Sail into the summer with the June—July Monthly. Game enthusiasts will love our lead article "Collapse: A Fibonacci and Sturmian Game," by Dennis Epple and Jason Siefken. In addition, to help us celebrate the bicentennial of Poncelet's Theorem, Lorenz Halbeisen and Norbert Hungerbühler offer a new and simple proof of this result. Also, don't miss in our Notes Section a terrific new proof of quadratic reciprocity from Virgil Barnard. In our Book Reviews Section, Marion Cohen reviews "*Mathematicians on Creativity" *by Peter Borwein, Peter Liljedahl, and Helen Zhai. Heading out for vacation—take our Problem Section along to help pass the time. Stay tuned for October, when Jack Cassidy teaches us all about early round bluffing in poker. - *Scott T. Chapman, Editor*

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Volume 122, Issue 06, pp. 515 - 614

## Table of Contents

## Articles

### Collapse: A Fibonacci and Sturmian Game

Dennis D. A. Epple and Jason Siefken

We explore the properties of Collapse, a number game closely related to Fibonacci words. In doing so, we fully classify the set of periods (minimal or not) of finite Fibonacci words via careful examination of the Exceptional (sometimes called singular) finite Fibonacci words.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.515

### A “Four Integers” Theorem and a “Five Integers” Theorem

Kenneth S. Williams

The recent exciting results by Bhargava, Conway, Hanke, Kaplansky, Rouse, and Schneeberger concerning the representabiltity of integers by positive integral quadratic forms in any number of variables are presented. These results build on the earlier work of Dickson, Halmos, Ramanujan, and Willerding on quadratic forms. Two results of this type for positive diagonal ternary forms are proved. These are the “four integers” and “five integers” theorems of the title.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.528

### A Simple Proof of Poncelet's Theorem (on the Occasion of Its Bicentennial)

Lorenz Halbeisen and Norbert Hungerbühler

We present a proof of Poncelet's theorem in the real projective plane which relies only on Pascal's theorem.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.537

### Covering Numbers of Finite Rings

Nicholas J. Werner

A noncyclic finite group is always equal to a union of its proper subgroups, and the minimum number of subgroups necessary to achieve this union is called the covering number of the group. Here, we investigate the analogous ideas for finite rings. We say an associative ring *R* with unity is coverable if it is equal to a union of its proper subrings. If this can be done using a finite number of proper subrings, then the covering number of *R* is the minimum number of subrings required to cover *R*. Not every ring is coverable, and even when *R* is finite it is nontrivial to decide whether *R* is coverable. We present a classification theorem for finite coverable semisimple rings and determine the covering number for *R* when *R* is coverable and equal to a direct product of finite fields.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.552

### Diophantine Approximation and Coloring

Alan Haynes and Sara Munday

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.567

### Periodic Points of Some Discontinuous Mappings

Satomi Murakami and Hiroki Ohwa

The purpose of this paper is to investigate periodic points of discontinuous mappings. For some discontinuous mappings, we prove an analogue of the Sharkovsky theorem.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.581

## Notes

### A Proof of Quadratic Reciprocity

Virgil Barnard

This paper gives an alternative proof of the law of quadratic reciprocity that hinges on some well-known facts about Euler's criterion, the existence of primitive roots, and basic properties of the floor function.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.588

### On the Hölder and Cauchy–Schwarz Inequalities

Iosif Pinelis

A generalization of the Hölder inequality is considered. Its relations with a previously obtained improvement of the Cauchy–Schwarz inequality are discussed.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.593

### Another Proof That the Real Numbers ℝ Are Uncountable

José Gascón

We give a proof, based on Cousin's lemma, that the real numbers ℝ are uncountable.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.596

### An Alternative to Faulhaber's Formula

Mircea Merca

In this note, the author proves that sums of powers of the first n positive integers can be expressed as finite discrete convolutions.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.599

### A New Proof of the Equality of Mixed Second Partial Derivatives

Tommaso Drammatico

A new proof for the equality of mixed second partial derivatives is provided using the increasing function theorem rather than the mean value theorem.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.602

## Problems and Solutions

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.604

## Book Review

*Mathematicians on Creativity*By Peter Borwein, Peter Liljedahl, and Helen Zhai

Reviewed by Marion Cohen

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.6.613

## MathBits

### Alternate Derivation for the Parity of the nth Triangular Number

Aaditya Salgarkar and Rishi Mehta

### A Geometric Proof of the Beppo–Levi Inequality

J. Rooin and A. Alikhani

### A Proof of the Replacement Theorem by the Notion of Maximal Elements

Haryono Tandra

### Asymptotic Formula for (1 + 1 /* x*) ^{x}, Revisited

Vadim Ponomarenko

### Uniform Continuity of Continuous Functions on Compact Metric Spaces

Daniel Daners

### A Simple Modified Version for Ferguson's Proof of the Extreme Value Theorem

Haryono Tandra

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

Edited by Vadim Ponomarenko