College Mathematics Journal Contents—September 2016

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