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 CreativityBy 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