AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 11 OLD AND NEW UNSOLVED PROBLEMS in Plane Geometry and Number Theory VICTOR KLEE STAN WAGON OLD AND NEW UNSOLVED PROBLEMS IN PLANE GEOMETRY AND NUMBER THEORY VICTOR KLEE AND STAN WAGON THE DOLCIANI MATHEMATICAL EXPOSITIONS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committeeon Publications DONALD ALBERS, Chairman DolcianiMathematical Expositions EditorialBoard JOE P. BUHLER, Chairman CHRISTINE AYOUB JOHN H. EWING JOHNR.LUX DolcianiMathematical Expositions series design ROBERT ISHI + DESIGN TEXmacros RON WHITNEY 10.1090/dol/011 The Dolciani Mathematical Expositions NUMBER ELEVEN OLD AND NEW UNSOLVED PROBLEMS IN PLANE GEOMETRY AND NUMBER THEORY UICTOR KLEE University ofWashington STAN WAGON Maca/ester College Publishedand Distributedby THE MATHEMATICAL ASSOCIATION OF AMERICA ©1991 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 91-61591 Print ISBN 978-0-88385-315-3 Electronic ISBN 978-1-61444-219-6 Printed in the United States of America Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 Figures 7.5, 7.6, and 7.8 are reprinted by permission of the publisher from Figures 3.8.6, 10.3.16, and 10.3.28, respectively, by B. Grünbaum and G. C. Shephard, Tilings and Patterns, pp. 156,542, and 548. Copyright 1987 by W.H. Freeman and Company. Figure 12.3 is reprinted by permission of the publisher from Figure 1.15 by J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, p. 22. Copyright 1976 by Elsevier Science Publishing Co., Inc. The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical As­ sociation of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successful expos­ itor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture iniating the revolving fund for this series from one who has served the Association with distinction, both as a member of the Committee on Publications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimulating mathematical content. Typically, they contain an ample supply of exercises, many with ac­ companying solutions. They are intended to be sufficientlyelementary for the undergraduate and even the mathematically inclined high-school student to understand and enjoy, but also to be interesting and sometimes challenging to the more advanced mathematician. DOLCIANI MATHEMATICAL EXPOSITIONS 1. Mathematical Gems, Ross Honsberger 2. Mathematical Gems II, Ross Honsberger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus, Ivan Niven 7. Great Moments in Mathematics (After 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette 9. Mathematical Gems III, Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger 11. Old and New Unsolved Problems in Plane Geometry and Number Theory, Victor Klee and Stan Wagon CONTENTS Preface xi Chapter 1. 1\vo-Dimensional Geometry Introduction 1 1. Illuminating a Polygon 3-11, 71-79 Is each reflecting polygonal region illuminable? 2. Equichordal Points 12-15, 80-85 Can a plane convex body have two equichordal points? 3. Pushing Disks Together 16-20, 86-90 When congruent disks are pushed closer together, can the area of their union increase? 4. Universal Covers 21-24, 91-94 Ifa convex body C contains a translate ofeach plane set of unit diameter, how small can C's area be? 5. Forming Convex Polygons 25-28,95-102 How many points are needed to guarantee a convex n-gon? 6. Points on Lines 29-35,103-110 Ifn points are not collinear, must one ofthem lie on at least ~n connect- ing lines? 7. Tiling the Plane 36-44,111-119 Is there a polygon that tiles the plane but cannot do so periodically? 8. Painting the Plane 45-49,120-127 What is the minimum number ofcolors for painting the plane so that no two points at unit distance receive the same color? 9. Squaring the Circle 50-53,128-131 Can a circle be decomposed into finitely many sets that can be rearranged to form a square? 10. Approximation by Rational Sets 54-57,132-136 Does the plane contain a dense rational set? 11. Inscribed Squares 58-65,137-144 Does every simple closed curve in the plane contain all four vertices of some square? 12. Fixed Points 66-70,145-150 Does each nonseparating plane continuum have the fixed-point prop- erty? References 151 vii viii CONTENTS Chapter 2. Number Theory Introduction 167 13. Fermat's Last Theorem 168-172,199-202 Do there exist positive integers x, y, and z and an integer n ::::: 3 such that x n + yn = zn? 14. A Perfect Box 173-174,203-205 Does there exist a box with integer sides such that the three face diagonals and the main diagonal all have integer lengths? 