P 1 Q
APQR * ATSP
:. 1 + r + r2 + ... = τ
—Benjamin G. Klein and Irl С Bivens

Sequences & Series
121
Geometric Series III
1/2
1/2
1/4
[. · : ,'*'■';■ §§
1/8
111
w?
Ц
1/8 I
1/4 1
1/2
1/2
4 + 16 + 64 + 256
1
3
СЧ
1
1 1
1 ^
lllij^ei^ei
CO
I-*
CM
Я
r2(l-r) r3 1
r(l-r) r2 1
1-r
(1-r)2 + r2(l-r)2 + r^l-r)2 +
1 + r2 + r4 +
α + ar + ar· +
(1 - rY
(1 - r)2 + 2r(l - r)
1
1-r
1+r
1-r2
д
= 1-r
—Sunday A. Ajose

122
Proofs without Words
Geometric Series IV
Till
AM
^^^^Ш
Pes!
Ιl Д
27 Др]
1
243
<i**i
+ 9· -*-
27 243
00 -ι
и=1 3
00 -ι -ι
У — = λ
я=13 1
)■
Hi 4 Hi
οο ^
3Σ^=ι
и=1 4
У — = χ
я=1 4
[:$:$:·
Ρίίίίί
:>·:·:·:·:
Λ-ΛήΧ&χ':*:^^
§1 1 iill
1 5 Ш1
ШШшШШШ
:5:?::::ή·
■;:|:Йй:х':
ίίίίίίήϊίίίίίίίίίίίίίή
ШхЖ&х&йЖ
χ
25
ШхвЖ
ί·χ·χ·χ·χ·χ·χ·χ·
Ρίίίίίίίίίίίίή·
ШШШЩ
|g§||;;||
||||||
хШхШЗД
•ΧνΧ·Χ·Χ·Χ·Χ·]
::Xx|xjx|:|:|:|x1
^*Υΐ:*ί*ί*"*:*:':**Ί
¥ή¥:¥ί:¥ίίί:3
^и
·χ·χ·:·χ|:·:|ί:·::Η
:::х:х:х:х:х:::н
ίή:ίί:::%·ί:·:·>3
11Ш11
1
125
00 1
я=1 5
я=1 5 *
—Elizabeth M. Markham

Sequences & Series
123
Gabriel's Staircase
Σ krk = . Г ,2 for 0 < r < 1
1
r
1
r2
к
r3
A
• · · 1
1
r
1
r2
A
• I
A
3 Τ ·
r J |
oo oo
Σ^ - Σ Σ
A: = 1 A: = 11 = A:
rl =
(1 - r)2
—Stuart G. Swain

124
Proofs without Words
Differentiated Geometric Series
1 + 2(|) + з4)+4&+... = 4
A
2
▼
1
2
ι ! ι
8 ! 8
1
4
1
1 | 1
16 J 16
1
4
1
4
1
1
1
16
1
16
1
16
•
•
ι Ι
8
1 1
8
1
2

Sequences & Series
125
1 \2
1 + 2r + Зг2 + 4Г3 + ... = (j—f)> 0 < r < 1
1-r r(l-r)
1-r
r
1-r I
"T
1-r
Ί
r2
1
1-r
—RBN

126
Proofs without Words
1-2 + 2-3 +
η
n(n+l) n+1
и/(и + 1)
(я-1)/я
и vn + 1 /
—Roman W. Wong

Sequences & Series
127
The Series of Reciprocals of Triangular
Numbers
ι ι l
l+3+6+
. +
rn
= 2
0-1—V
\n n+\) c^>
n+1·
(T)
f—V-ь
0
η
w+1
—RBN

128
Proofs without Words
The Alternating Harmonic Series
in2 = 1-4 + 4-4 +
► X
ιίλ _ i\ = ι _ i.
2\3 4/ 3 4'
lf!_4\el_l l/i _ l\ = I _ 1.
4\5 6/ 5 6' 4V7 8J 7 8'
1/2" 2" \ 1 1 ifc-l,*,...^*-1;
2Я\2Я+2Л-1 2%2A:/ 2%2fc-l 2"+2fc' « = 1,2,.··.
In2 = f2^ = (l - \) + (\ - 4) +..· = 1- 4 + 4 - 4 +.·. .
Jl * V 2/ \3 4/ 2 3 4
—Mark Finkelstein

