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TIME TRAVEL AND OTHER MATHEMATICAL BEWILDERMENTS TIME TRAVEL AND OTHER MATHEMATICAL BEWILDERMENTS MARTIN GARDNER me W. H. FREEMAN AND COMPANY NEW YORK Library of'Congress Cataloguing-in-Publication Data Gardner, Martin, 1914- Time travel and other mathematical bewilderments. 1nclud~:s index. 1. Mathematical recreations. I. Title. QA95.G325 1987 793.7'4 87-11849 ISBN 0-7167-1924-X ISBN 0-7167-1925-8 @bk.) Copyright la1988 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 photographic 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 34567890 VB 654321089 To David A. Klarner for his many splendid contributions to recreational mathematics, for his friendship over the years, and in !gratitude for many other things. CONTENTS CHAPTER ONE Time Travel 1 CHAPTER TWO Hexes. and Stars 15 CHAPTER THREE Tangrams, Part 1 27 CHAPTER FOUR Tangrams, Part 2 39 CHAPTER FIVE Nontransitive Paradoxes 55 CHAPTER SIX Combinatorial Card Problems 71 CHAPTER SEVEN Melody-Making Machines 85 CHAPTER EIGHT Anamorphic. Art 97 CHAPTER NINE The Rubber Rope and Other Problems 11 1 viii CONTENTS CHAPTER TEN Six Sensational Discoveries 125 CHAPTER ELEVEN The Csaszar Polyhedron 139 CHAPTER TWELVE Dodgem and Other Simple Games 153 CHAPTER THIRTEEN Tiling with Convex Polygons 163 CHAPTER FOURTEEN Tiling with Polyominoes, Polyiamonds, and Polyhexes 177 CHAPTER FIFTEEN Curious Maps 189 CHAPTER SIXTEEN The Sixth Symbol and Other Problems 205 CHAPTER SEVENTEEN Magic Squares and Cubes 213 CHAPTER EIGHTEEN Block Packing 227 CHAPTER NINETEEN Induction and ~robabilit~ 24 1 CONTENTS ix CHAPTER TWENTY Catalan Numbers 253 CHAPTER TWENTY-ONE Fun with a Pocket Calculator 267 CHAPTER TWENTY-TWO Tree-Plant Problems 22 7 INDEX OF NAMES 291 Herewith the twelfth collection of my columns from Scientijic American. As usual, they have been corrected, updated, and expanded, mostly on the basis of letters from knowledgeable and alert readers. TIME TRAVEL AND OTHER MATHEMATICAL BEWILDERMENTS 2 • CHAPTER ONE Schools Journal under the horrendous title “The Chronic Argonauts.” Wells himself was so ashamed of this clumsily written tale that he broke it off after three installments and later destroyed all the copies he could find. A com- pletely rewritten version, “The Time Traveller’s Story,” was serialized in The New Review beginning in 1894. When it came out as a book in 1895, it brought Wells instant recognition. One of the many remarkable aspects of Wells’s novella is the introduction in which the Time Traveller (his name is not revealed, but in Wells’s first ver- sion he is called Dr. Nebo-gipfel) explains the theory behind his invention. Time is a fourth dimension. An instantaneous cube cannot exist. The cube we see is at each instant a cross section of a “fixed and unalterable” four-dimen- sional cube having length, breadth, thickness, and duration. “There is no dif- ference between Time and any of the three dimensions of Space,” says the Time Traveller, “except that our consciousness moves along it.” If we could view a person from outside our space-time (the way human history is viewed by the Eternals in Isaac Asimov’s The End of Eternity or by the Tralfamadorians in Kurt Vonnegut’s Slaughterhouse-Five), we would see that person’s past, pres- ent, and future all at once, just as in 3-space we see all parts of a wavy line that traces on a time chart the one-dimensional spatial movements of mercury in a barometer. Reading these remarks today, one might suppose that Wells had been famil- iar with Hermann Minkowski’s great work of tidying up Einstein’s special the- ory of relativity. The line along which our consciousness crawls is, of course, our “world line”: the line that traces our movements in 3-space on a four- dimensional Minkowski space-time graph. (My World Line is the title of George Gamow’s autobiography.) But Wells’s story appeared in its final form ten years before Einstein published his first paper on relativity! When Wells wrote his story, he regarded the Time Traveller’s theories as lit- tle more than metaphysical hanky-panky designed to make his fantasy more plausible. A few decades later physicists were taking such hanky-panky with the utmost seriousness. The notion of an absolute cosmic time, with absolute simultaneity between distant events, was swept out of physics by Einstein’s equations. Virtually all physicists now agree that if an astronaut were to trav- el to a distant star and back, moving at a velocity close to that of light, he could in