You can begin your summer *Monthly* reading with the text of Francis Su’s moving MAA Retiring Presidential Address, “Mathematics for Human Flourishing.” And, of course, the mathematics continues: you can survey some beautiful results and conjectures concerning covering convex bodies with planks that have applications to simultaneous approximations of polynomials; see Nash equilibria (and other aspects of game theory) illustrated via actual play of the online game “Dice City Roller”; study limiting behavior of nonzero solutions to linear difference equations; learn how to share candy fairly; and study functions that satisfy a weighted mean value property, together with their connections to harmonic functions, and see some multivariable generalizations.

In the Notes, you can learn how a result about the genera of algebraic curves enables one to study the action of the Klein four group on sets and graphs; see a matrix generalization of the Gauss congruence for numbers; determine the value of the Riemann zeta function at even integers as sums over partitions of integers; and see new elementary proofs of the error bounds for standard numerical integration methods.

You can while away the lazy summer hours with problems to solve. And Jeffrey Nunemacher reviews *Vector Calculus, Linear Algebra, and Differential Forms* by J. H. and B. B. Hubbard.

Happy Reading!

— *Susan Jane Colley, Editor*

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## Table of Contents

### Mathematics for Human Flourishing

p. 483.

Francis Edward Su

How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish?

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.483

### From Tarski’s Plank Problem to Simultaneous Approximation

p. 494.

Andrey Kupavskii and János Pach

A *slab* (or plank) is the part of the *d*-dimensional Euclidean space that lies between two parallel hyperplanes. The distance between the these hyperplanes is called the *width* of the slab. It is conjectured that the members of any infinite family of slabs with divergent total width can be translated so that the translates together cover the whole *d*-dimensional space. We prove a slightly weaker version of this conjecture, which can be regarded as a converse of Bang's theorem, also known as Tarski’s plank problem.

This result enables us to settle an old conjecture of Makai and Pach on simultaneous approximation of polynomials. We say that an infinite sequence* S* of positive numbers controls all polynomials of degree at most *d* if there exists a sequence of points in the plane whose *x*-coordinates form the sequence *S*, such that the graph of every polynomial of degree at most *d* passes within vertical distance 1 from at least one of the points. We prove that a sequence *S* has this property if and only if the sum of the reciprocals of the *d*th powers of its elements is divergent.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.494

### Introducing Nash Equilibria via an Online Casual Game That People Actually Play

p. 506.

David Aldous and Weijian Han

This is an extended write-up of a lecture introducing the concept of *Nash equilibrium* in the context of an auction-type game that one can observe being played by "ordinary people" in real time. In a simplified model, we give an explicit formula for the Nash equilibrium. The actual game is more complicated and more interesting; players place a bid on one item (among several) during a time window; they can see the numbers but not the values of previous bids on each item. A complete theoretical analysis of the Nash equilibrium now seems a challenging research problem. We give an informal analysis and compare with data from the actual game.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.506

### Equitable Candy Sharing

p. 518.

Grant Cairns

Children, sitting in a circle, each have a nonnegative number of candies in front of them. A whistle is blown and each child with more than one candy passes one candy to the left and one to the right. The sharing process is repeated until a fixed state is attained, or the system enters a periodic cycle. This paper treats the case where the total number of candies equals the number of children. For a given initial distribution of candies, a necessary and sufficient condition is given for the system to ultimately attain the equitable distribution in which each child has one candy.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.518

### On the Limiting Behavior of Solutions of Linear Difference Equations

p. 527.

Finbarr Holland

The limit superior of the *n*th root of |*x*(*n*)|, where* x* is any nonzero solution of a scalar homogeneous linear difference equation of finite order, is identified in an elementary manner with the modulus of a root of the associated characteristic polynomial. Similar results are obtained for sequences of determinants of certain finite Hankel matrices of the same order generated by such solutions.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.527

### On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions

p. 535.

Paul Carter and David Lowry-Duda

We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogs and the existence of “weighted harmonic functions.”

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.535

## Notes

### Klein Four Actions on Graphs and Sets

p. 543.

Darren B. Glass

We consider how a standard theorem in algebraic geometry relating properties of a curve with a (ℤ/2ℤ)^{2}-action to the properties of its quotients generalizes to results about sets and graphs that admit (ℤ/2ℤ)^{2}-actions.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.543

### Fermat’s Little Theorem and Gauss Congruence: Matrix Versions and Cyclic Permutations

p. 548.

Heinrich Steinlein

In recent years, several papers appeared on generalizations of Fermat’s little theorem and the Gauss congruence to integer square matrices. We will give simple and intuitive proofs deriving these matrix versions from the corresponding number-theoretic results.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.548

### The Riemann Zeta Function With Even Arguments as Sums Over Integer Partitions

p. 554.

Mircea Merca

In this note, we build on recent work in [**7**] to establish formulas for ζ(2*n*) as sums over all the unrestricted integer partitions of *n*.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.554

### An Elementary Derivation of the Numerical Integration Bounds in Beginning Calculus

p. 558.

Lee K. Jones

Best error bounds are given for the trapezoidal, midpoint, and Simpson methods of numerical integration using only elementary techniques that a beginning calculus student can comprehend.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.558

## Problems and Solutions

p. 563.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.563

## Book Review

p. 572.

*Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach* by John H. Hubbard and Barbara Burke Hubbard

Reviewed by Jeffrey Nunemacher

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.56.572

## MathBits

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

p. 534.

### Dissection of an Equilateral Triangle into Five Congruent Open Sets

p. 547.

### On Owning Mathematics

p. 557

### A Very Short Proof of the Infinitude of Primes

p. 562.