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College Mathematics Journal Contents—January 2016

The first issue of volume 47 of The College Mathematics Journal leads off with a Markov chain analysis of the new overtime rules for the National Football League. The authors, Jacqueline Leake and Nicholas Pritchard, are one of three student-mentor teams whose work is featured in this issue. While the double blind referee system does not favor such student-faculty collaborations, we are pleased to end up publishing so many. The seven articles of this issue nicely represent the breadth of the standard undergraduate curriculum: calculus, differential equations, geometry, probability, and group theory. But in a nod to the continual change in personal technology and its impact on the academy, the review in this issue looks at various graph theory smartphone apps rather than a book. -Brian Hopkins

Vol 47 No 1, pp 1-80


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The Advantage of the Coin Toss for the New Overtime System in the National Football League

Jacqueline Leake and Nicholas Pritchard

In the National Football League, the coin toss winner has had a clear advantage during an overtime period since 1994. The NFL instituted a new overtime system during the 2011 playoffs and the 2012 season in hopes of reducing the advantage of winning the coin toss. To analyze this change, with only a few seasons of data, we use absorbing state Markov chains to create a model and examine the effect of the coin toss in this new system.

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Proof Without Words: The Pentagon-Hexagon-Decagon Identity

Roger B. Nelsen

Using Ptolemy′s theorem and similar triangles, we prove the pentagon–hexagon–decagon identity, Proposition 10 in Book XIII of Euclid′s Elements.

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Yet More Ways to Skin a Definite Integral

Brian Bradie

Continuing a topic from two recent articles, we present three additional methods for evaluating a particular definite integral, using partial fractions from complex roots, an infinite series representation beyond Taylor series, and special functions. We discuss what further topics students can explore after becoming familiar with each method. We also present a generalization of the original problem and extensions.

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Area and Perimeter Bisecting Lines of a Triangle

Allan Berele and Stefan Catoiu

Any triangle has between one and three lines that bisect both the area and the perimeter. We determine the conditions for each possible number.

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A Characterization of the Cyclic Groups by Subgroup Indices

Greg Oman and Victoria Slattum

We provide a new characterization of cyclic groups using the index of a subgroup. We show that a group, finite or infinite, is cyclic exactly when its distinct subgroups have distinct indices.

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Boundary Value Problems and Finite Differences

Paul T. Allen

The solvability of boundary value problems differs greatly from that of initial value problems and can be somewhat difficult to make sense of in the context of a sophomore-level differential equations course. We present an approach that uses finite difference approximations to motivate and understand the theory governing the existence and uniqueness of solutions to boundary value problems at an elementary level.

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William Neile′s Contribution to Calculus

Andrew Leahy

In 1657, the Oxford University student William Neile, using infinitesimal techniques developed by Bonaventura Cavalieri, Evangelista Torricelli, and John Wallis, successfully found the arc length of the semicubical parabola. We give an exposition and modern interpretation of his result and discuss some of the historical outcomes that stemmed from it.

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Simplified Expectations in the Birthday Problem

Leonard Littleton and Russell May

We present a simplified derivation for obtaining asymptotic approximations of the expected number of people needed to find two with the same birthday in a year with a specified number of days. This derivation of an expected value in the birthday problem uses only techniques common to a standard calculus course and avoids higher-level methods used in previous derivations.

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Classroom Capsules

Weighted AM-GM Inequality via Elementary Multivariable Calculus

Heiko Hoffmann

We provide a short proof for the weighted inequality of arithmetic and geometric mean based on elementary facts from multivariable calculus.

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Waiter! One Classic Calculus Problem, Hold the Calculus

Ricardo E. Rojas

Using the AM-GM inequality, we solve the classic open-top box problem from first-semester calculus.

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Problems and Solutions

Problems and Solutions: 61-66

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Book Review

Technology Review: Illustrating Planar Graphs and Kuratowski's Theorem on Smartphone Apps

Anne Quinn

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Media Highlights

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