theory travel thousands of years into the earth’s future. Kurt Gödel constructed a rotating cosmological model in which one can, in principle, travel to any point in the world’s past as well as future, although travel to the past is ruled out as physically impossible. In 1965 Richard P. Feynman received a Nobel prize for his space-time approach to quantum mechanics in which antiparticles are viewed as particles momentarily moving into the past. TIME TRAVEL 3 Hundreds of science-fiction stories have been written about time travel, many of them raising questions about time and causality that are as profound as they are sometimes funny. To give the most hackneyed example, suppose you traveled back to last month and shot yourself through the head. Not only do you know before making the trip that nothing like this happened but, assum- ing that somehow you could murder your earlier self, how could you exist next month to make the trip? Fredric Brown's "First Time Machine" opens with Dr. Grainger exhibiting his machine to three friends. One of them uses the device to go back sixty years and kill his hated grandfather when the man was a youth. The story ends sixty years later with Dr. Grainger showing his time machine to two friends. It must not be thought that logical contradictions arise only when people travel in time. The transportation of anything can lead to paradox. There is a hint of this in Wells's story. When the Time Traveller sends a small model of his machine into the past or the future (he does not know which), his guests raise two objections. If the time machine went into the future, why do they not see it now, moving along its world line? If it went into the past, why did they not see it there before the Time Traveller brought it into the room? One of the guests suggests that perhaps the model moves so fast in time it becomes invisible, like the spokes of a rotating wheel. But what if a time-trav- eling object stops moving? If you have no memory of a cube on the table Monday, how could you send it back to Monday's table on Tuesday? And if on Tuesday you go into the future, put the cube on the table Wednesday, then return to Tuesday, what happens on Wednesday if on Tuesday you destroy the cube? Objects carried back and forth in time are sources of endless confusion in certain science-fiction tales. Sam Mines once summarized the plot of his own story, "Find the Sculptor," as follows: "A scientist builds a time machine, goes 500 years into the future. He finds a statue of himself commemorating the first time traveler. He brings it back to his own time and it is subsequently set up in his honor. You see the catch here? It had to be set up in his own time so that it would be there waiting for him when he went into the future to find it. He had to go into the future to bring it back so it could be set up in his own time. Somewhere a piece of the cycle is missing. When was the statue made?" A splendid example of how paradox arises, even when nothing more than messages go back in time, is provided by the conjecture that tachyons, particles moving faster than light, might actually exist. Relativity theory leaves no escape from the fact that anything moving faster than light would move backward in time. This is what inspired A. H. Reginald Buller, a Canadian botanist, to write his often quoted limerick: 4 CHAPTER ONE There was a young lady named Bright Who traveled much faster than light. She started one day In the relative way, And returned on the previous night. Tachyons, if they exist, clearly cannot be used for communication. G. A. Benford, D. L. Book, and W. A. Newcomb (of "Newcomb's paradox," the topic of two chapters in my Knotted Doughnuts ad Other Mathematical Enter- tainments, W.H. Freeman and Company, 1986), have chided physicists who are searching for tachyons for overlooking this. In "The Tachyonic Antitele- phone," they point out that certain methods of looking for tachyons are based on interactions that make possible, in theory, communication by tachyons. Suppose physicist Jones on the earth is in communication by tachyonic antite- lephones with physicist Alpha in another galaxy. They make the following agreement. When Alpha receives a message from Jones, he will reply immedi- ately. Jones promises to send a message to Alpha at three o'clock earth time, if and only if he has not received a message from Alpha by one o'clock. Do you see the difficulty? Both messages go back in time. IfJones sends his message at three, Alpha's reply could reach him before one.