Nontransitive dice, long a favorite topic in recreational mathematics, are featured in the two lead articles of this first issue of volume 48, including a set of "new Grime dice" by popular speaker and YouTube personality James Grime (singingbanana.com, numberphile.com). With these new dice and a particular game, you would have a 44% chance of beating both Bill Gates and Warren Buffett (if they agreed to play).

Among the other articles is a historical treatment of "the first whisper" of Galois theory through symmetric polynomials. While the modern treatment of this revolutionary material has been streamlined, Ben Blum-Smith and Samuel Coskey show the value of tracing through this earlier viewpoint.

In the reviews, Jenna Carpenter reflects on the surprising connections she found with the characters of Margot Lee Shetterly's powerful book *Hidden Figures* (see the next *Math Horizons* for her review of the celebrated movie). And Craig Kaplan, editor of the *Journal of Mathematics and the Arts*, reviews Henry Segerman's beautiful new book *Visualizing Mathematics with 3D Printing*. —*Brian Hopkins*

Vol. 48, No. 1, pp. 1-80

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## ARTICLES

### The Bizarre World of Nontransitive Dice: Games for Two or More Players

p. 2.

James Grime

With nontransitive dice, you can always pick a dice with a better chance of winning than your opponent. There are well-known sets of three or sets of four nontransitive dice. Here, we explore designing a set of nontransitive dice that allows the player to beat two opponents at the same time. Three-player games have been designed before using seven dice. We introduce an improved three-player game using five dice, exploiting a reversing property of some nontransitive dice.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.2

### Balanced Nontransitive Dice

p. 10.

Alex Schaefer and Jay Schweig

We study triples of labeled dice in which the relation “is a better die than” is nontransitive. Focusing on such triples with an additional symmetry we call balance, we prove that such triples of dice exist for all dice having at least three faces.We then examine the sums of the labels of such dice and use these results to construct an algorithm for verifying whether or not a triple of dice is balanced and nontransitive. We also consider generalizations to larger sets of dice and other related ideas.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.10

### Proof Without Words: Perfect Numbers Modulo 7

p. 17.

Roger B. Nelsen

We partition triangular numbers to show wordlessly that every even perfect number except 28 is congruent to 1 or 6 modulo 7.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.17

### The Fundamental Theorem on Symmetric Polynomials: History's First Whiff of Galois Theory

p. 18.

Ben Blum-Smith and Samuel Coskey

We describe the fundamental theorem on symmetric polynomials (FTSP), exposit a classical proof, and offer a novel proof that arose out of an informal course on group theory. The paper develops this proof in tandem with the pedagogical context that led to it. We also discuss the role of the FTSP both as a lemma in the original historical development of Galois theory and as an early example of the connection between symmetry and expressibility that is described by the theory.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.18

### A Plane Angle Poem

p. 30.

Jordie Albiston

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.30

### Existence of Limits and Continuity

p. 31.

Julie Millett and Xingping Sun

In this article we prove the following result. If a function defined on an interval has a finite one-sided limit at each point of a dense subset of the interval, then the set of points where the function is continuous is dense in the interval and uncountable. Our proof is accessible to undergraduate students.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.31

### Proof Without Words: Tangents of 15 and 75 Degrees

p. 35.

García Capitán Francisco Javier

We provide a figure showing the values for the tangents of 15 and 75 degrees.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.35

### Divisibility Tests, Old and New

p. 36.

Sandy Ganzell

This article reviews some of the history of divisibility tests. Based on an elementary idea by Lagrange, the author describes a new test that not only detects when one number divides another, it also determines the remainder.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.36

### Proof Without Words: An Arithmetic-Geometric Series

p. 41.

Óscar Ciaurri

We give a visual proof that a series with squares and powers of two sums to six.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.41

### Covariances Between Transient States in Finite Absorbing Markov Chains

p. 42.

Michael A. Carchidi and Robert L. Higgins

It is well known that, starting from a transient state in a finite absorbing Markov chain, the mean and variance in the number of visits to any transient state can be expressed in terms of entries in the chain's normal matrix. We show that, starting from a transient state, the covariance in the number of visits to any two transient states can also be expressed in terms of entries in the normal matrix, as can the mean and variance in the total number of visits to all transient states.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.42

### Proof Without Words: The Triangle with Maximum Area for a Given Base and Perimeter

p. 51.

Ángel Plaza

By using the ellipse with foci at the extreme points of the base, we show wordlessly that the triangle with maximum area for a given base and perimeter is the isosceles triangle where the different edge is the base.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.51

## Classroom Capsules

### A Powerful Method of Non-Proof

p. 52.

John Beam

Although truth tables can be used in a legitimate way to justify arguments, one should exercise caution when doing so. We demonstrate by suggesting a method of proof that is too good to be true.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.52

### A Function Worth a Second Look

p. 55.

Michael Maltenfort

We take a closer look at an interesting function introduced in a recent Classroom Capsule by Denis Bell.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.55

## Problems and Solutions

p. 58.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.58

## Book Reviews

*Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race* By Margot Lee Shetterly

p. 64.

Reviewed by: Jenna P. Carpenter

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.64

*Visualizing Mathematics with 3D Printing *By Henry Segerman

p. 69.

Reviewed by: Craig S. Kaplan

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.69

## Media Highlights

p. 73.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.48.1.73