The Liar Paradox A N D T H E Towers of Hanoi Marcel Danesi John Wiley & Sons, Inc. THE 10 GREA TEST M A TH PUZZLES OF ALL TIME The Liar Paradox A N D T H E Towers of Hanoi Marcel Danesi John Wiley & Sons, Inc. THE 10 GREA TEST M A TH PUZZLES OF ALL TIME Copyright © 2004 by Marcel Danesi. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada Design and production by Navta Associates, Inc. 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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, 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 10 9 8 7 6 5 4 3 2 1 visit our web site at www.wiley.com. To Alex and Sarah; their existence is a gift and their life is a blessing. ACKNOWLEDGMENTS vii Introduction 1 The Riddle of the Sphinx 5 Alcuin’s River-Crossing Puzzle 27 Fibonacci’s Rabbit Puzzle 47 Euler’s Königsberg Bridges Puzzle 67 Guthrie’s Four-Color Problem 85 Lucas’s Towers of Hanoi Puzzle 105 Loyd’s Get Off the Earth Puzzle 125 Epimenides’ Liar Paradox 141 The Lo Shu Magic Square 159 The Cretan Labyrinth 177 ANSWERS AND EXPLANATIONS 191 GLOSSARY 237 INDEX 243 ᮤ v ᮣ CONTENTS 1 2 3 4 5 6 7 8 9 10 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. ᮤ vii ᮣ ACKNOWLEDGMENTS [...]... with mathematics The answer to this will become obvious as they work their way through this chapter Simply put, in its basic structure, the Riddle ᮤ 5ᮣ 6 ᮣ The Liar Paradox and the Towers of Hanoi of the Sphinx is a model of how so-called insight thinking unfolds And this form of thinking undergirds all mathematical discoveries The Puzzle According to legend, when Oedipus approached the city of Thebes,... 20 ᮣ The Liar Paradox and the Towers of Hanoi Since the four powers of 3 represent our weights, all we have to do is “translate” addition in the previous layout as the action of putting weights on the left pan and subtraction as the action of putting weights on the right pan (along with the sugar) The following chart gives an indication of how this can be done Readers may wish to complete it on their... least number of straight lines required to do so? 14 Finally, connect sixteen dots without lifting your pencil What is the least number of straight lines required this time around? 24 ᮣ The Liar Paradox and the Towers of Hanoi Further Reading The following list contains collections of puzzles and complementary treatments of the role of puzzles in the development of mathematics Averbach, Bonnie, and Orin... bewilderment and confusion, at the same time that they challenge our wits As Helene 18 ᮣ The Liar Paradox and the Towers of Hanoi Hovanec has stated in her delectable book The Puzzler’s Paradise (see Further Reading), the lure of puzzles lies in the fact that they “simultaneously conceal the answers yet cry out to be solved,” piquing solvers to pit “their own ingenuity against that of the constructors.” Consider... granted 4 ᮣ The Liar Paradox and the Towers of Hanoi Reflections After the mathematical annotations, I have added my own reflections on the puzzle or its mathematical implications Explorations This section provides follow-up exploratory exercises that allow readers to engage directly in puzzle-solving Answers and detailed explanations to the exercises are found at the back of the book A word of advice... through which they could gain knowledge This explains why the Greek priests and priestesses (called oracles) expressed their prophecies in the form of riddles The implicit idea was, evidently, that only people who could penetrate the language of the message would unravel its concealed prophecy 8 ᮣ The Liar Paradox and the Towers of Hanoi However, not all riddles were devised to test the acumen of mythic... ——— The Colossal Book of Mathematics New York: Norton, 2001 ——— Gotcha! Paradoxes to Puzzle and Delight San Francisco: Freeman, 1982 ——— The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications New York: Copernicus, 1997 The Riddle of the Sphinx ᮤ 25 ——— Mathematics, Magic, and Mystery New York: Dover, 1956 ——— Riddles of the Sphinx and Other Mathematical Tales Washington, D.C.: Mathematical... ABCD is one such figure: 12 ᮣ The Liar Paradox and the Towers of Hanoi Notice that this figure can be divided into two triangles as shown (triangle ABC and triangle ADC) By doing this, we have discovered that the sum of the angles in the quadrilateral is equivalent to the sum of the angles in two triangles, namely, 180° + 180° = 360° Next, consider the case of a pentagon (a five-sided figure) ABCDE, as follows,... that I chose the ten best In reality, I went on a mathematical dig to unearth ten puzzles that ᮤ 1ᮣ 2 ᮣ The Liar Paradox and the Towers of Hanoi were demonstrably pivotal in shaping mathematical history and that, I believe, most mathematicians would also identify as among the most important ever devised The Uses of This Book Above all else, this book can be read to gain a basic understanding of what puzzles... with the sugar, is equivalent to taking its weight away from the total weight on the left pan Think about this for a moment For example, if 2 pounds of sugar are to be weighed, we would put the 3-pound weight on the left pan and the 1-pound weight on the right pan The result is that there are 2 pounds less on the right pan We will therefore get a balance when we pour the missing 2 pounds of sugar on the . The Liar Paradox A N D T H E Towers of Hanoi Marcel Danesi John Wiley & Sons, Inc. THE 10 GREA TEST M A TH PUZZLES OF ALL TIME The Liar Paradox A N D T H E Towers of Hanoi Marcel Danesi John. products, 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. 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