Sequences & Series
129
И
sin(2n + l)0 = sir\0+2sin0 Σ cos2fc0
jt = i
2sin0
βίη(2«+1)β
2sin0
—J. Chris Fisher and E. L. Koh

130
Proofs without Words
An Arctangent Identity and Series
(я2+я+1,я3+я2+я)
I h/(\,n)
Y-* ►
(я2+1,я3+я2+я+1)
Ч._л.
arctan
ι—я2+я+1-
(я2+1,(я+1)(я2+1))
arctan я + arctan -5 7 = arctanfo + 1)
η1 + η + 1
1
arctan -5 7 = arctanfa + 1) - arctan я
я2 + я + 1
π
Σ arctan -5 7 = hm.,^ arctan (N + 1) = ;*-
n я2 + я +1 Ν-»00 2
я = 0
-RBN

Miscellaneous
A 2 χ 2 Determinant is the Area of a Parallelogram . . .133
Area of the Parallelogram Determined by Vectors (a,b) and (c,d) . 134
The Characteristic Polynomials of Λ Β and ΒΑ are Equal . .135
The Gaussian Quadrature as the Area of Either Trapezoid . .136
Inductive Construction of an Infinite Chessboard with
Maximal Placement of Nonattacking Queens .137
Combinatorial Identities . . .138
3Σ?= 0 ( 3/) = 8" + 2(-1)n' ЬУ Inclusion-Exclusion
in Pascal's Triangle 139
The Existence of Infinitely Many Primitive Pythagorean Triples . 140
Pythagorean Triples via Double Angle Formulas . .141
The Problem of the Calissons . .142
Recursion ......... 143
η
J] **·(*!) = (и!)*+1 144
fc = l

Miscellaneous
133
A 2 χ 2 Determinant is the Area
of a Parallelogram
(a,b + d) (a + cjb + d)
(a + c,d)
a b\
с d\
ad-he =
Π
□I
•VA
—Solomon W. Golomb

134
Proofs without Words
Area of the Parallelogram Determined by
Vectors (а,Ъ) and (c,d) - ±
с d
= ±(ad-bc)
(erf)
ш
(a,b)
—Yihnan David Gau

Miscellaneous
135
The Characteristic Polynomials of
AB and В A are Equal
-λη\ΑΒ-λΙ\ =
_\(a ab-xi\ _\(a i\(i b\\_\a ι
~\\λΙ 0 / " \A/ b)\0 -λΐ) " \λ1 Β
(-λ)η
-λη\ΒΑ-λΙ\ =
_|/ ο αΛΜ/λ ι\ί-ι о\|_|л /
" \ΒΛ-λί ЯВ/ ~ \λί Β/\Α λΐ)\~\λΙ Β
(-Я)"
—Sidney Η. Kung

136
Proofs without Words
The Gaussian Quadrature as the Area of
Either Trapezoid
\{h - a)(f(fl) + /(b)) = \(b - a)(h(a) + Ш)
y = h(x)
► χ
—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.
2. M. Kraitchik, La Mathematique des Jeux ou Recreations Mathema-
tiaues, Imprimerie Stevens Freres, Bruxelles, 1930, 349-353.
—Dean S. Clark and Oved Shisha

138
Proofs without Words
Combinatorial Identities
я-1
(2")=^2-«)=(.Σ.·
.Я,
( 2 ) = (2)+"
—James О. Chilaka

Miscellaneous
139
3 Σ qv = 8" + 2(-1)η, by Inclusion-Exclusion in
Pascal's Triangle
ι
1 1
1 2 1
13 3 1
• + · +· + · + · + · + · + · + ...+ ·
-у V V ""^
• oo¥oo¥oo¥o... ·
/ и , η
- 2(-l)
—Dean S. Clark

