The Mathematics of Various Entertaining Subjects THE MATHEMATICS OF VARIOUS ENTERTAINING SUBJECTS Volume 2 RESEARCH IN GAMES, GRAPHS, COUNTING, AND COMPLEXITY EDITED BY Jennifer Beineke & Jason Rosenhouse WITH A FOREWORD BY RON GRAHAM National Museum of Mathematics, New York • Princeton University Press, Princeton and Oxford Copyright c 2017 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu In association with the National Museum of Mathematics, 11 East 26th Street, New York, New York 10010 Jacket art: Top row (left to right) Fig. 1: Courtesy of Eric Demaine and William S. Moses. Fig. 2: Courtesy of Aviv Adler, Erik Demaine, Adam Hesterberg, Quanquan Liu, and Mikhail Rudoy. Fig. 3: Courtesy of Peter Winkler. Middle row (left to right) Fig. 1: Courtesy of Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Quanquan Liu, Ron Taylor, and Ryuhei Uehara. Fig. 2: Courtesy of Robert Bosch, Robert Fathauer, and Henry Segerman. Fig. 3: Courtesy of Jason Rosenhouse. Bottom row (left to right) Fig. 1: Courtesy of Noam Elkies. Fig. 2: Courtesy of Richard K. Guy. Fig. 3: Courtesy of Jill Bigley Dunham and Gwyn Whieldon. Excerpt from “Macavity: The Myster Cat” from Old Possum’s Book of Cats by T. S. Eliot. Copyright 1939 by T. S. Eliot. Copyright c Renewed 1967 by Esme Valerie Eliot. Reprinted by permission of Houghton Mifflin Harcourt Publishing Company and Faber & Faber Ltd. All rights reserved. All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Beineke, Jennifer Elaine, 1969– editor. | Rosenhouse, Jason, editor. Title: The mathematics of various entertaining subjects : research in games, graphs, counting, and complexity / edited by Jennifer Beineke & Jason Rosenhouse ; with a foreword by Ron Graham. Description: Princeton : Princeton University Press ; New York : Published in association with the National Museum of Mathematics, [2017] | Copyright 2017 by Princeton University Press. | Includes bibliographical references and index. Identifiers: LCCN 2017003240 | ISBN 9780691171920 (hardcover : alk. paper) Subjects: LCSH: Mathematical recreations-Research. Classification: LCC QA95 .M36874 2017 | DDC 793.74–dc23 LC record available at https://lccn.loc.gov/2017003240 British Library Cataloging-in-Publication Data is available This book has been composed in Minion Pro Printed on acid-free paper. ∞ Typeset by Nova Techset Private Limited, Bangalore, India Printed in the United States of America 13579108642 Contents Foreword by Ron Graham vii Preface and Acknowledgments xi PART I PUZZLES AND BRAINTEASERS 1 The Cyclic Prisoners 3 Peter Winkler 2 Dragons and Kasha 11 Tanya Khovanova 3 The History and Future of Logic Puzzles 23 Jason Rosenhouse 4 The Tower of Hanoi for Humans 52 Paul K. Stockmeyer 5 Frenicle’s 880 Magic Squares 71 John Conway, Simon Norton, and Alex Ryba PART II GEOMETRY AND TOPOLOGY 6 A Triangle Has Eight Vertices But Only One Center 85 Richard K. Guy 7 Enumeration of Solutions to Gardner’s Paper Cutting and Folding Problem 108 Jill Bigley Dunham and Gwyneth R. Whieldon 8 The Color Cubes Puzzle with Two and Three Colors 125 Ethan Berkove, David Cervantes-Nava, Daniel Condon, Andrew Eickemeyer, Rachel Katz, and Michael J. Schulman 9 Tangled Tangles 141 Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Quanauan Liu, Ron Taylor, and Ryuhei Uehara vi • Contents PART III GRAPH THEORY 10 Making Walks Count: From Silent Circles to Hamiltonian Cycles 157 Max A. Alekseyev and Gérard P. Michon 11 Duels, Truels, Gruels, and Survival of the Unfittest 169 Dominic Lanphier 12 Trees, Trees, So Many Trees 195 Allen J. Schwenk 13 Crossing Numbers of Complete Graphs 218 Noam D. Elkies PART IV GAMES OF CHANCE 14 Numerically Balanced Dice 253 Robert Bosch, Robert Fathauer, and Henry Segerman 15 A TROUBLE-some Simulation 269 Geoffrey D. Dietz 16 A Sequence Game on a Roulette Wheel 286 Robert W. Vallin PART V COMPUTATIONAL COMPLEXITY 17 Multinational War Is Hard 301 Jonathan Weed 18 Clickomania Is Hard, Even with Two Colors and Columns 325 Aviv Adler, Erik D. Demaine, Adam Hesterberg, Quanquan Liu, and Mikhail Rudoy 19 Computational Complexity of Arranging Music 364 Erik D. Demaine and William S. Moses About the Editors 379 About the Contributors 381 Index 387 Foreword Ron Graham recreation—something people do to relax or have fun. —Merriam–Webster Dictionary One of the strongest human instincts is the overwhelming urge to “solve puzzles.” Whether this means how to make fire, avoid being eaten by wolves, keep dry in the rain, or predict solar eclipses, these “puzzles” have been with us since before civilization. Of course, people who were better at successfully dealing with such problems had a better chance of surviving, and then, as a consequence, so did their