MARTIN GARDNER KNOTTED DOUGHNUTS KNOTTED DOUGHNUTS AND OTHER MATHEMATICAL ENTERTAINMENTS MARTIN GARDNER KNOTTED DOUGHNUTS AND OTHER MATHEMATICAL ENTERTAINMENTS W. H. Freeman and Company New York Library of Congress Cataloging-in-Publication Data Gardner. Martin, 1914 - Knotted doughnuts and other mathematical entertainments Includes bibliographies and index. 1. Mathematical recreations. I. Title. QA95.GZ7 1986 793.7'4 85-31134 ISBN 0-7 167-1794-8 ISBN 0-7 167-1799-9 (pbk.) Copyright O 1986 by W. H. Freeman and Company No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use, without written permission from the publisher. Printed in the United States of America To Gerry Pie1 and Dennis Flanagan and all my other good friends at Scientific American during the 25 years that I had the great privilege of writing the magazine's Mathematical Games column contents Preface xi CHAPTER ONE Coincidence 1 CHAPTER TWO The Binary Gray Code 11 CHAPTER THREE Polycubes 2 8 CHAPTER FOUR Bacon's Cipher 45 CHAPTER FIVE Doughnuts: Linked and Knotted 5 5 CONTENTS x CHAPTER SIX The Tour of the Arrows and Other Problems 6 8 CHAPTER SEVEN Napier's Bones 85 Napier's Abacus CHAPTER NINE Sim, Chomp and Racetrack CHAPTER TEN Elevators 123 CHAPTER ELEVEN Crossing Numbers 133 CHAPTER TWELVE Point Sets on the Sphere 145 CONTENTS xi Newcomb's Paradox 155 CHAPTER FOURTEEN Reflections on Newcomb's Paradox CHAPTER FIFTEEN Reverse the Fish and Other Problems 176 Look-See Proofs 192 CHAPTER SEVENTEEN Worm Paths 205 Waring's Problems CHAPTER NINETEEN Cram, Bynum and Quadraphage 232 CONTENTS xii CHAPTER TWENTY The I Ching 244 CHAPTER TWENTY-ONE The La$er Curve 257 Index of Names 273 Because this is the eleventh collection of my Scientijc American columns, there is little to say in a preface that I have not said before. As in earlier volumes, I have made corrections and additions throughout and included addendurns to material sent by readers and to update chapters in ways that were not easy to squeeze into the earlier text. References cited in the chapters are given more fully in the bibliographies that follow the chapters. Martin Gardner KNOTTED DOUGHNUTS AND OTHER MATHEMATICAL ENTERTAINMENTS CHAPTER ONE Coincidence Don't worry. Lightning never strikes twice in the same -BILLY BEE Since the beginning of history, unusual coincidences have strengthened belief in the influence on life of occult forces. Events that seemed to miraculously violate the laws of probability were attributed to the will of gods or devils, God or Satan, or at the very least to mysterious laws unknown to science and mathematics. On the other hand, skeptics have argued that in the unthinkably intricate snarls of human history, with billions on billions of events unfolding every second around the globe, the situation is really the other way around. It is surprising that more strange coincidences are not publicized. "Life," wrote G. K. Chesterton in Alarms and Discursions, "is full of a ceaseless shower of small coincidences. It is this that lends a frightful plausibility to all false doc- trines and evil fads. There are always such props of accidental arguments upon anything. If I said suddenly that historical truth is generally told by red-haired men, I have no doubt that ten minutes' reflection (in which I decline to indulge) would provide me with a handsome list of instances in support of it." "We trip over these trivial repetitions and exactitudes at every turn," Ches- terton continued, "only they are too trivial even for conversation. A man named Williams did walk into a strange house and murder a man named William- son. A journalist of my acquaintance did move quite unconsciously from a place called Overstrand to a place called Overroads." CHAPTER ONE 2 In his Poetics, Aristotle attributes to Agathon the remark that it is probable that the improbable will sometimes happen. All the same, most coincidences surely go unrecognized. For instance, would you notice it if the license plate of a car just ahead of you bore digits that, read backward, gave your telephone number? Who except a numerologist or logophile would see the letters U,S, A symmetrically placed in LOUISIANA or at the end O~JOHNPHILIP SOUSA, the name ofthe composer of our greatest patriotic marches? It takes an odd sort of mind to discover that Newton was born the same year that Galileo died, or that Bobby Fischer was born under the sign of Pisces (the Fish). That's not all. "Fish" is chess slang for a mediocre player. In 1972, when Bobby Fischer's blunder cost him the first game in his famous match in Iceland with Boris Spassky, he