The annual games and puzzles issue considers various mathematical aspects of several sports, including the dynamics of rankings in tennis & golf and the possibility of betting in horse racing to beat the track. There are also two articles related to what most of the world calls football: collecting picture stickers of World Cup players and predicting whether professional teams will play one another in England's FA Cup. For those preferring indoor recreations, the issue includes explorations related to SET and The Wheel of Fortune. To engage immediately, we present a challenge on the cover to complete a semimagic knight's tour and offer four more David Nacin puzzles with multiplicity that require counting strokes (but not quite as in golf). -Brian Hopkins
Vol 47 No 4, pp 240-319
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ARTICLES
Rankings over Time
Michael A. Jones, Alexander Webb and Jennifer Wilson
In 2010, Kim Clijsters won the U.S. Open but had her world ranking drop from #3 to #5 by the Women′s Tennis Assocation (WTA). How can a tennis player win a tournament but drop in the rankings? The WTA uses a moving window to determine the rankings. Discounting older results in the window can prevent such counterintuitive behavior. We consider geometric and arithmetic discounting methods. We examine real data from the WTA and comment on discounting methods already in use by the Fédération Internationale de Football Association (FIFA) for ranking national teams for the World Cup and the Official World Golf Rankings.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.242
MAA 101st Anniversary CMJ Puzzle A
David Nacin
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.249
Statistics on the Bonus Round of Wheel of Fortune
Kathleen Ryan and Brittany Shelton
We show several examples of how the puzzles from Wheel of Fortune can be used as a data set in an introductory statistics course. We use descriptive statistics, a permutation test, and a hypothesis test to answer three different questions about the percentage of letters in each puzzle that are R, S, T, L, N, and E, the letters that are revealed to the contestant in the bonus round of the game.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.250
MAA 101st Anniversary CMJ Puzzle C
David Nacin
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.254
The Sticker Collector′s Problem
M. A. Diniz, D. Lopes, A. Polpo and L. E. B. Salasar
We present a generalization of the coupon collector′s problem called the sticker collector′s problem. We cover four different ways to handle the problem, illustrating the results with the stickers of the FIFA World Cup album. This material could be used as motivating examples in undergraduate courses.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.255
MAA 101st Anniversary CMJ Puzzle J
David Nacin
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.264
Algebra From Geometry in the Card Game SET
Timothy E. Goldberg
The card game SET has often been studied as a rich source of combinatorial and probabilistic questions and also as a beautiful and hands-on example of a finite geometry. In fact, SET also possesses an interesting algebraic structure: There is a natural binary operation on the cards in SET that is commutative but possesses no identity and is not even associative. This structure was previously introduced and studied in a paper by Holdener in 2005 by assigning coordinates to the SET cards using the integers modulo 3. Here, we obtain similar results with an entirely different approach, coordinate free and based solely on the geometric structure of SET. The algebraic structure is defined and many of its properties demonstrated, including a proof that SET has the structure of an involutary quandle.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.265
MAA 101st Anniversary CMJ Puzzle M
David Nacin
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.274
Horse Racing Odds: Can You Beat the Track by Hedging Your Bets?
Joel Pasternack and Stewart Venit
In this article, we consider the question “Given a list of odds on a horse race, is it possible, by betting the right amount on each horse, to win money regardless of the outcome of the race?” Converting the given odds to probabilities and summing those probabilities yields an easily calculated parameter that indicates whether the answer to this question is “yes” or “no.” This parameter also determines the percentage of the total amount bet on the race returned to the winning bettors and the percentage retained by the track.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.275
Multiplying by 9
Arthur Benjamin and Rohan Chandra
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.281
The FA Cup Draw and Pairing Up Probabilities
Patrick Sullivan
We use the draw for the third round of the FA cup as a starting point for examining probabilities of certain pairings during a random draw. More generally, we consider where the elements of a set are paired up at random and find the probability distribution of the number of pairs consisting of two elements of a specified subset. This gives rise to combinatorial identities, especially when we consider multiple subsets.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.282
MAA 101st Anniversary CMJ Puzzles Solutions
David Nacin
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.293
Classroom Capsules
The Generalized Birthday Problem
Stephen Scheinberg
How many independent repetitions of an experiment are required so that the probability of a repeated outcome is at least one-half? We find a simple formula that determines the number of repetitions in terms of the number of outcomes to within one of the exact answer.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.294
A Direct Proof of the Integral Formulae for the Inverse Hyperbolic Functions
John Engbers and Adam Hammett
In the same vein as a Classroom Capsule of Arnold Insel on the arctangent, we present a direct geometric derivation of the integral formulae for the inverse hyperbolic functions. We then use these formulae to obtain the derivatives of the various hyperbolic trigonometric functions.
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.297
Problems and Solutions
Problems and Solutions: 300-306
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.300
Book Review
Review: The Magic of Math: Solving for x and Figuring Out Why By Arthur Benjamin
Reviewed by: Raymond N. Greenwell
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.307
Media Highlights
To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.4.312