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Mathematics Magazine - June 2015

As the June issue of the Magazine is the summer issue, it coincides with the MAA MathFest celebration of the 100th anniversary of the MAA. The issue kicks off the celebration with an interview with MAA past president Lynn Steen and concludes with a special anniversary crossword. In between are articles on a wide range of topics. One article relates sums to Cantor sets, another uses lattice-path counting to determine the average height of Catalan trees, and a third considers the mathematics behind a recent lottery scam investigation. Of course, there is more, including the Reviews and Problems sections. Happy summer reading. —Michael A. Jones, Editor

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Vol. 88, No. 3, pp 161 – 244

Articles

The Idea Man: An Interview with Lynn Steen

Deanna Haunsperger and Stephen Kennedy

An interview with Lynn Arthur Steen, this article questions this consummate idea man about his childhood, editing MATHEMATICS MAGAZINE with Arthur Seebach, being a member of the St. Olaf Math Department through its period of phenomenal growth, serving as MAA President, being on the front lines of math education—writing and being involved in the conversations about the NCTM Standards and Quantitative Literacy, and also commenting on what he sees as the most important issue facing the MAA today.

To purchase from JSTOR: 10.4169/math.mag.88.3.163

Proof Without Words: Sums of Triangular Numbers

Hasan Unal

To purchase from JSTOR: 10.4169/math.mag.88.3.177

Selective Sums of an Infinite Series

R. John Ferdinands

The sum of a subset of the terms of an infinite series is called a selective sum of the series. We describe the set of all selective sums for some series, and we show that for a series which satisfies certain conditions, the set of selective sums can be obtained in a way analogous to the construction of the Cantor set.

To purchase from JSTOR: 10.4169/math.mag.88.3.179

Proof Without Words: An Elegant Property of the Equilateral Triangle

Victor Oxman and Moshe Stupel

Figures are used to show that for an equilateral triangle ABC and some point D on the ray CB, the distances from the point D to the vertices of the triangle satisfy different polynomial equations depending on whether the point D is between B and C or not.

To purchase from JSTOR: 10.4169/math.mag.88.3.186

The Average Height of Catalan Trees by Counting Lattice Paths

Nachum Dershowitz and Christian Rinderknecht

The average height of Catalan trees of a given size is a structural parameter important in the analysis of algorithms, as it measures the expected maximum cost of a search in a tree. This parameter has been studied first with generating functions and complex variable theory, yielding an asymptotic approximation. Later on, real analysis was used instead of complex analysis. We have further reduced the conceptual difficulty by replacing generating functions with the enumeration of monotonic lattice paths, whose graphical representations make the derivation much more intuitive.

To purchase from JSTOR: 10.4169/math.mag.88.3.187

Some People Have All the Luck

Richard Arratia, Skip Garibaldi, Lawrence Mower, and Philip B. Stark

We looked at the Florida Lottery records of winners of prizes worth $600 or more. Some individuals claimed large numbers of prizes. Were they lucky, or up to something? We distinguished the “plausibly lucky” from the “implausibly lucky” by solving optimization problems that took into account the particular games each gambler won. Plausibility was determined by finding the minimum expenditure so that if every Florida resident spent that much, the chance that any of them would win as often as the gambler did would still be less than one in a million. Dealing with dependent bets relied on the BKR inequality; solving the optimization problem numerically relied on the log-concavity of the regularized Beta function. Subsequent investigation by law enforcement confirmed that the gamblers we identified as “implausibly lucky” were indeed behaving illegally.

To purchase from JSTOR: 10.4169/math.mag.88.3.196

Anniversary Crossword: Editors Past and President, Part I

Tracy Bennett

To purchase from JSTOR: 10.4169/math.mag.88.3.212

Pythagoras via Cavalieri

Zsolt Lengvárszky

We present a proof of the Pythagorean theorem via Cavalieri's principle.

To purchase from JSTOR: 10.4169/math.mag.88.3.214

Optimal Defensive Strategies in One-Dimensional RISK

Darren Glass and Todd Neller

We consider a one-dimensional version of the board game RISK and discuss the problem of how a defending player might choose to distribute his armies along a chain of territories in order to maximize the probability of survival. In particular, we analyze a Markov chain model of this situation and run computer simulations in order to make conjectures as to the optimal strategies. The latter sections of the paper analyze this strategy rigorously and use results on recurrence relations and probability theory in order to prove a related result.

To purchase from JSTOR: 10.4169/math.mag.88.3.217

Non-Integrality of Binomial Sums and Fermat's Little Theorem

Daniel López-Aguayo

Problem 1942 from the April 2014 issue of the Magazine asks whether a particular binomial sum is nonintegral. We pose an open question whether there are infinitely many integers r for which the requisite sum is nonintegral when k + 1 is replaced by k + r. We prove that the sum is nonintegral for r = 2, 3, and 4 by an application of Fermat's little theorem.

To purchase from JSTOR: 10.4169/math.mag.88.3.231

Problems and Solutions

Proposals, 1971-1975

Quickies, 1051-1052

Solutions, 1941-1945

Answers, 1051-1052

To purchase from JSTOR: 10.4169/math.mag.88.3.235

Reviews

The Imitation Game: not history; mathematics: unreasonable effectiveness vs. reasonable ineffectiveness; a statistics language for calculus?

To purchase from JSTOR: 10.4169/math.mag.88.3.243

Dummy View - NOT TO BE DELETED