The liar paradox and the towers of Hanoi: the 10 greatest puzzles of all time

257 Pages·2004·1.309 MB·English

by  Marcel Danesi

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The Liar Paradox
AND THE
Towers of Hanoi
THE 10 GREATEST MATH PUZZLES OF ALL TIME
Marcel Danesi
John Wiley & Sons, Inc.
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Copyright © 2004 by Marcel Danesi. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Danesi, Marcel, date.
The liar paradox and the towers of Hanoi: the ten greatest
math puzzles of all time / Marcel Danesi.
p. cm.
Includes bibliographical references and index.
ISBN 0-471-64816-7 (paper)
1. Mathematical recreations. I. Title.
QA95 .D29 2004
793.74—dc22
2003027191
Printed in the United States of America
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ToAlex and Sarah;
their existence is a gift and their life is a blessing.
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CONTENTS
ACKNOWLEDGMENTS vii
Introduction 1
1 The Riddle of the Sphinx 5
2 Alcuin’s River-Crossing Puzzle 27
3 Fibonacci’s Rabbit Puzzle 47
4 Euler’s Königsberg Bridges Puzzle 67
5 Guthrie’s Four-Color Problem 85
6 Lucas’s Towers of Hanoi Puzzle 105
7 Loyd’s Get Off the Earth Puzzle 125
8 Epimenides’ Liar Paradox 141
9 The Lo Shu Magic Square 159
10 The Cretan Labyrinth 177
ANSWERS AND EXPLANATIONS 191
GLOSSARY 237
INDEX 243
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ACKNOWLEDGMENTS
I wish to thank the many people who have helped me, influenced me, and
critiqued me over the years. First and foremost, I must thank all of the stu-
dents I have had the privilege of teaching at the University of Toronto. They
were a constant source of intellectual animation and enrichment. I must
also thank Professor Frank Nuessel of the University of Louisville, for his
unflagging help over many years. I am, of course, grateful to the editors at
John Wiley for encouraging me to submit this manuscript to a publishing
house that is renowned for its interest in mathematics education. It is my
second book for Wiley. I am particularly grateful to Stephen Power, Jeff
Golick, and Michael Thompson for their expert advice, and to Kimberly
Monroe-Hill and Patricia Waldygo for superbly editing my manuscript,
greatly enhancing its readability. Needless to say, any infelicities that this
book may contain are my sole responsibility.
Finally, a heartfelt thanks goes out to my family, which includes Lucy
(my wife), Alexander and Sarah (my grandchildren), Danila (my daughter),
Chris (my son-in-law), and Danilo (my father), for the patience they have
shown me during the research and the writing of this book. I truly must beg
their forgiveness for my having been so cantankerous and heedless of fam-
ily duties.
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Introduction
P
UZZLESAREASOLDASHUMANHISTORY. They are found in cultures through-
out the ages. Why is this so? What are puzzles? What do they reveal
about the human mind? Do they have any implications for the study of
mathematics?
This book attempts to answer some of these questions. Its main focus is
on showing how certain ideas in mathematics originated in the form of
puzzles. I use the word puzzle in its basic sense, to mean a challenging
problem that conceals a nonobvious answer. I do not use it in the figurative
sense of “anything that remains unsolved,” even though the two meanings
share a lot of semantic territory, as the mathematician Keith Devlin recently
demonstrated in his fascinating book on the seven greatest unsolved math-
ematical puzzles of our time (The Millennium Problems, Basic Books, 2002).
In the humanities and the arts, there is a long-standing tradition of iden-
tifying the masterpieces—the great novels, the great symphonies, and the
great paintings—as the most illuminating things to study. Books are written
and courses taught on them. Mathematics, too, has its “great” problems.
Significantly, most of these were originally devised as clever puzzles. So, in
line with teaching practices in literature, music, and the fine arts, this book
introduces basic mathematical ideas through ten puzzle masterpieces.
Needless to say, so many ingenious puzzles have been invented throughout
history that it would be brazenly presumptuous to claim that I chose the ten
best. In reality, I went on a mathematical dig to unearth ten puzzles that
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