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MARTIN GARDNER
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WHEELS, LIFE
AND OTHER
MATHEMATICAL
AMUSEMENTS
W. H. Freeman and Company
New York
Lil~rar! of Cor~gr-e,sC ataloging in Public,ition Data
Gardner, Xlartin, 191.1-
Wheels, life, and othrr mathenlatical aniusemenrs.
Inclutles bibliograpliirs .~ndi ntlex.
1. Rlathernatical recl-eations. I. Title.
(LA9.5.(;333 1983 793.7'4 83-1 1.592
ISBN 0-7167-1588-0
ISBN 0-5 167- 1.589-1) (pbk.)
Life configurations conrtes! of R. LV~lliarnG oaper of S\nit)olic~.
Jacket photograp11 b\ Jenn~terL 17alsli
Cop>right C 1983 h! 1V.H. Freeman and Cornpan)
No part of ttiis book ma! be reprodrlced h\- an! mechanical, pl~otographic.o r
electronic process, or it1 the form of a phonog-I-apl~irce cortling, nur nla\ it be
stored in a retrieval s!stem, trari~rriitted.o r other\vise copird for public or 1)l.i-
vate use, \vithout witten permissio~lf rom the puhli5her.
PRINTED IS THE L'SITEI) S'I'AI'ES OF AAlt-KIC .a
Ninth printing 1996, VB
For Ronald L. Graham
who juggles nunlbers
and other mathematical objects
as elegantly
as he juggles balls and clubs, and trvirls himself
on the trampoline
CONTENTS
Introduction vii
Wheels 1
Diophantine Analysis and Fermat's Last
Theorem 10
The Knotted Molecule and Other Problems 20
Alephs and Supertasks 31
Nontransitive Dice and Other Probability
Paradoxes 40
Geometrical Fallacies 51
The Combinatorics of Paper Folding 60
A Set of Quickies 74
Ticktacktoe Games 94
Plaiting Polyhedrons 106
The Game of Halma 115
Advertising Premiums 124
Salmon on Austin's Dog 134
Nim and Hackenbush 142
Golomb's Graceful Graphs 152
Charles Addams' Skier and other Problems 166
Chess Tasks 183
+
Slither, 3X 1, and Other Curious Questions 194
19. Mathematical Tricks With Cards 206
20. The Game of Life, Part l 214
21. The Game of Life, Part 11 226
22. The Game of Life, Part Ill 241
Name Index 259
INTRODUCTION
"There remains one more game."
"LVhat is it?"
"Ennui," I said. "The easiest of all. No
rules, no boards, no equipment."
"LVhat is Ennui?" Amanda asked.
"Ennui is the abqence of games."
-Donald Barthelme, Guzltj Pltatulti
Unfortunately, as recent studies of' education in this country
have made clear, one of the chief characteristics of mathemat-.
ical classes, especially on the lo~verle vels of public education, is
ennui. Some teachers may be poorly trained in mathematics
and others not trained at all. If mathematics bores them, can
you blame their students for being bored?
Like science, mathematics is a kind of game that we play ~vith
the universe. The best mathematicians and the best teachers of
mathematics obviously are those who both understand the
rules of the game, and who relish the excitement of playing it.
Raymond Smullyan, who has enormous zest for the games of
philosophy and mathematics, once taught an elementary
course in geometry. In his delightful book 5000 U.C. and 0 t h ~ ~
Philosophical Fantasies (1983) he tells how- he once introduced
his students to the Pythagorean theorem:
I drew a right triangle on the board with squares on the hy-
potenuse and legs and said, "Obviously, the square on the hy-
potenuse has a larger area than either of the other two
squares. Now suppose these three squares were made of
beaten gold, and you were offered either the one large
square or the two small squares. Which would you choose?"
Interestingly enough, about half the class opted for the one
large square and half for the two small ones. A lively argu-
ment began. Both groups were equally amazed when told
that it would make no difference.
INTRODUCTION
It is this sense of surprise that all great mathematicians feel,
and all great teachers of mathematics are able to communicate.
