AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 33 33 Sink or Float? Sink or Float: Thought Problems in Math and Physics is a collection of prob- lems drawn from mathematics and the real world. Its multiple-choice format Sink or Float? forces the reader to become actively involved in deciding upon the answer. Thought Problems in Math and Physics The book’s aim is to show just how much can be learned by using everyday Thought Problems in Math and Physics common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of Keith Kendig the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem s solution, with explanation, appears in the answer section at the end of the book. A notable feature is the generous sprinkling of boxes appearing throughout the text. These contain historical asides or A tub is filled Will a solid little-known facts. The problems themselves can easily turn into serious with lubricated aluminum or tin debate-starters, and the book will find a natural home in the classroom. ball bearings. cube sink... ... or float? Keith Kendig AMSMAA / PRESS 4-Color Process 395 page • 3/4” • 50lb stock • finish size: 7” x 10” i i \AAARoot" – 2009/1/21 – 13:22 – page i – #1 i i Sink or Float? Thought Problems in Math and Physics i i i i i i \AAARoot" – 2011/5/26 – 11:22 – page ii – #2 i i c 2008 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2008929711 Print ISBN 978-0-88385-339-9 Electronic ISBN 978-1-61444-207-3 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 3 2 1 i i i i i i \AAARoot" – 2009/1/21 – 13:22 – page iii – #3 i i 10.1090/dol/033 The Dolciani Mathematical Expositions NUMBER THIRTY-THREE Sink or Float? Thought Problems in Math and Physics Keith Kendig Cleveland State University ® Published and Distributed by The Mathematical Association of America i i i i i i \AAARoot" – 2009/1/21 – 13:22 – page iv – #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Publications James Daniel, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Tevian Dray Robert L. Devaney Jerrold W. Grossman Virginia E. Knight Mark A. Peterson Jonathan Rogness Joe Alyn Stickles James S. Tanton i i i i i i \AAARoot" – 2009/1/21 – 13:22 – page v – #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association 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 initiating 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 understand 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, 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 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 Mysteries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon 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. Bashmakova and G. S. Smirnova (Translated by Abe Shenitzer) 24. Mathematical Chestnuts from Around the World, Ross Honsberger i i i i i i \AAARoot" – 2009/1/21 – 13:22 – page vi – #6 i i 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 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 \AAARoot" – 2009/1/21 – 13:22 – page vii – #7 i i Preface In his book I Want to be a Mathematician, Paul Halmos reveals a secret: . it's examples, examples, examples that, for me, all mathematics is based on, and I always look for them. I look for them first when I begin to study, I keep looking for them, and I cherish them all. Sink or Float? is actually a collection of examples in the guise of problems. The problems are multiple-choice, challenging you to think through each one. You don't need fancy terminology or familiarity with a mathematical specialty to understand them. They are concrete, and most are understandable to anyone. The willingness and confidence to apply common sense to the real world can be very rewarding. It's one of the aims of this book to show that it works, that thinking can enlighten us and bring perspective to the world we live in. The range of problems to which common sense applies is almost limitless. The example is one of the most powerful tools for discovering new things and for sug- gesting how to solve problems. Examples come in different degrees of concreteness and often, the more concrete, the more informative. The following tale provides an example of —concreter can be better," and is the story of what started this book. It begins in the dentist's chair. After putting generous amounts of cotton in my mouth, my dentist commences his customary free-wheeling question and answer session. One time, things go like this: Dentist. I've been wondering for some time, what was it that convinced you to become a mathematician? Me. Waa-gaa ook ha! Dentist. I see. I was sort of OK with math up through trigonometry, but when it came to calculus, the teacher went fast, I got lost and never caught up. From then on, I can't say I've liked math. Me. Aagh hinc wah ad. Dentist. Yes. Is there anything that you could say to give me some sense of the attraction? Me. Eh he eee . vii i i i i i i \AAARoot" – 2009/1/21 – 13:22 – page viii – #8 i i viii Preface At last the cotton came out of my mouth, and so did this: Me. You asked for some sense of the attraction in math. Here's a concrete example: 10 10 is 100, right? Now, add 1 and subtract 1 to change 10 10 into 11 9. That's 99, 1 less that 100. Dentist (already sounding bored). Sure is. Me. OK, now try that on 8 8, which is 64. Adding 1 and subtracting 1 from 8 changes it to 9 7. That's 63, again one less than 64. Dentist. Funny coincidence, huh? Me. Well, try this on 5 5, which is 25. Adding and subtracting 1 gives 6 4 which is 24, one less–again! Dentist (whose expression has changed slightly). You're not telling me this always works? Me. Try one yourself! Dentist (possibly wanting to burst this little bubble). How about 11 11? Me. OK, do the math . Dentist. That would be 12 10. That's 120. Geez . 11 11 is 121, isn't it? Me. This always does work! Dentist. Well . not for really small numbers, I'll bet . Me. What do you mean? Dentist.