Spend some of your time off over the holidays with the December *Monthly*. If you like to factor things, then you are in luck: both of our lead articles, "Sets of Lengths" by Alfred Geroldinger, and "Balanced Factorizations" by Anton A. Klyachko and Anton N. Vassilyev, deal with factorizations in various algebraic structures.

Chris Marx reviews *A First Course in the Calculus of Variations* by Mark Kot, and our Problems and Solutions section is as challenging as ever. Stay tuned for the January *Monthly*, which will be the inaugural issue of the new *Monthly* Editor Susan Colley!

— *Scott T. Chapman, Editor*

##### JOURNAL SUBSCRIBERS AND MAA MEMBERS:

To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.

## Table of Contents

### A Letter from the Editor: The Long and Winding Road

p. 955.

Scott T. Chapman

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.955

### Sets of Lengths

p. 960.

Alfred Geroldinger

Oftentimes the elements of a ring or semigroup can be written as finite products of irreducible elements. An element a can be a product of *k* irreducibles and a product *l* of irreducibles. The set L(*a*) of all possible factorization lengths of *a* is called the set of lengths of *a*, and the system consisting of all these sets L(*a*) is a well-studied means of describing the nonuniqueness of factorizations of a ring or semigroup. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.960

### Balanced Factorizations

p. 989.

Anton A. Klyachko and Anton N. Vassilyev

*Any rational number can be factored into a product of several rationals whose sum vanishes. *This simple but nontrivial fact was suggested as a problem on a mathematical olympiad for high school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.989

### Cantor-Polynomials and the Fueter-Pólya Theorem

p. 1001.

Melvyn B. Nathanson

A packing polynomial is a polynomial that maps the set N_{0}^{2} of lattice points with nonnegative coordinates bijectively onto N_{0}. Cantor constructed two quadratic packing polynomials, and Fueter and Pólya proved analytically that the Cantor polynomials are the only quadratic packing polynomials. The purpose of this paper is to present a beautiful elementary proof of Vsemirnov of the Fueter–Pólya theorem. It is a century-old conjecture that the Cantor polynomials are the only packing polynomials on N_{0}^{2}.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1001

### Optimizing the Video Game Multi-Jump: Player Strategy, AI, and Level Design

p. 1013.

Aaron M. Broussard, Martin E. Malandro, and Abagayle Serreyn

This article initiates the mathematical study of multi-jumping in video games. We begin by proving a necessary, and frequently sufficient, condition for a multi-jump to be *optimal*, i.e., achieve the highest possible height after traveling a given horizontal distance. We then give strategies that can be used by human players and by AI to select successful multi- jumps in real time. We also show how a video game designer can build the ground around a platform to guarantee that the platform is reachable—or unreachable—by a multi-jump beginning at any point on the ground.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1013

## Notes

### Dilated Floor Functions That Commute

p. 1033.

Jeffrey C. Lagarias, Takumi Murayama, and D. Harry Richman

We determine all pairs of real numbers (*α*, *β*) such that the dilated floor functions ⌊*αx*⌋ and ⌊*βx*⌋ commute under composition, i.e., such that ⌊*α*⌊*βx*⌋⌋ = ⌊*β*⌊*αx*⌋⌋ holds for all real *x*.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1033

### Quotients of Fibonacci Numbers

p. 1039.

Stephan Ramon Garcia and Florian Luca

There have been many articles in the *Monthly* on quotient sets over the years. We take a first step here into the *p*-adic setting, which we hope will spur further research. We show that the set of quotients of nonzero Fibonacci numbers is dense in the *p*-adic numbers for every prime *p*.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1039

### Parking Cars of Different Sizes

p. 1045.

Richard Ehrenborg and Alex Happ

We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1045

## Problems and Solutions

p. 1050.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1050

## Book Review

p. 1058.

*A First Course in the Calculus of Variations* by Mark Kot

Reviewed by Chris A. Marx

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1058

### Referee Thank You

p. 1062.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1062

## MathBits

### Single Digit NFL Scores

p. 959.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.959

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

p. 1032.

### Corrigendum to "A New Proof of the Change of Variable Theorem for the Riemann Integral"

p. 1049.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.10.1049