The *Monthly* welcomes you to autumn. The October issue kicks off with a report on the 77th Putnam Competition, followed by Soham Basu and Daniel Velleman’s filling of the gap left by Gauss in his 1799 proof (or, should it be “proof”?) of the fundamental theorem of algebra, measurement of distances between different domino tilings of a surface, results about plane curves generated by rolling one curve along another that recall and extend a century-old theory of such “roulettes,” and a study of the denominators appearing in the polynomials that express the sums of the powers of the first n natural numbers.

In the Notes, you will find that there are infinitely many integers that are not expressible as the sum of two squares and at most two powers of 2; a lattice-based proof that there can be at most there can be at most three different gap lengths in the fractional parts of the sequence *a*, 2*a*,…, *Na* where a is any real number; an example of a one-to-one function that is continuous at all irrational numbers and discontinuous at all rationals; and a lower bound on the Minkowski complexity of a finite set of nonnegative integer vectors (i.e., the minimum number of union and Minkowski sums needed to build the set beginning with the singleton sets of vectors).

Finally, besides an assortment of problems to absorb your free time, Peter Winkler offers a discussion of mathematical puzzles in his joint review of David Singmaster’s *Problems for Metagrobologists* and Hans van Ditmarsch and Barteld Kooi’s *One Hundred Prisoners and A Light Bulb*.

— *Susan Jane Colley, 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

### The Seventy-Seventh William Lowell Putnam Mathematical Competition

p. 675.

Leonard F. Klosinski, Gerald L. Alexanderson, and Mark Krusemeyer

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.675

### On Gauss’s First Proof of the Fundamental Theorem of Algebra

p. 688.

Soham Basu and Daniel J. Velleman

Carl Friedrich Gauss is often given credit for providing the first correct proof of the fundamental theorem of algebra in his 1799 doctoral dissertation. However, Gauss’s proof contained a significant gap. In this paper, we give an elementary way of filling the gap in Gauss’s proof.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.688

### Power-Sum Denominators

p. 695.

Bernd C. Kellner and Jonathan Sondow

The *power sum* 1^{n} + 2^{n} + · · · + *x*^{n} has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in *x* of degree *n* + 1 with rational coefficients. Here, we consider the denominators of these polynomials and prove some of their properties. A remarkable one is that such a denominator equals *n* + 1 times the squarefree product of certain primes *p* obeying the condition that the sum of the base-*p* digits of *n* + 1 is at least *p*. As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.695

### Distances in Domino Flip Graphs

p. 710.

Hugo Parlier and Samuel Zappa

This article is about measuring and visualizing distances between domino tilings. Given two tilings of a simply-connected square-tiled surface, their distance is the minimum number of flips between them. For a given shape of surface, we’re interested in computing the diameters of the flip graphs, meaning the maximal distance between any two of its tilings. Building on work of Thurston and others, we give geometric interpretations of distances that result in formulas for the diameters of the flip graphs of rectangles or Aztec diamonds.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.710

### Rolling Curves: An Old Proof of the Roulette Lemma

p. 723.

Ulrich Abel, Linda Beukemann, and Vitaliy Kushnirevych

The purpose of this note is to bring to light a nearly forgotten theory of roulettes that was developed more than 100 years ago. It leads to a differential equation which contains the roulette lemma as a special case and uses complex-valued functions to describe curves in a plane.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.723

## Notes

### On the Sum of Two Squares and At Most Two Powers of 2

p. 737.

David J. Platt and Timothy S. Trudgian

We demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two nonnegative integer powers of 2.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.737

### The Three Gap Theorem and the Space of Lattices

p. 741.

Jens Marklof and Andreas Strömbergsson

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence *α*, 2*α*, . . . , *Nα*, for any integer *N* and real number *α*. This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.741

### A One-to-One Popcorn Function

p. 746.

Zoltán Kánnai

A palpable example for a one-to-one Thomae-type popcorn continuous function of the real line is constructed.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.746

### Minkowski Complexity of Sets: An Easy Lower Bound

p. 749.

Stasys Jukna

The Minkowski complexity of a finite set of vectors is the minimum number of set-theoretic union and Minkowski sum operations needed to create this set when starting from single-element sets, each containing only one vector. We give an amazingly simple proof of a general lower bound on this complexity.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.749

## Problems and Solutions

p. 754.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.754

## Book Review

p. 763.

*Problems for Metagrobologists: A Collection of Puzzles with Real Mathematical, Logical, or Scientific Content* by David Singmaster and

*One Hundred Prisoners and A Light Bulb* by Hans van Ditmarsch and Barteld Kooi

Reviewed by Peter Winkler

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.8.763

## MathBits

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

p. 687.

### Linearly Independent Spanning Sets

p. 722.