Mathematics Magazine - February 2007

ARTICLES

The River Crossing Game
3
The River Crossing Game is easy to describe: Each of two players distributes 11 boats (chips) at docks numbered from 2 to 12. More than one boat may be placed at a dock. Two dice are rolled and each player removes one boat from the dock whose number matches the dice sum. This continues until one player has removed all his boats and is declared the winner (or the game may be tied). Despite the game’s apparent simplicity, the underlying mathematics makes the choice of an optimal initial distribution for the boats (chips) anything but simple. This article analyzes optimal strategies for distributing one’s chips, and highlights several of the interesting and counter-intuitive results encountered along the way.

A Fresh Look at Peg Solitaire
George I. Bell
Page 16
Most readers are familiar with the one-person game of peg solitaire, usually played on a cross-shaped board with 33 holes. We consider this game on a board of arbitrary shape; our goal is to classify all peg solitaire boards with square symmetry. Along the way we uncover many interesting boards that the game can be played on, as well as a connection to the solitaire army problem. We prove that the shape of the standard 33-hole board is unique in a certain well-defined sense.

Counting Cyclic Binary Strings
Alice McLeod and William Moser
Page 29
Our objective is to illustrate by examples a particularly simple technique for enumerating some restricted subsets of Some of these examples have appeared in the literature, enumerated there by more complicated methods; others are new and lead in some cases to combinatorial identities.

A Sequence of Polynomials Related to the Evaluation of the Riemann Zeta Function
Javier Duoandikoetxea
Page 38
There are several ways to calculate the values of the Riemann zeta function at even integers. Some of them use moments of trigonometric functions. A modification of this approach leads in a natural way to a sequence of polynomials, and the value of the Riemann zeta function appears evaluating those polynomials at a given point. The sequence has some nice properties, which provide an inductive formula to get the desired values. The Fourier cosine series of the polynomials shows how they are related to Euler polynomials.

Proof Without Words: The Area of a Right Triangle
Roger B. Nelsen
Page 45
A proof without words of the following theorem: The area Kof a right triangle is equal to the product of the lengths of the segments of the hypotenuse determined by the point of tangency of the inscribed circle.

Brussels Sprouts and Cloves
Page 46
Cloves is a two person combinatorial game. It is a generalization of John Conway’s game of Brussels Sprouts, which is in turn a variation on the well-known game of Sprouts, invented by John Conway and Michael Paterson in the 1960’s. Like Brussels Sprouts, Cloves is trivial when played on the plane, or on any orientable surface. The game is much more amusing when played on nonorientable surfaces. We explain how to visualize games of cloves on nonorientable surfaces, just using pencil-and-paper, and we show how the required strategy is determined by the topology of the surface.

NOTES

Crackpot Angle Bisectors!
Robert J. MacG. Dawson
Page 59
Taxicab geometry, in which the distance between two points is the sum of their horizontal and vertical separations, is sometimes used in geometry courses as a Â“foil'Â” for Euclidean geometry, to illustrate the role of the SAS congruence and related properties. Here we examine this approach, and show in particular that bisecting an angle (suitably defined) in this geometry is as impossible as trisecting one in Eucldean geometry.

Quadratic Residues and the Frobenius Coin Problem
Michael Z. Spivey
Page 64
An odd prime p has (p-1)/2 quadratic residues mod p, and for relatively prime p and q there are (p-1)(q-1)/2 non-representable Frobenius numbers. Is there some relationship between quadratic residues and the Frobenius numbers that accounts for the presence of (p-1)/2 in both expressions? Yes, there is. We establish this relationship, and we show that, in fact, the non-representable Frobenius numbers are members of a wider class of subsets of {1, 2, ..., pq} that have the relationship.

Factoring Quartic Polynomials: A Lost Art
Gary Brookfield
Page 67
We revive an old and forgotten method for determining whether a quartic polynomial over the rations is reducible.

Butterflies in Quadrilaterals: A Comment on a Note
Sidney Kung
Page 70
We offer a slight generalization of the usual Butterfly Theorem using techniques from projective geometry, specifically the notion of an involution. Then we show that the Butterfly Theorem suggested by Sidney Kung in last year's issue of this MAGAZINE (pp. 314-316) really is a special case of this generalized Butterfly Theorem.

The Dottie Number
Samuel R. Kaplan
Page 73
The unique root of cos(x)=x is dubbed the Dottie number. This root is a simple non-trivial example of a universal attracting fixed point. The story of how the Dottie number got its name and mathematical concepts relating to this value can be used as teaching tools. Pedagogical examples are given for several courses ranging from Calculus I to Complex Analysis.

PROOF WITHOUT WORDS

Proof Without Words: Alternating Sums of Squares of Odd Numbers
Ángel Plaza
Page 74
We provide a visual solution for an alternating sum of odd squares.

NOTE

An Integral Domain Lacking unique Factorization into Ireducibles
Gerald Wildenberg
Page 75
We display an elegant, but not well known, example of a domain where factorization is not unique.

Proof Without Words: A Series Involving Harmonic Sums
Steven J. Kifowit
Page 83
Let Hn be the nth harmonic sum. This is a visual proof that the infinite series with terms converges to 2.