140
Proofs without Words
The Existence of Infinitely Many Primitive
Pythagorean Triples
t ©
e
©
Φ
Q
L k J
n2 = 2k+l => ** + η2 = (fc+1)2 & (k,k+l) = l
—Charles Vanden Eynden

Miscellaneous
141
Pythagorean Triples via Double Angle
Formulas
r
<
m> w>0
m,ne I
sine =
COS0 =
JL
V.
m
/~~2 2
vwi + и
sin20 =
cos20 =
2mn
2 2
w + и
Л
2 2
m - η
2 2 J
tn + η
>
2mn
—David Houston

142
Proofs without Words
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 и 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
A2 = 3
A3 = 2A2 + l A4 = IA3 + I A5 = 2A4
a2 = 3 & An = гл^ + ι <=> л„ = га"-1)-! = 2"-ι
—Shirley Wakin

144
Proofs without Words
η к П + 1
Π к к\ = (η!)
к=1
п-2 п-2 п-2
и-1 п-\ п-\
1
2
3
п-2
1
2
3
п-2
и-1 я-1
1
2
3
п-2
и-1
и и и
НИИ
и + 1
—Edward Т. Н. Wang

Sources
page 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 College Mathematics Journal, vol. 17, no. 5 (Nov. 1986), p. 422.
9 College Mathematics Journal, vol. 20, no. 1 (Jan. 1989), p. 58.
10 Mathematics Magazine, vol. 50, no. 3 (May 1977), p. 162.
11 Mathematics Magazine, vol. 48, no. 4 (Sept.-Oct. 1975), p. 198.
12 College Mathematics Journal, vol. 21, no. 5 (Nov. 1990), p. 393.
13 Mathematics Magazine, in press.
14 College Mathematics Journal, vol. 17, no. 4 (Sept. 1986), p. 338.
15 Mathematics Magazine, vol. 62, no. 3 (June 1989), p. 190.
16 College Mathematics Journal, vol. 22, no. 5 (Nov. 1991), p. 420.
17 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 Mathematics Magazine, vol. 57, no. 4 (Sept. 1984), p. 231.
21 Mathematics Magazine, vol. 64, no. 2 (April 1991), p. 138.
22 Mathematics Magazine, vol. 66, no. 3 (June 1993), p. 180.
23 Mathematics Magazine, vol. 64, no. 2 (April 1991), p. 114.
24 Mathematics Magazine, in press.
25 Mathematics Magazine, in press.

146
Proofs without Words
page source
Trigonometry/ Calculus & Analytic Geometry
29 Mathematics Magazine, vol. 64, no. 2 (April 1991), p. 97.
30 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 317.
31 Mathematics Magazine, vol. 61, no. 2 (April 1988), p. 113.
32 Mathematics Magazine, vol. 63, no. 5 (Dec. 1990), p. 342.
33 Mathematics Magazine, vol. 64, no. 2 (April 1992), p. 103.
34 College Mathematics Journal, vol. 20, no. 1 (Jan. 1989), p. 51.
35 American Mathematical Monthly, vol. 49, no. 5 (May 1942),
p. 325.
36 Mathematics Magazine, vol. 61, no. 5 (Dec. 1988), p. 281.
37 Mathematics Magazine, vol. 61, no. 2 (April 1988), p. 113.
38 Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 267.
39 College Mathematics Journal, vol. 18, no. 2 (March 1987), p. 141.
40 Mathematics Magazine, vol. 42, no. 1 (Jan.-Feb. 1969), pp. 40-41.
41 College Mathematics Journal, vol. 16, no. 1 (Jan. 1985), p. 56.
42 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 Mathematics Magazine, vol. 64, no. 3 (June 1991), p. 175.
45 Mathematics Magazine, vol. 66, no. 1 (Feb. 1993), p. 39.
Inequalities
49 Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 98.
50 Mathematics Magazine, vol. 59, no. 1 (Feb. 1986), p. 11.
51 College Mathematics Journal, vol. 16, no. 3 (June 1985), p. 208.
52 College Mathematics Journal, vol. 19, no. 4 (Sept. 1988), p. 347.
53 Mathematics Magazine, vol. 60, no. 3 (June 1987), p. 158.
54 College Mathematics Journal, vol. 21, no. 3 (May 1990), p. 227.
55 College Mathematics Journal, vol. 20, no. 3 (May 1989), p. 231.
56 Mathematics Magazine, in press.
58 Mathematics Magazine, vol. 60, no. 3 (June 1987), p. 165.
59 Mathematics Magazine, vol. 64, no. 1 (Feb. 1991), p. 31.
60 Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 172.
61 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 of Mathematics, Inc.

