The December *Monthly* begins with a pair of articles about taking things apart: decompose a cube into nearly equal smaller cubes, and discuss and show how to construct self-similar polygonal tilings of the plane. As the sun sets earlier this season, consider the solvability of “Lights Out”-type puzzles.

In the Notes, read a demonstration of the infinitude of primes using theorems of Fermat and van der Waerden on arithmetic progressions, and how to throw a ball (far).

Finally, while visions of problems dance in your head, read a review of Gareth Roberts’s From Music to Mathematics: Exploring the Connections.

Happy holidays and happy reading!

— Susan Jane Colley, Editor

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## Table of Contents

### Decomposition of a Cube into Nearly Equal Smaller Cubes

p. 895.

Peter Frankl, Amram Meir, and áJanos Pach

Let *d* be a fixed positive integer and let *ε* > 0. It is shown that for every sufficiently large *n* ≥ *n*_{0}(*d*, *ε*), the *d*-dimensional unit cube can be decomposed into exactly *n* smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most 1 + *ε*. Moreover, for every *n* ≥ *n*_{0}, there is a decomposition with the required properties, using cubes of at most *d* + 2 different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of three different sizes.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.895

### Self-Similar Polygonal Tiling

p. 905.

Michael Barnsley and Andrew Vince

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting and to provide a novel method for the construction of such polygonal tilings.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.905

### Brunn-Minkowski Theory and Cauchy's Surface Area Formula

p. 922.

Emmanuel Tsukerman and Ellen Veomett

We use Brunn–Minkowski theory and well-known integration tools to prove Cauchy’s surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension. Using the same technique, we give a straightforward extension to intrinsic moment vectors of a convex body.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.922

### Absolute Real Root Separation

p. 930.

Yann Bugeaud, Andrej Dujella, Tomislave Pejković, and Bruno Salvy

While the separation (the minimal nonzero distance) between roots of a polynomial is a classical topic, its absolute counterpart (the minimal nonzero distance between their absolute values) does not seem to have been studied much.We present the general context and give tight bounds for the case of real roots.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.930

### "Lights Out" and Variants

p. 937.

Martin Kreh

In this article, we investigate the puzzle “Lights Out” as well as some variants of it (in particular, varying board size and number of colors).We discuss the complete solvability of such games, i.e., we are interested in the cases such that all starting boards can be solved. We will model the problem with basic linear algebra and develop a criterion for the unsolvability depending on the board size modulo 30. Further, we will discuss two ways of handling the solvability that will rely on algebraic number theory.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.937

## NOTES

### Squares in Arithmetic Progressions and Infinitely Many Primes

p. 951.

Andrew Granville

We give a new proof that there are infinitely many primes, relying on van der Waerden’s theorem for coloring the integers, and Fermat’s theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.951

### Throwing a Ball as Far as Possible, Revisited

p. 960.

Joshua Cooper and Anton Swifton

What initial trajectory angle maximizes the arc length of an ideal projectile? We show the optimal angle, which depends neither on the initial speed nor on the acceleration of gravity, is the solution *θ* to a surprising transcendental equation: csc(*θ*) = coth(csc(*θ*)), i.e., *θ* = csc^{−1}(*α*) where α is the unique positive fixed point of coth(*x*). Numerically, *θ* ≈ 0.9855 ≈ 56.47^{◦}. The derivation involves a nice application of differentiation under the integral sign.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.960

### How Many Units Can a Commutative Ring Have?

p. 960.

Sunil Chebolu and Keir Lockridge

László Fuchs posed the following problem in 1960, which remains open: determine whether a given abelian group can occur as the group of units in a commutative ring. In this note, we provide an elementary solution to a simpler, related problem: find all cardinal numbers occurring as the cardinality of the group of units in a commutative ring. As a byproduct, we obtain a solution to Fuchs’s problem for the class of finite abelian *p*-groups when *p* is an odd prime.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.960

### A Constructive Elementary Proof of the Skolem-Noether Theorem for Matrix Algebras

p. 966.

Jenő Szigeti and Leonard Van Wyk

We give a constructive elementary proof for the fact that any *K*-automorphism of the full *n* × *n* matrix algebra over a field* K* is conjugation by some invertible *n* × *n* matrix over *K*.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.966

## Problems and Solutions

p. 970.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.970

## Book Review

p. 979.

*From Music to Mathematics: Exploring the Connections* by Gareth E. Roberts

Reviewed by Evelyn Lamb

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.10.979

### Editor's Endnotes

p. 983.

### Monthly Referees for 2017

p. 986.

## MathBits

### The Student with All the Answers

p. 950.

### 100 Years Ago This Month in *The American Mathematical Monthly*

p. 965.

### If Primes are Finite, All of Them Divide the Number One

p. 969.