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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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.
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.
A visual proof of an identity for alternating sums of squares.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Codebreaker, directed by Clare Beavan, 81 minutes, Story Center Productions, 2012.
Reviewed by Daniel King