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AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 18 Which Way Did the Bicycle Go? ... and Other Intriguing Mathematical Mysteries ©1996 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 95-81495 Print ISBN 978-0-88385-325-2 Electronic ISBN 978-1-61444-220-2 Printed in the United States of America Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 DID 10.1090/dol/018 T~~ ICV(l~ O~ ....nHD OT-U.c.A IHT.AIGUIHG m.nr-u.cm.nTlc.nl mVSH.AI.cS Jm~p~ D.~.~D"~J1m~~ ~" ~Ul~mJl" ~J1" WJlGD" MATHEMATICAL ASSOCIATION OF AMERICA DOLCIANI MATHEMATICALEXPOSITIONS - NO.18 THE DOLCIANI MATHEMATICAL EXPOSITIONS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications JAMES W. DANIEL, Chair Dolciani Mathematical Expositions Editorial Board BRUCE P. PALKA, Editor CHRISTINE W. AYOUB EDWARD 1. BARBEAU IRL C. BIVENS The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Associ ation of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College ofthe City University of New York. In making the gift, Professor Dolciani, herselfan exceptionally talented and successful expositor 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 initiating the revolving fund for this series from one who has served the Association with distinction, both as a member ofthe 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 accompanying solutions. They are intended to be sufficiently elementary for the undergrad uate 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. I. Mathematical Gems, Ross Honsberger 2. Mathematical Gems IL 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 II. Old and New Unsolved Problems in Plane Geometry and Number Theory, Victor Klee and Stan Wagon 12. Problems for Mathematicians, Young and Old, Paul R. Halmos 13. Excursions in Calculus: An Interplay ofthe Continuous and the Discrete, Robert M. Young 14. The Wohascum County problem Book, George T. Gilbert, Mark Krusemeyer, and Loren C. Larson 15. Lion Hunting and Other Mathematical Pursuits: A Collection ofMathematics, Verse, and Stories by Ralph P. Boas, Jr., edited by Gerald L. Alexanderson and Dale H. Mugler 16. Linear Algebra Problem Book, Paul R. Halmos 17. From Erdos to Kiev: Problems ofOlympiad Caliber, Ross Honsberger 18. Which Way Did the Bicycle Go? ... and Other Intriguing Mathematical Mysteries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon Preface While realizing that the solution of problems is one of the lowest forms of mathematical research, and that, in general, it has no scientific value, yet its educational value cannot be overestimated. It is the ladder by which the mind ascends into the higher fields oforiginal research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem. - Benjamin Finkel and John M. Colaw, founders of the American Mathematical Monthly, 1894 (Amer. Math. Monthly, 1894, volume I, number I, page I) Joe Konhauser (1924-1992) was a great believer in the value of problem-solving activity. For 25 years he posted a Problem ofthe Week at Macalester College, and this book consists of 190 problems chosen from Joe's and, later, Stan Wagon's, posted problems. In order to encourage student participation, Joe developed a definite style in the problems he posted. They had to involve almost no prerequisites and be succinctly stated and inherently attractive. In short, his plan was to hook students immediately into thinking about the problem. This book is aimed at everyone with an interest in problems, but in particular at teachers who want a source ofproblems for their students. We have taken what we believe to be the most attractive problems from the Konhauser collection. Because his files contain many student solutions and attempts at solutions, we are confident that they will appeal to students at the advanced high-school or beginning college level. We have followed the traditional problem book layout. Part I contains the problems in no particular order except for the chapter and section groupings by field, Part II then contains the solution, historical and other notes, and often some auxiliary problems (without solution). While nothing could be more obvious than that a problem is more appreciated if one tries it without looking at the solution, there is something to be said for using the solution as a guide to whether one wishes to study the