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Mathematics Magazine - February 2015

The first three articles in the issue loosely have to do with paths. Kalman and Verdi relate a path based on coefficients of a polynomial to a divisibility condition on the polynomial. Kuczmarski considers the twists and turns a bicycle wheel makes while following a path. Crans, Rovetti, and Vega describe paths of play in the {\it KO Labyrinth} by applying graph theory and Markov chains. Other articles focus on the median value of a continuous function and a class of functions that satisfy conditions posed by sabermatrician Bill James. —Michael A. Jones, Editor

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Vol. 88, No. 1, pp 1 – 89

Articles

Polynomials with Closed Lill Paths

Dan Kalman and Mark Verdi

To purchase from JSTOR: 10.4169/math.mag.88.1.3

Bend, Twist, and Roll: Using Ribbons and Wheels to Visualize Curvature and Torsion

Fred Kuczmarski

We describe classical ways to think about the signed curvature and torsion of space curves using spherical indicatrices to record the twisting and turning of the Frenet frame. By constructing ribbons around curves, we show how to visualize their curvature and torsion. Finally, we use bicycle wheels to measure these quantities.

To purchase from JSTOR: 10.4169/math.mag.88.1.11

Solving the KO Labyrinth

Alissa S. Crans, Robert J. Rovetti, and Jessica Vega

The KO Labyrinth is a colorful spherical puzzle with 26 chambers, some of which can be connected via holes through which a small ball can pass when the chambers are aligned correctly. The puzzle can be realigned by performing physical rotations of the sphere in the same way one manipulates a Rubik’s Cube, which alters the configuration of the puzzle. The goal is to navigate the ball from the entrance chamber to the exit chamber. We find the shortest path through the puzzle using Dijkstra’s algorithm and explore questions related to connectivity of puzzle with the adjacency matrix, distance matrix, and first passage time analysis. We show that the shortest path through the maze takes only 10 moves, whereas a random walk through the maze requires, on average, about 340 moves before reaching the end. We pose an analogue of the gambler’s ruin problem and separately consider whether we are able to solve the puzzle if certain chambers are off limits. We conclude with comments and questions for future investigation.

To purchase from JSTOR: 10.4169/math.mag.88.1.27

Proof Without Words: Sums of Products of Three Consecutive Integers

Hasan Unal

To purchase from JSTOR: 10.4169/math.mag.88.1.37

The Median Value of a Continuous Function

Irl C. Bivens and Benjamin G. Klein

Using a limit we define the median value of a continuous function on a bounded interval and we explicate the connection between the median and area minimization. With the help of Lebesgue measure we show that this connection can be viewed as a special case of an interesting minimization principle in probability.

To purchase from JSTOR: 10.4169/math.mag.88.1.39

2D or Not 2D

Brendan W. Sullivan

The February crossword puzzle.

To purchase from JSTOR: 10.4169/math.mag.88.1.52

The James Function

Christopher N. B. Hammond, Warren P. Johnson, Steven J. Miller

We investigate the properties of the James function, associated with Bill James’s so-called “log5 method,” which assigns a probability to the result of a game between two teams based on their respective winning percentages. We also introduce and study a class of functions, which we call Jamesian, that satisfy the same a priori conditions that were originally used to describe the James function.

To purchase from JSTOR: 10.4169/math.mag.88.1.54

Problems and Solutions

Proposals, 1961-1965

Quickies, 1047-1048

Solutions, 1936-1940

Answers, 1047-1048

To purchase from JSTOR: 10.4169/math.mag.88.1.72

Reviews

Is computational science mathematics?; 400 years of logarithms; baking schemes

To purchase from JSTOR: 10.4169/math.mag.88.1.79

News and Letters

75th Annual William Lowell Putnam Mathematical Competition

To purchase from JSTOR: 10.4169/math.mag.88.1.81

Dummy View - NOT TO BE DELETED