I know of no better way to do this, especially fbr-b eginning stu-
dents, than by way of games, puzzles, paradoxes, magic tricks,
and all the other curious paraphernalia of' "recreational
mathematics."
"Puzzles and ganies provide a rich source of example
problems useful for illustrating and testing problem-solving
methods," wrote Nils Nilsson in his widely used textbook
Problem-Solving Met!~od,si n ArtlJicial Intelligence. He quotes Mar-
vin Minsky: "It is not that the games and mathematical prob-
lems are chosen because they are clear and simple; rather it is
that they giveus, for the smallest initial structures, the greatest com-
plexity, so that one can engage some really formidable situations
after a relatively rninirnal diversion into programming."
Nilsson and Minsky had in mind the value of recreational
mathematics in learning how to solve pr-ol~lernsb y computers,
but its value in learning how to solve problems by hand is just
as great. In this book, the tenth collection of the Mathenlatical
Games colurrins that I wrote for- Scic.ntzjic American, you will
find an assortment of mathematical recreations of every vari-
ety. 'I'he last three chapters (the third was written especially for
this volume) deal with John H. C;onway's fantastic game of
Life, the full wonders of which are still being explored.
The two previously published articles on Life, in which I had
the privilege of introducing this game for the first time,
aroused more interest among computer buffs around the
world than any other columns I have written. Now that Life
software is beconlirlg available for home-computer screens,
there has been a renewed interest in this remarkable recrea-
tion. Although Life rules are incredibly simple, the complexity
of its structure is so awesome that no one can experiment with
its "life forms" without being overwhelmed by a sense of the
infinite range and depth and mystery of mathematical struc-
ture. Few have expressed this emotion Inore colorfully than the
British-American mathematician .James J. Sylvester:
Mathematics is not a book confined within a cover and bound
between brazen clasps, whose contents it needs orily patience
. .
to ransack; it is not a rnine, whose treasures rnay take long to
reduce into possession, but which fill only a limited number
of veins and lodes; it is not a soil, whose terrility can be ex-
hausted by the yield of successive harvests; it is not a conti-
nent or an ocean, whose area can be mapped out and its ron-
tour defined: it is linlitless as that space which it finds too
INTRODUCTION
narrow for its aspirations; its possibilities are as infinite as the
worlds which are forever crowding in and multiplying upon
the astronomer's gaze; it is as incapable of being restricted
within assigned boundaries or being reduced to definitions of'
permanent validity, as' the consciousness, the life, which
seems to slumber in each monad, in every atom of matter, in
each leaf and bud and cell, and is forever ready to burst forth
into new forms of vegetable and animal existence.
Martin Gardner
WHEELS
The miraculous paradox of smooth round
objects conquering space by simply tumbling
over and over, instead of laboriously lifting
heavy limbs in order to progress, must have
given young mankind a most salutary shock.
Things would be very different without the wheel. Transpor-
tation aside, if we consider wheels as simple machines-pulleys,
gears, gyroscopes and so on-it is hard to imagine an) ad-
vanced society without them. H. G. Wells, in The War of the
Worlds, describes a Martian civilization far ahead of ours but
using no wheels in its intricate machinery. Wells may have in-
tended this to be a put-on; one can easily understand how the
American Indian could have missed discovering the wheel, but
a society capable of sending spaceships from Mars to the earth?
Until recently the wheel was believed to have originated in
Mesopotamia. Pictures of wheeled Mesopotamian carts date
back to 3000 B.C. and actual remains of massive disk wheels
have been unearthed that date back to 2700 B.C. Since FYorld
War 11, however, Russian archaeologists have found potter\
models of wheeled carts in the Caucasus that suggest the wheel
may have originated in southern Russia even earlier than it did
in Mesopotamia. There could have been two or more inde-
pendent inventions. Or it may have spread by cultural diffu-
sion as John Updike describes it in a stanza of his poem, Whe~l:
Description:Like science, mathematics is a kind of game that we play ~vith . models of wheeled carts in the Caucasus that suggest the wheel may have