Sources
147
page
source
Inequalities (continued)
61 II. Mathematics Magazine, in press.
62 Mathematics Magazine, in press.
63 Mathematics Magazine, vol. 66, no. 1 (Feb. 1993), p. 65.
64 Mathematics Magazine, in press.
65 Mathematics Magazine, in press.
66 College Mathematics Journal, vol. 24, no. 2 (March 1993). p. 165.
Integer Sums
69 Scientific American, vol. 229, no. 4 (Oct. 1973), p. 114.
70 Mathematics Magazine, vol. 57, no. 2 (March 1984), p. 104.
71 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.
II. 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 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 199.
85 Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 74.
86 Mathematics Magazine, vol. 58, no. 1 (Jan. 1985), p. 11.
87 Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 259.
Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 200.
88 Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 178.
89 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 323.
90 Mathematics Magazine, vol. 65, no. 3 (June 1992), p. 185.

148 Proofs without Words
page source
Integer Sums (continued)
91 Mathematics Magazine, in press.
92 Mathematics Magazine, vol. 65, no. 1 (Feb. 1992), p. 55.
93 Mathematics Magazine, vol. 66, no. 5 (Dec. 1993), p. 329.
94 Mathematics Magazine, vol. 63, no. 5 (Dec. 1990), p. 314.
95 College Mathematics Journal, in press.
96 Richard K. Guy, written communication.
97 Mathematics Magazine, vol. 62, no. 1 (Feb. 1989), p. 27.
98 Mathematics Magazine, vol. 62, no. 2 (April 1989), p. 96.
99 College Mathematics Journal, vol. 18, no. 4 (Sept. 1987), p. 318.
100 Mathematics Magazine, vol. 66, no. 5 (Dec. 1993), p. 325.
101 Mathematics Magazine, vol. 58, no. 4 (Sept. 1985), pp. 203, 236.
102 Mathematics Magazine, vol. 63, no. 1 (Feb. 1990), p. 25.
103 Mathematics Magazine, vol. 55, no. 2 (March 1982), p. 97.
104 Richard K. Guy, written communication.
105 Mathematics Magazine, in press.
106 Richard K. Guy, written communication.
107 Mathematics Magazine, vol. 60, no. 5 (Dec. 1987), p. 327.
108 Mathematics Magazine, vol. 63, no. 4 (Oct. 1990), p. 225.
109 American Mathematical Monthly, vol. 95, no. 8 (Oct. 1988),
pp. 701,709.
110 Mathematics Magazine, vol. 66, no. 5 (Dec. 1993), p. 316.
111 Mathematics Magazine, vol. 63, no. 5 (Dec. 1990), p. 349.
Sequences & Series
115 Mathematics Magazine, in press.
116 Mathematics Magazine, in press.
117 Mathematics Magazine, vol. 66, no. 3 (June 1993), p. 179.
118 Mathematics Magazine, vol. 54, no. 4 (Sept. 1981), p. 201.
119 Mathematics Magazine, vol. 60, no. 3 (June 1987), p. 177.
120 Mathematics Magazine, vol. 61, no. 4 (Oct. 1988), p. 219.
121 Mathematics Magazine, in press.
122 Mathematics Magazine, vol. 66, no. 4 (Oct. 1993), p. 242.
123 Mathematics Magazine, in press.
124 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), pp. 332-333.
126 Mathematics Magazine, vol. 65, no. 5 (Dec. 1992), p. 338.
127 Mathematics Magazine, vol. 64, no. 3 (June 1991), p. 167.

