The final College Mathematics Journal issue of volume 45 opens with Jeff Suzuki considering when a gerrymander goes awry into the realm of a dummymander. There are two articles on the popular game Nim: Lisa Rougetet examines the European recreations that preceded it, while Fillers, Linderman, and Simoson explore its connections to the game Mancala. Interspersed with a combinatorial proof, a statistics perspective on calculus, and new results on confocal conics are a record six Proofs Without Words. The cover art by Marc Frantz illustrates a "found math" example of a hyperbola with horizontal and vertical asymptotes detailed in his Classroom Capsule. The issue closes with citations for the George Pólya Awards given at MathFest and a list of the generous referees whose work is essential to the CMJ. —Brian Hopkins
Vol 45 no 5, pp 335-416
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The Self-Limiting Partisan Gerrymander: An Optimization Approach
Applying optimization techniques to partisan gerrymandering yields profoundly disturbing results. If equipopulation is the only enforceable constraint, then an optimal redistricting would allow the party in control of the districting apparatus (the in-party) to produce a substantial disparity between the fraction of the total vote won by the party and the fraction of the legislative seats won by the party. Moreover, unrestricted partisan gerrymandering allows the in-party to engage in extremist politics without electoral consequences.
To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.5.338
Proof Without Words: Summing Squares by Counting Triangles
A visual proof of an identity for squares using the inclusion-exclusion principle.
To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.5.349
Mancala as Nim
Whitney Rhianna Fillers, Bill Linderman, and Andrew Simoson
We show that a simple version of Mancala is a somewhat camouflaged version of Nim and use game trees to analyze certain cases.
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Proof Without Words: An Arctangent Identity
A visual proof of an arctangent identity involving π.
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A Prehistory of Nim
The first occurrence of Nim dates back to 1901 when the mathematician Charles Leonard Bouton published an article on its solution. But what are the origins of the Nim game? This article offers a survey of European ancestors of Nim.
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Proof Without Words: Limit of a Recursive Arithmetic Mean
Visual proof of the limit of a recursive arithmetic mean sequence.
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A Combinatorial Proof of a Theorem of Katsuura
Brian K. Miceli
We give a combinatorial proof of an algebraic result of Katsuura’s. Moreover, we use the proof of this result to shed some light on an interesting property of the result itself.
To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.5.365
Proof Without Words: Cosine Difference Identity
We give a visual proof of the cosine difference formula.
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Unfamiliar Properties of Familiar Shapes
We present a surprising property of intersecting confocal conics: Their intersection points and the intersection point of their directrices are collinear. We give geometric proofs for various cases of two ellipses.
To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.5.371
Proof Without Words: A Sine Identity for Triangles
Roger B. Nelsen
We wordlessly prove a sine result for the angles of a triangle.
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Calculus from a Statistics Perspective
Kimberly Leung, Chris Rasmussen, Samuel S. P. Shen, and Dov Zazkis
We present a statistics perspective on calculus, which can help students understand calculus concepts and analyze a function defined by data or sampling values from a given function, rather than an explicit mathematical formula. Our approach uses the arithmetic and graphic means for the notions of integral and derivative, respectively. An intuitive approach to the law of large numbers replaces the notion of limit for this introductory level.
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Proof Without Words: A Formula of Hermann
A visual proof of a formula for π proved by the Swiss mathematician Jakob Hermann in 1706.
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Hyperbola: Under Construction!
We show how a perspective illusion in a picture of a building under construction leads to the design of a classical linkage for drawing a hyperbola.
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A Squeeze for Two Common Sequences That Converge to e
The note gives a simple proof that two common sequences, that are traditionally used to define the number e, converge to the same limit.
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Problems and Solutions
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Origins of Mathematical Words by Anthony Lo Bello
Reviewed by Brian Hopkins
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George Pólya Awards for 2014 and Referees in 2014
To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.5.414