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Mathematics Magazine - April 2005

ARTICLES

Groups of Arithmetical Functions
James E. Delany, Emeritus
83-97
The functions from the positive integers to the real numbers or complex numbers form a ring under pointwise addition and the Dirichlet product. Ideas from Abelian group theory are applied to the group of units of this ring, with interesting results. It is shown that the functions withf(1) = 1 form a vector space over the rationals, and that the multiplicative functions form a subspace. Bell series are used to study linear independence in these spaces.

LETTER TO THE EDITOR

Sury’s Parent of Binet’s Formula
Arthur T. Benjamin
97
In this Letter to the Editor, the author gives a combinatorial interpretation to a polynomial identity presented by B. Sury in his October 2004 paper called "A Parent of Binet's Formula?" The interpretation involves counting the ways to tile a strip of length n with i dominoes of length two and n - 2i squares of length one, where the tiles can be colored. Each tile can be colored in X + Y ways, where X of the colors are light and Y of the colors are dark, and when a domino is colored, the left half is always light and the right half is always dark. Counting the number of tilings with an even number of dominoes minus the number with an odd number of dominoes in two ways gives Sury's identity.

Outwitting the Lying Oracle
Robb T. Koether and John K. Osoinach, Jr.
98-109
An oracle, who can predict the outcome of the flip of a coin, engages you in a game where you bet on the outcome of the coin flip. After you announce the size of your bet, the oracle tells you the outcome of the coin flip, but warns you that it could be a lie. You then state your prediction of the coin flip. How should you place your bets in order to maximize your winnings? Can you outwit the oracle by disagreeing with the prediction when you think the oracle is lying to you? This article finds your optimal wagers when your strategy is simply to agree with the oracle’s prediction and then considers your optimal wagers and strategy when you try to outwit the oracle by occasionally disagreeing.

Twentieth-Century Gems from Mathematics Magazine
Gerald L. Alexanderson and Peter Ross
110-123
The general Stone-Weierstrass theorem was first published, surprisingly, in Mathematics Magazine. We discuss Stone’s two-part article and many other less famous gems that have appeared in the Magazine since it was first published in 1926. The gems are organized by mathematical themes, such as calculus and analysis, geometry, and number theory, with additional sections on topics such as the history of mathematics, mathematical culture, and the problem section. The authors range from those equally famous as Stone, like Paul Erdos and George Pólya, to those you’ve never heard of.

NOTES

Transposition Graphs: An Intuitive Approach to the Parity Theorem for Permutations
Dean Clark
124-130
Our proof of the classical parity theorem for permutations is pictorial, constructive, intuitive, and requires no knowledge of group theory or combinatorics. The applications (Three Card Monte and the newspaper word game Jumble©) are ideal for use in a liberal arts math course or undergraduate abstract algebra.

How to Maximize Your Chances of Getting a Color Match
Ramin Naimi and Roberto Carlos Pelayo
132-137
Suppose in a classroom with seats arranged in a rectangular grid each student is given a list of n colors, from which they are to choose one at random. Each pair of lists may have colors in common, possibly all n. Our intuition might suggest that the probability that two adjacent students will pick the same color is greatest if all the lists are identical. By representing this as a graph coloring problem, we see that it is not true! We then explore the question: for which graphs does assigning identical n-color lists to every vertex maximize the probability of obtaining adjacent vertices with the same color?

Why Euclidean Area Measure Fails in the Noneuclidean Plane
Dieter Ruoff
137-139
In noneuclidean geometry the area of a polygon can be measured in a much simpler way than in Euclidean geometry. Indeed, the angles of a polygon alone determine its area. The Euclidean approach to area with its reliance on the formula (base x altitude)/2 for triangles is definitely more clumsy. But can it nonetheless be used in the noneuclidean context? The answer, as will be shown, is  “no.”

PROOF WITHOUT WORDS

Candido’s Identity
Roger B. Nelsen
131
A proof without words of the identity [x2+y2+(x+y) 2] 2 = 2[x4+y4+(x+y) 4]. This identity was employed by Giacomo Candido (1871-1941) to establish [Fn2+F{n+1}2+F{n+2}2] 2 = 2[Fn4+F{n+1}4+F{n+2}4], where Fn denotes the nth Fibonacci number.

 

The Slope Mean and Its Invariance Properties
Jun Ji and Charles Kicey
139-144
A mean that we call the slope mean is motivated and introduced in this note. In the case of two numbers, the slope mean S(a,b) of a and b returns the slope of the line that bisects the angle formed by lines with slopes a and b. Although the slope mean is not homogeneous, we will see how it is closely related to three classic means. We will discuss how this mean, as well as three classic means, can be uniquely determined by their own invariance under two sets (algebraic groups) of functions.

A Carpenter’s Rule of Thumb from a Mathematical Viewpoint
Robert Fakler
144-146
We show that there is a mathematical basis for a commonly used carpenter’s rule of thumb to determine the diagonal length needed to square up a nearly rectangular wooden frame from its diagonal measurements.

Chess: A Cover Up
Eric K. Henderson, Douglas M. Campbell, Douglas Cook, and Erik Tennant
146-158
The game of chess provides a rich source of challenging combinatorial problems. We explore the application of computing techniques to decide one such problem, and ask the deeper question of whether these methods can substitute for traditional proofs by logic.

POEM

Stopping by Euclid’s Proof of the Infinitude of Primes (with apologies to Robert Frost)
Brian D. Beasley
171

 

Whose proof this is I think I know.
I can’t improve upon it, though.
You will not see me trying here
To offer up a better show.

Dummy View - NOT TO BE DELETED