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College Mathematics Journal Contents—January 2014

Volume 45 of The College Mathematics Journal begins with a Roger Nelsen cover showing three representations of 45 as a polygonal number. Several articles deal with the integers: John Bonomo considers arithmetic series representations of integers, Yonah Cherniasky and Artour Mouftakhov remind us of Zbikowski’s divisibility rules, and Vince Matsko tweaks the standard recurrence relation problems of discrete mathematics by considering non-consecutive initial values. Although we all know games like American Roulette favor the house, Jennifer Switkes and Louis Bohorquez explore interesting questions on some details of gambler’s ruin. Dan King pens the review for this issue, based on Codebreaker, an Alan Turing drama-documentary. Brian Hopkins 

Vol. 45, No. 1, pp.2-79.


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From the Editor

Constancy and Change

Brian Hopkins



Not All Numbers Can Be Created Equally

John P. Bonomo

We take a well-known problem—which numbers can or cannot be written as a sum of consecutive integers—and generalize it for summations involving any arithmetic sequence. Various general and specific theorems involving these summations are proven.


Correspondence Between Geometric and Differential Definitions of the Sine and Cosine

Horia I. Petrache

In textbooks, the familiar sine and cosine functions appear in two forms: geometrical, in the treatment of unit circles and triangles, and differential, as solutions of differential equations. These two forms correspond to two different definitions of trigonometric functions. By using elementary geometry and elementary calculus, it is shown that the two definitions are equivalent.


Proof  Without Words: Alternating Sums of Consecutive Squares

Roger Nelsen

A visual proof of an identity for alternating sums of squares.


Zbikowski’s Divisibility Criterion

Yonah Cherniavsky and Artour Mouftakhov

We present a simple, quick method to obtain a criterion that determines whether one integer is divisible by another. This method is simpler than using long division or Pascal’s test of divisibility, and it can be explained to students at any level.


Proof Without Words: Completing the Square via the Difference of Squares

Munir Mahmood

The difference of squares is shown as two trapezoids, which are rearranged to a rectangle. This demonstrates the algebraic identity known as completing the square.


An Acute Case of Discontinuity

Sam Vandervelde

We create a function of two variables defined at all points in the first quadrant with the curious property that this function is continuous at points on lines through the origin with irrational slope, but discontinuous at points on lines with rational slope. The definition of the function involves both the Euclidean algorithm and an infinite series. This construction is of interest because it does not single out lines based on their slope; rather, it applies uniformly to all points, and the unexpected set of discontinuities arises naturally.


Traveling Waves and Taylor Series: Do They Have Something in Common?

Ádám Besenyei

The phenomenon of traveling waves on a rope and the notion of Taylor series are well known. Surprisingly, these two seemingly distant concepts have something in common. This connection can be used to derive the Taylor series representation for some real functions.


American Roulette: A Gambler’s Ruin

Louis Bohorquez and Jennifer Switkes

American Roulette provides rich examples of Gambler’s Ruin. We use the mathematics of difference equations, and then Markov chain matrix methods, to explore success probabilities under various betting strategies. Ultimately, ruin is likely for the gambler under each betting strategy, but we see interesting results along the way.


Second-Order Recurrences with Nonconsecutive Initial Conditions

Vincent J. Matsko

When consecutive initial conditions for second-order linear homogeneous recurrence relations with constant coefficients are given, the resulting sequence is uniquely determined. However, if the initial conditions are not consecutive, it may be the case that no sequence is possible, or that infinitely many sequences satisfy the recurrence.


Proof Without Words: Ensphering Three Capped Prisms

David Seppala-Holtzman

For n = 3, 4, or 5, a unit sphere is the smallest sphere that encloses a right prism with square sides on a regular n-gon capped by a regular pyramid with all sides of unit length. The points of incidence are the apex of the pyramid and the opposite n vertices of the prism.



Green Jello World and Escher’s World

Ruth I. Berger

Introducing models of hyperbolic geometry informally in the setting of different worlds lets students naturally come up with the idea that lines (shortest paths) can look like parts of Euclidean circles. By learning to think like inhabitants of these worlds, students are able to abandon their totally Euclidean view of lines and take ownership of hyperbolic geometry.


Continuously Differentiable Curves Detect Limits of Functions of Two Variables

Ollie Nanyes

We present a result of Rosenthal, which shows that if a two-variable function has the same limit when evaluated over the graph of every convex function (in one variable or the other) with continuous first derivative, then the two-variable function is continuous. We give a new method of constructing the “testing curves” by using complex splines.



Problems 1016-1020
Solutions 991-995



Codebreaker, directed by Clare Beavan, 81 minutes, Story Center Productions, 2012.

Reviewed by Daniel King




George Pólya Awards for 2013