problem in more detail. The choice is yours. But many of vii viii WHICH WAY DID THE BICYCLE GO? these problems have surprising little twists in them and for the majority ofthem the solutions are quite short; thus even ifyou choose not to solve a problem yourself, we encourage you to at least take a guess at the answer before turning to the solution. We find the computer to be very valuable in analyzing certain types ofproblems, and there are several in this collection for which a computer is essential for the solution. A good software package is just one more tool in the problemist's arsenal, and we encourage its use. In several ofthe solutions we have included a few lines of Mathematica code so that the reader can see what is involved in generating and analyzing computer output. We have included reference information in all cases for which we knew it. Posing a good problem is harder than solving it, and we would like to fill in any missing attributions, so do let us know ifyou have information ofthis sort. Finally, we must acknowledge our gigantic and, sadly, unpayable debt to Joe Konhauser. In his hands, the Problem of the Week was an opportunity to share with students the many surprising twists and patterns that can be found even in very elementary mathematics. And the students responded over the years with humor, perseverance, and occasional brilliance. Many of the solutions in this book come from the yellowed sheets of student submissions in the Konhauser files, and we therefore express our thanks to several generations of Macalester students. Faculty also contributed mightily to the solution files, and we gratefully acknowledge the many fine contributions of three Macalester professors: John Schue, Emil Slowinski, and the late John Howe Scott. The last two are chemists and their indefatigable efforts over the years show once again the universal appeal ofattractive puzzles. We are also indebted to those who have provided us with problems for inclusion; we would like to especially mention Lee Sallows, an electrical engineer from Nijmegen, the Netherlands, who has invented many beautiful problems, three of which are included in this collection. We thank the many colleagues who have commented on our draft manuscript, providing many references or superior solutions: these include Frank Bernhart, Larry Carter, Hung Dinh, Woody Dudley, John Duncan, Curtis Greene, Jim Guilford, John Guilford, Richard Guy, John Hamilton, Joan Hutchinson, Murray Klamkin, Loren Larson, Jim Mauldon, Hugh Montgomery, Bruce Palka, and Stanley Rabinowitz. And we are grateful to Bev Ruedi at the MAA for her superb job of typesetting and layout. Dan Velleman Stan Wagon Amherst College, Amherst, Mass. Macalester College, St. Paul, Minn. [email protected] [email protected] P.S. Stan Wagon maintains the Problem of the Week tradition at Macalester College and sends out the weekly problems bye-mail; contact him if you would like to be added to the mailing list. Contents Preface vii 1 Plane Geometry 1 1.1 Locus 1. Which Way Did the Bicycle Go? 1 2. Where Can the Third Vertex Live? 1 3. Seeing a 45° Angle 2 4. A Narrow Path .. 2 5. Test Your Intuition. 2 6. Disks on a Circle 3 7. Don't Cut Corners-Fold Them. 3 1.2 Dissection 8. Straighten these Curves ...... 4 9. Cut the Triangle .... ...... 4 10. A Triangle Duplication Dissection 4 11. A Square Triangulation 4 12. Equilateral Into Isosceles .. 5 13. Solitaire on a Chessboard .. 5 14. A Notorious Tiling Problem. 5 1.3 Triangles 15. An 80°-80°-20°Triangle 6 16. A Decomposition of Unity 6 17. Find the Missing Altitude 6 18. An Isosceles Chain .... 7 jy x WHICH WAY DID THE BICYCLE GD? 19. Nested Triangles . 7 20. Avoiding Equilaterals . 7 21. The Square on the Hypotenuse 7 22. A Hexagon-Triangle Hinge . 7 23. How Many Isosceles Triangles? . 8 24. A Triangle Bisection. .... 8 25. A Cut Through the Centroid 8 26. A Common Ratio . 8 27. A 3-4-5 Triangle Problem .. 8 28. Triangles from a Triangle . 9 29. An Equilateral Triangle from a Circle and Hyperbola. 9 1.4 Circles 30. Abe Lincoln's Somersaults 9 31. Focus on This . 9 32. Circular Surprises 10 33. Tangent Cuts .. 10 34. Five Circles . 10 35. A Ring of Disks . 11 36. A Circle Inside an Angle 11 37. A Ray that Pierces Concentric Circles 11 38. If You Lose Your Compass 12 39.

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