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Math Bends It Like Beckham Tim Chartier Davidson College Front Cover 2/6/07 8:18 PM Page 1 Chartier 1/30/07 8:12 PM Page 2 Math Bends it like Beckham Tim Chartier Davidson College n the recent 2006 FIFA World Cup, soccer balls curved and swerved through the air in an attempt to confuse goalkeep- Iers and make their way to the back of the net. World class soccer players such as Brazil's Roberto Carlos, Germany's Michael Ballack and England's David Beckham have perfected “bending” the ball from a freekick. Digitizing a soccer ball (down to the stitching) and computing the resulting air flow over the kicked ball, researchers from the University of Sheffield's Sports Engineering Research Group and Fluent Europe recently found that the shape and surface of the ball, as well as its initial orientation, play a fundamental role in the ball's trajectory through the air. Wind tunnel measurements from the University of Tsukuba in Japan verified these studies. Among the researchers is PhD student Sarah Barber who commented, “As a soccer player, I feel this research is invaluable in order for players to be able to optimize their kicking strate- gies.” Research in computational fluid dynamics impacts various sports including improvements in the aerodynamics in Formula 1 motor race cars, the circulation within sports arenas, and the hydrodynamics of swimsuits. Further Reading http://www.fluent.com/solutions/sports/ http://www.shef.ac.uk/mecheng/sports/ Images courtesy of Fluent Inc. 2 FEBRUARY 2007 pp.03-04 2/6/07 8:30 PM Page 3 In this Issue Math Bends it like Beckham 2 Tim Chartier Beautiful images from computational fluid dynamics. Why are Games Exciting and Stimulating? 5 Math Horizons is for undergraduates and Aviezri S. Fraenkel others who are interested in mathematics. Its What separates a hard game like chess from an easy game like nim? purpose is to expand both the career and intellectual horizons of students. Mixing a Night out with Probability… & Making a Fortune 8 Kari Lock ARTHUR T. BENJAMIN How students earned more than just an “A” from their probability class. Harvey Mudd College JENNIFER J. QUINN An ESPeriment with Cards 10 Association for Women in Mathematics Colm Mulcahy Editors An amazing trick based on elegant mathematics. CAROL BAXTER Sudoku: Just for Fun or Is It Mathematics? 13 Managing Editor & Art Director Sam Cook, Cory Fujimoto, Leigh Mingle and Cami Sawyer KARA L. KELLER Some mathematical questions arising from the latest puzzle craze. Assistant Managing Editor Who Wants to Be a Half-Millionaire? 16 DEENA R. BENJAMIN Patrick Headley Assistant Editor How a mathematician made a fortune on the game show “Super Millionaire.” Math Horizons (ISSN 1072-4117) is pub- The Taxman Game 18 lished four times a year; September, Novem- Robert K. Moniot ber, February, and April by the Mathematical How to beat the taxman at his own game. Association of America, 1529 Eighteenth Street, NW, Washington, DC 20036. The Solutions to Elmsley’s Problem 22 February 2007 Volume XIV, Issue 3. Periodi- Persi Diaconis and Ron Graham cals postage paid at Washington, DC and additional mailing offices. Annual subscrip- The solution to a 50-year-old unsolved problem in recreational mathematics. tion rates are $29.00 for MAA members, Book Reviews $38.00 for nonmembers, and $49.00 for Steven Byrnes and Jacob McMillen libraries. Bulk subscriptions sent to a single 28 Students review Calculated Bets and Connection Games. address are encouraged. The minimum order is 20 copies ($190.00); additional subscrip- REU Profile: The Duluth REU Program tions may be ordered in units of 10. To order Joseph A. Gallian call (800) 331-1622. For advertising informa- 30 tion call (866) 821-1221. Printed in the Unit- Mathematical Survivor ed States of America. Copyright ©2007. The Steven Kahan Mathematical Association of America. 34 A mathematical game show that, surprisingly, never has a surprise ending. POSTMASTER: Send address changes to Math Horizons, MAA Service Center, PO SET, Affine Planes and Latin Squares Box 91112, Washington, DC 20090-1112. Anna Bickel and Zsuzsanna Szaniszlo 37 Underlying this simple game is a set of interesting mathematical questions. THE MATHEMATICAL ASSOCIATION OF AMERICA Problem Section 1529 Eighteenth Street, NW Andy Liu Washington, DC 20036 39 The Game of SET Anna Bickel 43 Are you READY to play SET, then GO here first! Cover image: Bag of Marbles by Dan Fletcher. Copyright iStockphoto.com. WWW.MAA.ORG/MATHHORIZONS 3 pp.03-04 2/6/07 8:30 PM Page 4 EDITORS Arthur T. Benjamin Jennifer J. Quinn Harvey Mudd College Association for Women in Mathematics EDITORIAL BOARD sarah-marie belcastro Dan Kalman Smith College American University Ezra Brown Frank Morgan Virginia Tech Williams College Beth Chance Colm Mulcahy Cal Poly, San Luis Obispo Spelman College Timothy P. Chartier Karen Saxe Davidson College