descendants. (A current (fictional) solver of problems like this is the character played by Matt Damon in the recent film The Martian). On a more theoretical level, mathematical puzzles have been around for thousands of years. The Palimpsest of Archimedes contains several pages de- voted to the so-called Stomachion, a geometrical puzzle consisting of fourteen polygonal pieces which are to be arranged into a 12 × 12 square. It is believed that the problem given was to enumerate the number of different ways this could be done, but since a number of the pages of the Palimpsest are missing, we are not quite sure. It is widely acknowledged by now that many recreational puzzles have led to quite deep mathematical developments as researchers delved more deeply into some of these problems. For example, the existence of Pythagorean triples, such as 32 + 42 = 52, and quartic quadruples, such as 26824404 + 153656394 + 187967604 = 206156734, led to questions, such as whether xn + yn = zn could ever hold for positive integers x, y,andz when n ≥ 3. (The answer: No! This was Andrew Wiles’ resolution of Fermat’s Last Theorem, which spurred the development of even more powerful tools for attacking even more difficult viii • Foreword TABLE 1. Numbers expressible in the form n = 6xy ± x ± y x y 6xy + x + y 6xy + x − y 6xy − x + y 6xy − x − y 1 1 8 6 6 4 2 1 15 13 11 9 2 2 28 24 24 20 3 1 22 20 16 14 3 2 41 37 35 31 3 3 60 54 54 48 4 1 29 27 21 19 . questions.) Similar stories could be told in a variety of other areas, such as the analysis of games of chance in the Middle Ages leading to the development of probability theory, and the study of knots leading to fundamental work on von Neumann algebras. In 1900, at the International Congress of Mathematicians in Paris, the legendary mathematician David Hilbert gave his celebrated list of twenty-three problems which he felt would keep the mathematicians busy for the remainder of the century. He was right! Many of these problems are still unsolved. (Actually, he only mentioned eight of the problems during his talk. The full list of twenty-three was only published later.) In that connection, Hilbert also wrote about the role of problems in mathematics. Paraphrasing, he said that problems are the core of any mathematical discipline. It is with problems that you can “test the temper of your steel.” However, it is often difficult to judge the difficulty (or importance) of a particular problem in advance. Let me give two of my favorite examples. Problem 1. Consider the set of positive integers n which can be represented as n = 6xy ± x ± y, where x ≥ y ≥ 0. Some such numbers are displayed in Table 1. It seems like most of the small numbers occur in the table, although some are missing. The list of the missing numbers begins {1, 2, 3, 5, 7, 10, 12, 17, 18, 23,...}. Are there infinitely many numbers m that are not in the table? I will give the answer at the end. Here is another problem. Foreword • ix Problem 2. A well-studied function in number theory is the divisor function d(n), which denotes the sum of the divisors of the integer n. For example, d(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28, and d(100) = 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 + 100 = 217. Another common function in mathematics is the harmonic number H(n).Itis defined by n 1 H(n) = . k k=1 In other words, H(n) is the sum of the reciprocals of the first n integers. Is it true that d(n) ≤ H(n) + e H(n) log H(n), for n ≥ 1? How hard could this be? Actually, pretty hard (or so it seems!). Readers of this volume will find an amazing assortment of brainteasers, challenges, problems, and “puzzles” arising in a variety of mathematical (and non-mathematical) domains. And who knows whether some of these problems will be the acorns from which mighty mathematical oaks will someday emerge! As for the problems, the answer to each is that no one knows! For Problem 1, each number m that is missing from the table corresponds to a pair of twin primes 6m − 1, 6m + 1. Furthermore, every pair of twin primes (except 3 and 5) occur this way. Recall, a pair of twin primes is a set of two prime numbers which differ by two. Thus, Problem 1 is really asking whether there are infinitely many pairs of twin primes. As Paul Erdos˝ liked to say, “Every right-thinking person knows the answer is yes,” but so far no one has been able to prove this. It is known that there exist infinitely many pairs of primes which differ by at most 246, the establishment of which was actually a major achievement in itself! For Problem 2, it is known that the answer is yes if and only if the Riemann Hypothesis holds! As I said, this appears to be a rather difficult problem at present (to say the least).