### ARTICLES

**Surprising Dynamics From a Simple Model**

James A Walsh

327-339

We present a simple model arising in both queueing theory and economic dynamics. Model behavior is determined via iteration of a two-parameter family of bimodal (two-hump) maps which take an interval to itself. Interestingly, both forward and reverse bifurcations occur as a parameter is varied. The model, which is appropriate for inclusion in a dynamical systems or modeling course, exhibits chaotic behavior for a range of parameter values.

**Proof Without Words: Surprising Property of Hyperbolas**

Tom A. Apostol and Mamikon A. Mnatsakanian

339-340

A proof without words is given of the following new property of hyperbolas: For each member of the family of hyperbolas with the same pair of vertices, every circle tangent to both branches intersects each asymptote along a chord of constant length, regardless of the location of the circle and the angle between the asymptotes. This constant length is the distance between vertices. The proof is revealed as a cross section of a three dimensional hyperboloid of revolution.

**Spectral Analysis of the Supreme Court**

Brian L. Lawson, Michael E. Orrison, and David T. Uminsky

340-346

We describe how basic voting data and simple ideas from linear algebra can be used to pinpoint voting coalitions in small voting bodies. The approach we use is an example of (generalized) spectral analysis, which is a nonmodel-based method for doing exploratory data analysis. As a proof of concept, we show how important coalitions of justices on the United States Supreme Court may be identified using only information about how often the justices split into various majority/minority configurations.

**Counting Train Track Layouts**

James D. Beaman, Erin J. Beyerstedt, and Mark R. Snavely

347-359

Train track layouts with switches not only provide children with hours of amusement, they provide us with a number of interesting mathematical problems. How many different layouts can be made from a set of track with two switches? What do we mean by different? We develop two different but reasonable definitions for equivalence of train track layouts, and in each case, count the number of equivalence classes. We use such tools as two-term recurrences, directed graphs, and eigenvalues of incidence matrices to learn about the fascinating mathematics of train track layouts.

**Proof Without Words: The Number of Unordered Selections with Repetitions **

Derek Christie

359-360

A short visual proof of the formula for selections with repetitions.

**Dials and Levers and Glyphs**

Jessica K. Sklar

360-367

Many puzzles in computer adventure games are obviously mathematical; others are math problems in disguise. In this paper, we use linear and abstract algebra to solve disguised mathematical puzzles in the computer games *Myst* and *Timelapse*.

### NOTES

**Disentangling Topological Puzzles by Using Knot Theory **

Matthew Horak

368-375

Recreational and professional mathematicians have found inspiration and enjoyment in all kinds of puzzles throughout the ages. The solutions of many types of puzzles have been thoroughly analyzed. Examples of these types include sequential movement puzzles, such as the Towers of Hanoi, and dissection/reassembly puzzles such as the Soma Cube. On the other hand, disentanglement puzzles in which two or more apparently linked pieces are to be unlinked, prove more difficult to analyze mathematically because the physical sizes and shapes of the puzzle pieces determine the solutions. By focusing on the disentanglement puzzle, "Quattro," we show how the ideas of knot theory can be used to begin to analyze disentanglement puzzles by ruling out "dead ends" and guiding the solver down a correct path towards a solution.

**Trigonometric Series via Laplace Transforms**

Costas J. Efthimiou

376-379

Previously we presented a method that uses the Laplace transform to allow one to find exact values for a large class of convergent series of rational terms. Lesko and Smith extended this method to include additional infinite series. In this note we apply the technique to trigonometric series.

**Two Methods for Approximating π**

Chien-Lih Hwang

380-385

Manipulating the Maclaurin series for π and truncating at various orders leads to various approximations for a number *x* in terms of the sine and cosine of *x*, and this leads to many ways to approximate π. In a second section the method is extended to develop a family of more accurate approximations; this time we approximate π in terms of a known approximation *P* and the sine of *P*. The approximations are visually similar to Fourier series; Fourier series are more accurate in a square-mean sense, but ours are better for particular approximations. This article is dedicated to the memory of Professor Dario Castellanos.

**A π-less Buffon's Needle Problem**

David Richeson

385-389

In 1733 the Compte de Buffon showed that if a needle of length L is dropped on a wood floor with boards of width D, then the probability that the need will cross a seam is 2*L*/*Dπ*. In this note we show that if the same needle is pushed into a clear rubber ball and the ball is dropped on the floor, then the probability that the needle lies above a seam is *L/2D*. Unlike the classical case π makes no appearance, and when *L=D* , the probability is the same as a coin toss.

**Conditions Equivalent to the Existence of Odd Perfect Numbers**

Judy A. Holdener

389-391

In 2000 P. A. Weiner proved that if the abundancy of an integer *n* is 5/3, then 5*n* is an odd perfect number. In 2003 R. F. Ryan generalized Weiner's result by proving that if there exists a positive integer *n* and an odd positive integer *m* such that 2*m*-1 is a prime not dividing *n* and *(I)n =2m-1/m *, then *n(2m-1)* is an odd perfect number. In this note we generalize both of these results, providing conditions that are equivalent to the existence of odd perfect numbers.

**The Associativity of the Symmetric Difference**

Majid Hosseini

391-392

We provide a short proof of the associativity of the symmetric difference of sets.