Sources
149
page source
Sequences & Series (continued)
128 American Mathematical Monthly, vol. 94, no. 6 (June-July 1988),
pp. 541-42.
129 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 Mathematics Magazine, vol. 64, no. 5 (Dec. 1991), p. 339.
135 Mathematics Magazine, vol. 61, no. 5 (Dec. 1988), p. 294.
136 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.
141 Mathematics Magazine, in press.
142 American Mathematical Monthly, vol. 96, no. 5 (May 1989),
pp. 429-30.
143 Mathematics Magazine, vol. 62, no. 3 (June 1989), p. 172.
144 College Mathematics Journal, vol. 20, no. 2 (March 1989), p. 152.

Index of Names
Ajose, Sunday A. 121
Akerman, Mike 136
Amir-Μοέζ, AH R. 99
Aristarchus of Samos 63
Ayoub, Ayoub B. 43,44
Beekman, Richard M. 45
Bernoulli, Jacques 65
Bhaskara 4
Bicknell, M. 147
Bivens, Irl C. 120
Bronson, Robert 111
Brousseau, Alfred 83
Brueningsen, Christopher 111
Burk, Frank 41
Cauchy, Augustin-Louis 67
Cavalieri, Bonaventura 25
Chilaka, James O. 76,80,138
Chuang, Pi-Chun 81
Chuquet, Nicolas 61
Clark, Dean S. 137,139
Coble, Scott 13
Courant, Richard 42
Cupillari, Antonella 87
David, Guy 142
DeKleine, H. Arthur 36
Dence, Thomas P. 117
Diophantus of Alexandria 22
Drude, K. Ann 21
Dudeney, Η. Ε. 6
Eddy, Roland H. 16,18,51
Eisenman, R. L. 40
Eisner, Thomas 10
Euclid 5
Eves, Howard 145
Fibonacci (Leonardo of Pisa) 83
Finkelstein, Mark 128
Fisher, J. Chris 129
Fry, Alan L. 86
Galileo Galilei 115
Gallant, Charles D. 19,49,59
Gardner, Martin 69, 74, 78
Garfield, James A. 7
Gau, Yihnan David 134
Gentile, Enzo R. 9
Gibbs, Richard A. 60
Golomb, Solomon W. 84,133
Gundlach, Bernard H. 147
Guy, Richard 148
Hardy, Michael 8
Harris, Edward M. 39
Hoggatt, V. E. 147
Houston, David 141
Isaacs, Rufus 11
Kalman, Dan 78
Kim, Нее Sik 75
Kincanon, Eric 12
Klein, Benjamin G. 120
Koh, E. L. 129
Kung, Sidney H. 17,25,29,30,
32,33,54,79,98,135
Landauer, Edwin G. 101
Lehel, Jen6 73
Li Changming 61
Logothetti, Dave 23,82
Love, J. Barry 85
Lushbaugh, Warren 87

152
Proofs without Words
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
Page, Warren 52,103,118
Pascal, Blaise 139
Pooley, Sam 21
Pouryoussefi, Farhood 89
Ptolemy, Claudius 33
Pythagoras 3-9,140,141
Richards, Ian 70
Romaine, William 37
Schattschneider, Doris 50
Schrage, Georg 90
Schwarz, Hermann A. 67
Shisha, Oved 137
Sipka, Timothy A. 31
Siu, Man-Keung 77
Snover, Steven L. 82
Swain, Stuart G. 123
Tomei, Carlos 142
Vanden Eynden, Charles 140
Viviani, Vincenzo 15
Wakin, Shirley 20,107,143
Wang, Edward Т. Н. 144
Walker, R. J. 35
Webb, J. H. 119
Wolf, Samuel 15
Wong, Roman W. 126
Wu, Т. С 97
Zerger, Monte J. 91,94
Technical Note
The manuscript for this book was edited and printed
using Microsoft® Word 5.1 on an Apple®
Macintosh™ Ilfx computer. The graphics were produced
in Claris™ MacDraw® Pro 1.5vl. 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.

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 His expository 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
and Related Fields, Statistics and Probability Letters, Sankhya, and
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 chat heip the
reader see why a particular mathematical statement is true, and
also to see how one could begin to go about proving it true.
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, as in "to see
the point of an argument."
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.
EXERCISES IN
SBNO 8838i 700-6