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College Mathematics Journal Contents—March 2015

 As suggested by Marc Roulstone's playful cover, this issue of The College Mathematics Journal continues the popular dogs and calculus series of articles.  Bacinski and Panaggio join Tim Pennings (a 2008 Pólya Award winner) in remembering Elvis, the Welsh Corgi who assisted Pennings in some 200 presentations, and comparing his optimal paths to a dog following a greedy approach.  Bell, Polson, and Richmond add to the series by considering how a swimming pool alters the optimal path between opposite corners of a square yard.  Among the articles with no canine motivation, Cotton, McLeman, and Pinchock explore what happens when you cross two fractals while Dae Hong and Ricardo Alfaro each make use of the beloved Fibonacci numbers in two very different ways.  Four Proofs Without Words, three Classroom Capsules, and one book review help complete this rich issue. —Brian Hopkins

Vol 46 No 2, pp 81-160


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Elvis Lives: Mathematical Surprises Inspired by Elvis, the Welsh Corgi

Steve J. Bacinski, Mark J. Panaggio, and Timothy J. Pennings

Elvis, the Welsh corgi, became famous when he found the quickest route down the beach and through the water to his ball. It was later discovered that Salsa, the Labrador, could achieve the same result by using a greedy approach, moving toward the ball as quickly as possible at each instant in time. We show that these paths coincide only under special conditions.

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The Fastest Path Between Two Points, with a Symmetric Obstacle

Kathleen Bell, Shania Polson, and Tom Richmond

Assume a square pool is positioned in a corner of a square courtyard. We find the fastest path between diagonally opposite corners of the courtyard, assuming that swimming speed through the pool is less than the running speed through the courtyard. A treatment of rectangular pools by scaling is shown not to be optimal.

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Proof Without Words: Sums of Every Third Triangular Number

Roger B. Nelsen

A visual proof relating sums of every third triangular number to a single triangular number.

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On Combining and Convolving Fractals

Nicholas Cotton, Cam McLeman, and Daniel Pinchock

Motivated by a construction in which we obtain an interesting one-dimensional fractal by merging the Cantor set with the Sierpinski triangle, we explore processes for merging two fractal constructions into one. In particular, we introduce a convolution operation on certain families of fractals and investigate the Hausdorff dimension of such a fractal in terms of the dimensions of its constituents.

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Proof Without Words: The Vertex Angle Sum of a Regular Star Polygon

Matthew Jakubowski and Raymond Viglione

We visually prove the formula for the vertex angle sum of a regular star polygon.

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When Is the Generating Function of the Fibonacci Numbers an Integer?

Dae S. Hong

For what values does the generating function of the Fibonacci numbers converge to an integer?We find families of such values for this and the related Lucas number generating function.

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Sequences of Power Lines

Ricardo Alfaro

It is known that the golden ratio is a common root of the Fibonacci polynomials. We consider the roots of similar polynomials which have coefficients from some related sequences, including the limit behavior of the roots as the degree increases. Furthermore, we show that these sequences of polynomials have a quadratic common factor exactly when the coefficient sequences are “Fibonacci like.”

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Proof Without Words: Cotangent Double Angle Identity

K. B. Subramaniam

We give a visual proof of an identity for the cotangent of twice a given angle.

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Groupoid Cardinality and Egyptian Fractions

Julia E. Bergner and Christopher D. Walker

Two very new questions about the cardinality of groupoids reduce to very old questions concerning the ancient Egyptians’ method for writing fractions. First, the question of whether any positive real number is the groupoid cardinality of some groupoid reduces to the question of whether any positive rational number has an Egyptian fraction decomposition. Second, the question of how many nonequivalent groupoids have a given cardinality can be answered via the number of distinct Egyptian fraction decompositions.

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Proof Without Words: Series of Reciprocals of Tetrahedral Numbers

Gunham Caglayan

We give a visual proof for the convergent sum of reciprocals of the tetrahedral numbers.

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Goldbach's Pigeonhole

Edward Early, Patrick Kim, and Michael Proulx

Goldbach’s conjecture states that every even integer greater than two can be written as the sum of two primes. For some even integers, we can prove the existence of two such primes nonconstructively via the pigeonhole principle. Using a computer search and asymptotic bounds on classic number-theoretic functions, we determine the greatest value for which this approach works.

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Classroom Capsules

A Very Elementary Proof of Bernoulli's Inequality

Cristinel Mortici

We present a proof of Bernoulli’s inequality and one of its extensions with very elementary methods.

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On an Identity Involving Powers of Binomial Coefficients

Ulrich Abel

We give an elementary proof of a combinatorial identity involving powers of binomial coefficients.

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Another Face of the Archimedean Property

Robert Kantrowitz and Michael Neumann

We show that the Archimedean property for an abstract ordered field is equivalent to several convergence conditions from calculus, most notably the validity of the geometric series test.

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Problems and Solutions

Problems and Solutions: 142-148

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Book Review

Taming the Unknown: A History from Antiquity to the Early Twentieth Century by Victor J. Katz and Karen Hunger Parshall

Reviewed by Jiang-Ping Jeff Chen

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Media Highlights

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