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Mathematics Magazine - April 2002

ARTICLES

The Many Names of (7,3,1)
Ezra Brown
This is a story about a single object, living in the world of discrete mathematics, that makes many connections. It is a difference set, a block design, a Steiner triple system, a finite projective plane, a complete set of orthogonal Latin squares, a doubly regular tournament, a skew-Hadamard matrix, and a graph consisting of seven mutually adjacent hexagons drawn on the torus. We investigate these connections and get to know this object, commonly known as (7,3,1), and its many other names quite well.

A Stirling Encounter with Harmonic Numbers
Arthur T. Benjamin, Gregory O. Preston, and Jennifer J. Quinn
The harmonic number Hn = 1 + 1/2 + 1/3 + . . . 1/n can be written as a fraction pn/n!. We present several proofs that pn is the Stirling number of the first kind that counts the number of permutations of n+1 elements with exactly 2 cycles. This relationship allows us to explain many interesting harmonic and Stirling identities combinatorially.

Plotting the Escape - An Animation of Parabolic Bifurcations in the Mandelbrot Set
Anne M. Burns
Is there a way to generalize the period-doubling transition from order to chaos that takes place as a parameter is varied in a real-valued quadratic function? Varying the parameter in a complex-valued quadratic function along different paths in the complex plane has new ramifications.

NOTES

Four person Envy free Chore Division
Elisha Peterson and Francis Edward Su
How to divide a cake fairly is a well known problem, but less well known is the dual problem of chore division: is there a procedure for dividing an undesirable set among several players in such a way that each feels she received the smallest piece? In this paper, we develop the first bounded procedure for envy free chore division among four players.

The Consecutive Integer Game
David M. Clark
The Consecutive Integer Game presents a problem, which can be understood and solved using nothing other than an ability to count and to reason together with a good bit of patience. Two players compete in a surreal game whose analysis appears at first glance to be impossible and at second glance to be almost trivial. Ultimately a correct analysis of the game challenges our ability to extricate a mathematical model from the game itself, as the two prove to be fiercely entangled.

Asymptotic Symmetry of Polynomials
Paul Deiermann and Richard D. Mabry
A typical classroom investigation of polynomials includes the use of technology to plot the functions at various scales. In particular, zooming out sufficiently far reveals certain asymptotic properties, such as a polynomial being asymptotic to its leading term. This note investigates a property revealed at certain “intermediate” scales, a property we call “asymptotic symmetry.” A definition for this property is formulated which is applied to polynomials of positive, even degree, but which can be applied to a much more general class of functions.

Duality and Symmetry in the Hypergeometric Distribution
James Jantosciak and William Barnier
A jury of 7 men and 5 women is hung, 9 to 3. What is the probability that 5 jurors among the majority are men? This problem can be solved using the hypergeometric probability distribution. Using this distribution, other problems such as selection by lot and lottery games are solved after designating successes and choosing a sample from a given population. For example, in a lottery game the player designates the successes and the house randomly chooses the sample. For the jury problem, and many similar examples, distinguishing successes from sample seems arbitrary and contrived. After pointing out the duality regarding the successes and the sample in the hypergeometric distribution, we develop a symmetric form of this probability distribution, which yields a natural solution to the jury problem.

Perfect Cyclic Quadrilaterals
Raymond A. Beauregard and Konstantine D. Zelator
Are there any quadrilaterals with integer sides having perimeter P equal to area A, or more generally with ratio of perimeter to area a fixed value, say, P/A = k? This paper discusses the number N(k) of cyclic quadrilaterals with integer sides satisfying P = kA, where k is a given positive real number. These quadrilaterals are said to be k-perfect. It is shown that N(k) is finite with N(k) = 0 for k > 4. Furthermore when k is an integer we have N(1) = 7, N(3) = 2, and N(2) = N(4) = 1. These results are analogous to known results for (perfect) triangles.

CARTOONS

Giraffes on the Internet
Richard Samuelson

Playing with Fire
Richard Samuelson

PROOFS WITHOUT WORDS

Every Triangle Has Infinitely Many Inscribed Equilateral Triangles
Sidney H. Kung
We start with an equilateral triangle touching two sides of a triangle and dilate to produce a new equilateral triangle that touches all three.

The Area of a Salinon
Roger B. Nelsen
A proof without words for Archimedes’ result on the area of a salinon.

The Area of an Arbelos
Roger B. Nelsen
A Proof Without Words for Archimedes’ result on the area of an arbelos.

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