College Mathematics Journal Contents—November 2017

November 2017 College Mathematics Journal Cover

In this final issue of volume 48, the lead article features students exploring the game Lazy Cops and Robbers on graphs; see this month's special issue of The American Mathematical Monthly focusing on undergraduate research for much more. Then every mathematician's favorite dog is in the spotlight again, with two articles considering new possibilities for just how Elvis managed to catch those frisbees so well. Other highlights include two Classroom Capsules on linear algebra topics. In closing the volume, we commend the winners of the 2017 George Pólya Award, offer comments and corrections on some 2016 articles, and devote some pages to publicly thanking recent referees—their gifts of close reading and suggestions for improvement are essential to the quality mathematical exposition you find here. —Brian Hopkins

Vol. 48, No. 5, pp. 321-400

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ARTICLES

An Introduction to Lazy Cops and Robbers on Graphs

p. 322.

Brendan W. Sullivan, Nikolas Townsend, and Mikayla L. Werzanski

Cops and Robbers is a classic pursuit–evasion game played on graphs. A new variant, Lazy Cops and Robbers, allows only one cop to move at a time, making the game’s mechanics more akin to chess. We investigate and describe a few examples, showing that on some graphs lazy cops can be as effective as ordinary cops, but on other graphs they are not (and sometimes they are much worse). Some examples are based on the cops and robber moving like particular chess pieces.We also mention related results and pose some open problems.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.322

Proof Without Words: The Pythagorean Theorem

p. 334.

John Molokach

We present a wordless proof of the Pythagorean theorem by dissection using three isosceles right triangles.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.334

Did Elvis Know Cauchy—Schwarz?

p. 335.

Li Zhou

Using the Cauchy–Schwarz inequality and without using calculus, we give an elementary algebraic derivation of Snell’s law.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.3.335

The Geometer Dog Who Did Not Know Calculus

p. 339.

Alda Carvalho, Carlos Pereira dos Santos, and Jorge Nuno Silva

We present a purely geometric solution for Elvis’s path problem. The geometric approach brings, for a constructible ratio of speeds, an easy compass-and-straightedge construction that visually explains some extreme cases. There is also a possible biological interpretation.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.339

Proof Without Words: A Pascal-Like Triangle With Pell Number Row Sums

p. 346.

Ángel Plaza

In a Pascal-like triangle, where each entry is the sum of the three numbers above them, we visually prove that the row sums are the Pell numbers, given by a two-step recursion.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.346

Bet(ch)a my Team Wins the Playoffs

p. 347.

Roger W. Johnson

We consider an arbitrary (odd) length playoff series between our team and an opponent where game outcomes are independent and our team has a fixed chance of winning any particular game. While the chance our team wins the playoff series is simple enough to write as a sum involving binomial coefficients, we see that expressing this probability in terms of an incomplete beta function allows us to easily show several elementary and intuitive properties about the playoff series.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.347

Proof Without Words: Varignon’s Theorem

p. 354.

Alik Palatnik

We present a visual proof of Vairgnon’s theorem by partitioning the Varignon parallelogram using a midpoint of the quadrilateral diagonal.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.354

Variations on an Archimedean Ground: The Generalized Salinon

p. 355.

Óscar Ciaurri and Emilio Fernández

We consider a geometric figure generalizing the Archimedean salinon defined in Liber Assumptorum and prove some of its metric properties, extending that classic work’s propositions related to the arbelos and the (symmetric) salinon.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.355

Classroom Capsules

A Short and Elementary Proof of the Two-Sidedness of the Matrix Inverse

p. 366.

Pietro Paparella

We give an elementary proof of the two-sidedness of the matrix inverse using only linear independence and the reduced row-echelon form of a matrix, concepts prominent in an elementary linear algebra course. In addition, we show that a matrix is invertible if and only if it is row-equivalent to the identity matrix without appealing to elementary matrices.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.366

On the Eigenvalues of Anticommuting Matrices

p. 368.

Wenti Zhong

In this note, we express the eigenvalues of the product and sum of two anticommuting matrices (where switching the order of multiplication is equivalent to multiplying by negative one) in terms of the eigenvalues of the constituent matrices under the assumption that one matrix is diagonalizable.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.5.368