15. Egyptian Fractions 175-177, 206-208 Does the greedy algorithm always succeed in expressing a fraction with odd denominator as a sum of unit fractions with odd denominator? 16. Perfect Numbers 178-181,209-214 Is there an odd perfect number? Are there infinitely many even perfect numbers? 17. The Riemann Hypothesis 182-185,215-220 Do the nontrivial zeros of the Riemann zeta function all have real part !?2 • 18. Prime Factorization 186-190, 221-224 Is there a polynomial-time algorithm for obtaining a number's prime fac­ torization? 19. The 3n + 1 Problem 191-194,225-229 Is every positive integer eventually taken to the value 1 by the 3n + 1 function? 20. Diophantine Equations and Computers 195-198,230-233 Is there an algorithm to decide whether a polynomial with integer coeffi- cients has a rational root? References 234 Chapter 3. Interesting Real Numbers Introduction 239 21. Patterns in Pi 240-242,251-254 Are the digits in the decimal expansion of tt devoid of any pattern? 22. Connections between tt and e 243, 255-257 Are 1r and e algebraically independent? Is their ratio rational? 23. Computing Algebraic Numbers 244-247, 258-260 Ifan irrational number is real-time computable, is it necessarily transcen- dental? Is v'2 real-time computable? 24. Summing Reciprocals of Powers 248-250,261-264 Is 1 + ~ + ~ + ~ + irrational? References 265 CONTENTS ix Hints and Solutions: Two-Dimensional Geometry 269 Hints and Solutions: Number Thoery 300 Hints and Solutions: Interesting Real Numbers 311 Glossary 315 Index of Names 325 Subject Index 331 PREFACE THE ROLE Of UNSOLVED PROBLEMS As mathematics becomes ever more complex, more structured, and more special­ ized, it is easy to lose sight of the fact that the research frontiers still contain many unsolved problems that are of immediate intuitive appeal and can be .understood very easily, at least in the sense of understanding what a problem asks. Parts of plane geometry and number theory are especially good sources of such intuitively appealing problems, and those are the areas emphasized in this book. For each of the presented problems, we look at its background, try to explain why some mathe­ maticians have found the problem of special interest, and tell about techniques that have been used to obtain partial results. Some of the exercises will help out readers establish partial results for themselves, and references are provided for those who want to learn more about the problems. All mathematical discoveries may be said to consist of the solution of unsolved problems. Sometimes a question is answered almost as soon as it is formulated. Then the question and its answer are published together, and the question never becomes known as an unsolved problem. In other cases, a mathematical question may be formulated long before anyone is able to answer it. Ifthe question becomes widely known, many researchers may direct their efforts toward answering it and the unsolved problem may have a significant influence on the development of math­ ematics. Some unsolved problems are conceptually very simple and specific-forexam­ ple, "Is the answer 'yes' or 'no'?", "What is the value of this function at a specified point of its domain?" Others are conceptually sophisticated-for example, "How xi xii PREFACE can such and such a theory or argument be extended so as to apply to a certain more general class of objects?" Both the simple and the sophisticated problems have played an important role in the development of mathematics, and often they are inextricably mingled. Some of the "simplest" unsolved problems, such as Fermat's last "theorem" (see Section 13) have led directly to some of the most sophisticated mathematical developments and, in tum, to sophisticated unsolved problems. On the other hand, the solution of a sophisticated problem often requires answering a number of simple, specific questions. Thus it is probably unreasonable to try to de­ cide whether the simple or the sophisticated problems have been more important in the development of mathematics. In any case, this book is devoted exclusivelyto problems of the simple sort--ones whose statements are short and easy to under­ stand. Some of the more modem areas of mathematics are poor sources for the sort of unsolved problem that is emphasized here. Ifone thinks of mathematical disci­ plines as buildings, some could be represented bybuildings that are very tall but also quite narrow, with a few important research problems sprouting out of the roof.