AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 42 CHARMING PROOFS A Journey into Elegant Mathematics Claudi Alsina and Roger B. Nelsen i i “MABK014-FM” — 2010/7/10 — 13:14 — page i — #1 i i 10.1090/dol/042 Charming Proofs A Journey into Elegant Mathematics i i i i ©2010 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2010927263 Print ISBN 978-0-88385-348-1 Electronic ISBN 978-1-61444-201-1 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 3 2 1 i i “MABK014-FM” — 2010/7/10 — 13:14 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY-TWO Charming Proofs A Journey into Elegant Mathematics Claudi Alsina Universitat Politecnica` de Catalunya Roger B. Nelsen Lewis & Clark College Published and Distributed by The Mathematical Association of America i i i i i i “master” — 2010/7/13 — 10:16 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Gerald Bryce, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Michael J. McAsey Mark A. Peterson Jonathan Rogness Thomas Q. Sibley i i i i i i “MABK014-FM” — 2010/7/10 — 13:14 — page v — #5 i i 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 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 initiat- ing 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 accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to under- stand and enjoy, but also to be interesting and sometimes challenging to the more advanced mathematician. 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,IvanNiven 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 12. Problems for Mathematicians, Young and Old, Paul R. Halmos 13. Excursions in Calculus: An Interplay of the 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 of Mathematics, 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 of Olympiad Caliber, Ross Honsberger 18. Which Way Did the Bicycle Go? . and Other Intriguing Mathematical Mys- teries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon i i i i i i “MABK014-FM” — 2010/7/10 — 13:14 — page vi — #6 i i 19. In Polya’s´ Footsteps: Miscellaneous Problems and Essays, Ross Honsberger 20. Diophantus and Diophantine Equations, I. G. Bashmakova (Updated by Joseph Silverman and translated by Abe Shenitzer) 21. Logic as Algebra, Paul Halmos and Steven Givant 22. Euler: The Master of Us All, William Dunham 23. The Beginnings and Evolution of Algebra,I.G.BashmakovaandG.S.Smirnova (Translated by Abe Shenitzer) 24. Mathematical Chestnuts from Around the World, Ross Honsberger 25. Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures, Jack E. Graver 26. Mathematical Diamonds, Ross Honsberger 27. Proofs that Really Count: The Art of Combinatorial Proof, Arthur T. Benjamin and Jennifer J. Quinn 28. Mathematical Delights, Ross Honsberger 29. Conics, Keith Kendig 30. Hesiod’s Anvil: falling and spinning through heaven and earth, Andrew J. Simoson 31. A Garden of Integrals, Frank E. Burk 32. A Guide to Complex Variables (MAA Guides #1), Steven G. Krantz 33. Sink or Float? Thought Problems in Math and Physics, Keith Kendig 34. Biscuits of Number Theory, Arthur T. Benjamin and Ezra Brown 35. Uncommon Mathematical Excursions: Polynomia and Related Realms,Dan Kalman 36. When Less is More: Visualizing Basic Inequalities, Claudi Alsina and Roger B. Nelsen 37. A Guide to Advanced Real Analysis (MAA Guides #2), Gerald B. Folland 38. A Guide to Real Variables (MAA Guides #3), Steven G. Krantz 39. Voltaire’s Riddle: Micromegas´ and the measure of all things, Andrew J. Simoson 40. A Guide to Topology, (MAA Guides #4), Steven G. Krantz 41. A Guide to Elementary Number Theory, (MAA Guides #5), Underwood Dudley 42. Charming Proofs: A Journey into Elegant Mathematics, Claudi Alsina and Roger B. Nelsen MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “MABK014-FM” — 2010/7/10 — 13:14 — page vii — #7 i i Dedicated to our many students, hoping they have enjoyed the beauty of mathematics, and who have (perhaps unknowingly) inspired us to write this book. i i i i i i “MABK014-FM” — 2010/7/10 — 13:14 — page viii — #8 i i i i i i i i “MABK014-FM” — 2010/7/10 — 13:14 — page ix — #9 i i Preface Having perceived the connexions, he seeks the proof, the clean revelation in its simplest form, never doubting that somewhere writing in the chaos is the unique elegance, the precise, airy structure, defined, swift-lined, and indestructible. Lillian Morrison Poet as Mathematician Theorems and their proofs lie at the heart of mathematics. In speaking of the “purely aesthetic” qualities of theorems and proofs in A Mathematician’s Apology [Hardy, 1969], G. H. Hardy wrote that in beautiful proofs “there is a very high degree of unexpectedness, combined with inevitability and economy.” These will be the charming proofs appearing in this book. The aim of this book is to present a collection of remarkable proofs in el- ementary mathematics (numbers, geometry, inequalities, functions, origami, tilings, ...) that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, we hope the charming proofs in our collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs. The remarkable Hungarian mathematician Paul Erdos˝ (1913–1996) was fond of saying that God has a transfinite Book that contains the best possible proofs of all mathematical theorems, proofs that are elegant and perfect. The highest compliment Erdos˝ could pay to a colleague’s work was to say “It’s straight from The Book.” Erdos˝ also remarked, “You don’t have to believe in God, but you should believe in The Book” [Hoffman, 1998]. In 1998 M. Aigner and G. Ziegler gave us a glimpse of what The Book might contain when they published Proofs from THE BOOK, now in its second edition [Aigner and Ziegler, 2001]. We hope that Charming Proofs complements Aigner and Ziegler’s work, presenting proofs that require at most calculus and some elementary discrete mathematics. ix i i i i i i “MABK014-FM” — 2010/7/10 — 13:14 — page x — #10 i i x Charming Proofs But we ask: Are there pictures in The Book? We believe the answer is yes, and you will find over 300 figures and illustrations in Charming Proofs. There is a long tradition in mathematics of using pictures to facilitate proofs. This tradition dates back over two thousand years to the mathematics of an- cient Greece and China, and continues today with the popular “proofs with- out words” that appear frequently in the pages of Mathematics Magazine, The College Mathematics Journal, and other publications. Many of these appear in two books published by the Mathematical Association of America [Nelsen, 1993 and 2000], and we have written two books [Alsina and Nelsen, 2006 and 2009] discussing the process of creating visual proofs, both also published by the MAA. Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, we present, in twelve chap- ters, a wide and varied selection of proofs we find charming, Each chapter concludes with some challenges for the reader which we hope will draw the reader into the process of creating charming proofs. There are over 130 such challenges. We begin our journey with a selection of theorems and proofs about the in- tegers and selected real numbers. We then visit topics in geometry, beginning with configurations of points in the plane. We consider polygons, polygons in general as well as special classes such as triangles, equilateral triangles, quadrilaterals, and squares. Next we discuss curves, both in the plane and in space, followed by some adventures into plane tilings, colorful proofs, and some three-dimensional geometry. We conclude with a small collection of theorems, problems, and proofs from various areas of mathematics.