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CLAiSIC BRAINTEASERS AND
OTHER TIMELESS MATHEMATICAL
GAMES OF THE LAST 1 0 CENTURIES
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Dominic Olivastro
BANTAM BOOKS
NEW YORK TORONTO LONDON SYDNEY AUCKLAND
ANCIENT PUZZLES
A Bantam Book/December 1993
See page 280 for acknowledgments.
All rights reserved.
Copyright © 1993 by Dominic Olivastro.
Book design by Glen M. Edelstein.
No part of this book may be reproduced or transmitted in any
form or by any means, electronic or mechanical, including
photocopying, recording, or by any information storage and
retrieval system, without permission in writing from the publisher.
For information address: Bantam Books.
Library of Congress Cataloging-in-Publication Data
Olivastro, Dominic.
Ancient puzzles : classic brainteasers and other timeless
mathematical games of the last ten centuries / Dominic Olivastro.
p. cm.
ISBN 0-553-37297-1
1. Mathematical recreations. 1. Title.
QA95.045 1994
793.7'4-dc2O 93-1985
CIP
Publisheds imultaneously in the United States and Canada
Bantam Books are published by Bantam Books, a division of Bantam
Doubleday Dell Publishing Group, Inc. Its trademark, consisting of the
words "Bantam Books" and the portrayal of a rooster, is Registered in
U.S. Patent and Trademark Office and in other countries. Marca
Registrada. Bantam Books, 1540 Broadway, New York, New York 10036.
PRINTED IN THE UNITED STATES OF AMERICA
0987654321
KING NEFERKIRE HAS BEGUN COUNTING
ON HIS FINGERS
- THE BOOK OF THE DEAD
To my Mother, Mary,
and my Father, Manfredo ...
... and to King Neferkire
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1,
T WOULD HAVE BEEN SIMPLE TO WRITE A BOOK CALLED THE
Classic Puzzles of All Time, and a second book called The Histories of Xl
Classic Puzzles. This book is neither. This book is an attempt to
9
merge the two into a single work. The obvious danger is that I will
disappoint readers who would have been interested in either of the two
books separately, but I hope I have struck such a note that everyone will All
find a familiar friend in an unfamiliar setting.
My obsession with ancient puzzles started early on. Like many in my
generation, I grew up on Martin Gardner's monthly essay on mathe-
matical games in Scientific American, and when a specific puzzle attracted
my attention I spent an improper amount of time tracking down its
1
origins in libraries. Often it turned up in the manuscripts of a pharaoh's
scribe or the letters of a medieval monk; in these cases the puzzle, once
merely interesting, became more like a relic. So much of this ancient
writing has an enduring charm, largely because the older writers were
able to find mysteries in simple things.
Consider the story of Eve's stay in paradise-here we have what the
author believes to be the origin of life and sin, yet there is no thunder or
Atl
lightning, Instead, it begins with a bone and it ends with a tree. All
deep and abiding literature is couched in simple terms like this. I hope ill
some of that charm can be garnered from this book. Certainly there are
I
puzzles enough to hold anyone's attention, especially novices; but even
2 I NTRO D UCTI O N
experts, or those who do not especially care to solve puzzles, will find
food for thought in the anecdotal sections.
In digging up the ruins of ancient puzzles, we are something like
archaeologists of logic. In this undertaking, we may have two experi-
ences that are as rewarding as, say, uncovering a lost city. First, we may
find a modern puzzle occurring only slightly changed at an improbably
early date. Second, we may find a dead puzzle, now hardly a puzzle at
all, attracting an inordinate amount of attention in a past civilization.
The Egyptians, for example, had a difficult time dividing five loaves of
bread among three workers. Is the latter type of puzzle uninteresting?
With our modern puzzle-solving methods, yes. But to anyone inter-
ested in the development of these methods, no. In our modern notation,
simply stating the problem is solving it: 5 divided by 3 is, well, 5/3. But
the Egyptians did not possess our notation. In cases like this, it is
important to keep in mind exactly how the ancient people themselves
went about solving their own problems, even if this forces us to
abandon our tried-and-true methods. Solving a problem in this ancient
way, without the essential tools, is actually a very difficult task-like
thinking without words. But it is well worth doing because it will tell
you a great deal about both thinking and words.