Macalester College Carl Cowen Francis Edward Su IUPUI Harvey Mudd College Joe Gallian James Tanton University of Minnesota Duluth St. Mark’s Institute of Sarah J. Greenwald Mathematics Appalachian State University STUDENT ADVISORY GROUP Sam Beck Lee Kennard Jackson Preparatory School Kenyon College Steven Byrnes James-Michael Leahy Harvard University Columbia University Moshe Cohen Greg Leffert Louisiana State University Maggie Walker Governors School Megan Cornman Melissa Mauck Fredonia University Sam Houston State University Diana Davis Jacob McMillen Williams College Emory University Patrick Dixon Sarah Reardon Occidental College Davidson College Michael Flake Ché Lena Smith Davidson College University of North Carolina Victoria Frost Natalya St. Clair Spelman College Scripps College Instructions for Authors Math Horizons is intended primarily for undergraduates interested in mathematics. Thus, while we especially value and desire to publish high quality exposition of beautiful mathematics, we also wish to publish lively articles about the culture of mathematics. We interpret this quite broadly—we welcome stories of mathematical people, the history of an idea or circle of ideas, applications, fiction, folklore, traditions, institutions, humor, puzzles, games, book reviews, student math club activities, and career opportunities and advice. Manuscripts may be submitted electronically to Editors Arthur Benjamin, [email protected], and Jennifer Quinn, [email protected]. If submitting by mail, please send two copies to Arthur Benjamin, Math Department, Harvey Mudd College, Claremont, CA, 91711. Subscription Inquiries e-mail: [email protected] Web: www.maa.org Fax: (301) 206-9789 Call: (800) 331-1622 or (301) 617-7800 Write: Math Horizons, MAA Service Center, P. O. Box 91112, Washington, DC 20090-1112 4 FEBRUARY 2007 Fraenkel 1/30/07 8:17 PM Page 5 “We ascend along scenic trails that lead from sea-level nim to alpine chess at a moderate gradient, via intermediate games sampled from various strategic viewpoints along the trails.” Why are Games Exciting and Stimulating? Aviezri S. Fraenkel Weizmann Institute of Science The lure of games Games have a natural appeal that entices both amateurs and 4 professionals to become addicted to the subject. What is the essence of this appeal? Perhaps the urge to play games is 3 rooted in our primal beastly instincts; the desire to corner, torture, or at least dominate our peers. A common expression of these dark desires is found in the passions roused by local, 2 national and international tournaments. An intellectually refined version, well hidden beneath the façade of scientific 1 research, is the consuming drive “to beat them all,” to be more clever than the most clever, in short—to create the tools to 0 Mathter them all in hot combinatorial combat! Reaching this goal is particularly satisfying and sweet in the context of com- binatorial games, in view of their inherent high complexity. Figure 1. The game of nim. The mainstream of the theory of combinatorial games consists of the analysis of two-player games with perfect Question 1. Can you win from the given position? If so, by information (unlike some card games where information is what move(s)? hidden), without chance moves (no dice), and outcome Nim has a very simple winning strategy. Write the number restricted to (lose, win), (tie, tie) and (draw, draw) for the two of each occupied circle in binary notation, then add them players who move alternately (no passing). A tie is an end without carry, such as a kindergarten child might do. If this so- position with no winner and no loser, as may occur in tic-tac- called nim-sum is zero you cannot win by beginning to play toe for example. A draw is a “dynamic tie,” i.e., a nonend from that position: every move necessarily makes the nim-sum position such that neither player can force a win, but each can nonzero. But if it’s nonzero, there always exists a move that find a next nonlosing move. makes it 0, which is also the value at the end of the game when Throughout we assume that the two players move all the marbles are in position 0, which is therefore a winning alternately (no passing), and that the player first unable to move. The nim-sum of the given position is displayed in move loses and the opponent wins. Figure 2(a). The move 4 → 3 causes the nim-sum to become Nim 0. It is a winning move—in this case, the only winning move. The simplest combinatorial game is nim: Place identical Fundamental question marbles on a directed graph, such as depicted in Figure 1, say Nim and chess and go belong to the same family of games. one on each of the circles 1, 2 and 4. Why does the former have such a simple strategy, whereas A move consists of selecting a marble and moving it to a chess and go seem to be so complex? neighboring circle, in the direction of an arrow.
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