said afterward, "I'm a fish! I played like a fish!" There are two other reasons why strange coincidences are seldom recorded. When trivial ones are noticed, it is easy to forget them, and when they are remarkable enough to be remembered, one may hesitate to speak about them for fear of being thought superstitious. Skeptics maintain that with all of this in mind the number of astonishing- coincidences that continually occur as the result of ordinary statistical laws is far greater than even occultists realize. The ancient view that many coincidences are too improbable to be explained by known laws has recently been revived by Arthur Koestler. In his book The Roots of Coincidence, he devotes many pages to a theory developed by Paul Kammerer, an eccentric Austrian biologist, whose Lamarckian convictions were much admired by T. D. Lysenko and who was the hero of Koestler's previous book, The Case of the Midwife Toad. Kammerer wrote a book, Das Gesetz der Serie (1919), about his theory of coincidences. It describes exactly 100 coincidences -concerning words, numbers, people, dreams and so on - that he had collected over a period of 20 years. Kammerer's seventh coincidence is typical. On September 18,1916, his wife was in a doctor's waiting room admiring magazine reproductions of ~aintings by a man named Schwalbach. A door opened and the receptionist asked if~riu Schwalbach was in the room. Kammerer's 10th coincidence is even more impressive. Two soldiers were separately admitted to the same hospital. They were 19,had pneumonia, were born in Silesia, were volunteers in the Transport Corps and were named Franz Richter. Kammerer was ~ersuadedthat such oddities could be accounted for only by assuming a universal law, independent of physical causality, that brought "like and like together." Koestler is sympathetic to this view. He suggests that some of the results of parapsychology, such as the tendency of falling dice to show a certain number more often than expected, can be explained not as the influence COINCIDENCE 3 of mind on matter but as coincidences produced by a transcendent "integrative tendency. " Estimating the probability that a hidden law is at work behind a series of apparent coincidences is a difficult task, and statisticians have developed so- phisticated techniques for doing so. How easy it is for our intuitions to go astray is illustrated by many familiar paradoxes. If 23 students are in a classroom and you pick two at random, the probability that their birthdays (month and day) match is about 11365. The probability that at least two ofthe 23 have the same birth date, however, is a trifle better than 112. The reason is that now there are 1 + 2 + 3 + . + 22 = 253 possible matching pairs, and figuring the exact probability of coincidence is a bit tricky. In a class of 35 students the probability of a birthday coincidence rises to about 85 percent. If students call out their birth dates one at a time until someone raises a hand to indicate that his birthday matches the one just called, you can expect a hand to go up after about nine calls (see "Note on the 'Birthday Problem,' " by Edmund A. Gehan in The American Statistician, April, 1968, page 28). William Moser has pointed out that the chances are better than even that two people in a group of 14 will have birth dates that either are identical or fall on consecutive days of the year. Among seven people, he calculates, the probability is about 60 percent that two will have birthdays within a week of each other, and among four people the probability is about 70 percent that two will have birthdays within 30 days of each other. Variants of the basic idea are endless. The next time you are in a gathering of a dozen or more people try checking on such things as the exact amount of change each person has, the first names of his parents, the street numbers of his home, the playing card each writes secretly on a slip of paper and so on. The number of coincidences may be scary. Another simple demonstration of an event that seems improbable but actu- ally is not can be given with a deck of playing cards. Shuffle the cards, then deal them while you recite their names in a predetermined order, say ace to king of spades followed by the same sequence for hearts, clubs and diamonds. The probability that a card named in advance, such as the queen of hearts, will be dealt when it is named is 1/52,but the probability that at least one card will be dealt when named is almost 213. Ifyou name only the values, the probability of a "hit" rises to 98 percent, or very close to certain.