My first attempt at writing this book was an article I wrote for The
Sciences, that marvelous, lively, and-this is unusual these days-
highly accurate journal of popular science.1 Even while writing the
article, I was struck by an inevitable question: Why do puzzles arise at
all? Some answer this with the analogy of a roller coaster. We invent
problems that do not exist in the real world-adding nothing to our
lives when we solve them-for the sheer pleasure of it, like seeking out
rides that rise and fall at breakneck speeds, taking us nowhere. I think a
better analogy is that of the earliest primitive carpenter. He has just
invented the first hammer. What does he do with it? Unfortunately, the
poor fellow lives in a village of grass huts, so there is nothing around
him that needs building. To pass his time, he bangs together crazy
lopsided wooden structures just for the sake of using his hammer. No
I "A sampler of Ancient Conundrums," The Sciences, January/February 1990. Interested readers may wish to obtain subscrip-
tions at $18.00 per year. Write to The Sciences, 2 East 63rd Street, New York, NY 10021. Or call 1-800-THE-NYAS.
INTRODUCTION 3
one asks to have them built; no one uses them after they are built. The
structures are junk, but if you don't understand them you might think
the carpenter, who is really a genius, is just a lunatic who makes a lot of
noise.
Puzzles are logical junk. They arise when our reasoning ability
outpaces any problem in the real world that needs to be reasoned
about. They are meaningless, profitless, unusable, silly, insignificant,
inconsequential-but without them highly intelligent people would
just be lunatics who make a lot of noise.
The hammer in our analogy is the number system-the ten digits 0,
1, 2, 3, 4, 5, 6, 7, 8, 9-and the notation, in which the value of a digit
depends on its position in the number. In the number 110, for example,
the middle "1" represents 10, while the left-most "1" represents 100.
When I was young, we were taught to call this the "Hindu-Arabic
number system," which not too inaccurately explained its historical
origins. Sometime later, it was decided that the numbers should be
given a functional name, and so they were denuded of their culture.
Most readers probably have been raised to call it simply the "positional
number system." In the course of human development, nothing is of
greater consequence-not the wheel, not fire, not nuclear energy-
than this number system. We, today, are a little jaded, so we think our
numbers are nothing more than a counting aid, no different from any
other number system. But the way in which our numbers tick off from 0
to 9, push the next digit up, then start all over, is actually an extraordi-
nary device that is capable of mirroring the purely logical workings of
the world. It is not farfetched to say that the history of puzzles is the
history of ancient people groping toward the positional number system.
Whenever appropriate, I have included in each chapter the numbers
and arithmetic that were used to solve that chapter's puzzles. This will
add flesh to the bare bones of the puzzles, and perhaps, too, it will
return some of the history that was lost.
This book is meant to be fun, but the introduction to any book, even
one that aspires only to entertain, is meant for pontificating. So, before
the fun begins, let me worry the reader about some thoughts that have
dogged me during the last few months.
There are two modern trends that may lead some to misinterpet this
book. The first is a movement that has coined the terrible words
4 I N T R O D U C T I O N
"multiculturalism" and "ethnocentrism." It is a movement that resents
the center that Europe, or the West, has occupied for so many years. By
way of correction, it has tried to emphasize the importance of other
parts of the world-thus, we have "multicultural science," even "ethno-
centric mathematics." Like most horrors, this started innocently
enough, but lately it has degenerated into a kind of snotty ancestor
worship. In the following chapters there will be many examples in
which Europe is compared unfavorably to other parts of the world. This
is unavoidable. One cannot go far in the history of anything "Western,"
especially science and mathematics, without finding that much of it
actually originated in places like China. But I hope I have never adopted
the scolding attitude of some writers. Reading history should be enter-
taining. In any case, the history of mathematics can never be more
important than mathematics itself, and for better or worse (I choose the
former) today and for the foreseeable future mathematics is largely a
Western affair.
The second trend is a movement toward irrationality, by which I
mean the disturbing rise in interest in such superstitions as astrology,
numerology, psychic phenomena, and so on. Just as you may find
examples of multiculturalism in this book, you may also find examples
of superstitions. In ancient times puzzles were intimately connected
with spiritual matters. This may seem strange at first, but actually it is
quite reasonable. Puzzles explain something that is invisible, an orderli-
ness that cannot actually be touched-the "obscure secrets" of the
world, as the scribe Ahmes once put it, believing he caught a glimpse of
the Deity's mind. One is reminded of what Gottfried Wilhelm Leibnitz
once said: "The Supreme Being is one who has created and solved all
possible games." There may be some truth in this. Perhaps God first
created all possible magic squares, then decided that every action
should have an equal and opposite reaction. Perhaps God first solved all
configuration games, then decided that space should have exactly three
dimensions. Perhaps God first solved all possible odd-coin problems,
then decided that every physical system would tend toward maximum
entropy. As we solve these puzzles, are we not really discovering the
workings of the world? It is likely that ancient people thought this way.
The superstitions that arose in ancient times should not be dismissed
out of hand; they are an important part of the puzzles themselves.
Consider the cult of Isis that flourished in Egypt around the time of
