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Vol. 38, No. 4, pp. 246-327

ARTICLES

**Epidemic Models for SARS and Measles**

Edward Rozema

246-259

Recent events have led to an increased interest in emerging infectious diseases. This article applies various deterministic models to the SARS epidemic of 2003 and a measles outbreak in the Netherlands in 1999-2000. We take a historical approach beginning with the well-known logistic curve and a lesser-known extension popularized by Pearl and Reed in the 1920s. We then considered Richards' generalization in 1959 using a shape parameter. This method remains popular today in the biological literature but is often neglected in mathematical texts. The paper is accessible to students finishing the second year of calculus.

**The Number Pad Game**

Alex Fink and Richard Guy

260-264

In this article, we explore the game based on the following problem from the 1st Mathematical Olympiad of Central America and the Caribbean, held in Costa Rica in 1999: Player *A* turns on the calculator, presses a digit key and then presses the + key. A second later, player *B* presses a digit key in the same row or column of the last digit key pressed by *A*, then presses the + key. The game proceeds with the two players taking turns alternately. The first player who reaches a sum greater than 30 loses. Which player has a winning strategy? Describe the strategy.

**Equiangular Surfaces, Self-Similar Surfaces, and the Geometry of Seashells**

Khristo N. Boyadzhiev

265-271

Logarithmic spirals are among the most fascinating curves in the plane, being the only curves that are equiangular, and the only ones that are self-similar. In this article, we show that in three dimensions, these two properties are independent. Although there are surfaces that have both properties, there are some that are equiangular, but not self-similar; and some that are self-similar, but not equiangular.

**The Normals to a Parabola and the Real Roots of a Cubic**

Majinder S. Bains and J. B. Thoo

272-277

The geometric problem of finding the number of normals to the parabola *y* = *x*^{2} through a given point is equivalent to the algebraic problem of finding the number of distinct real roots of a cubic equation. Apollonius solved the former problem, and Cardano gave a solution to the latter. The two problems are bridged by Neil's (semi-cubical) parabola.

**Surprising connections Between Partitions and Divisors**

Thomas J. Osler, Abdulkadir Hassen, and Tirupathi R. Chandrupatla

277-287

The sum of the divisors of a positive integer is one of the most interesting concepts in multiplicative number theory, while the number of ways of expressing a number as a sum is a primary topic in additive number theory. In this article, we describe some of the surprising connections between and similarities of these two concepts.

**Transcendental Functions and Initial Value Problems: A Different Approach to Calculus II**

Byungchul Cha

288-296

We present an approach of defining certain transcendental functions as solutions to initial value problems or systems of such problems. This material is suitable for use in a second-semester one-variable calculus course.

**Fallacies, Flaws, and Flimflam**

Ed Barbeau, editor

218-220

**CAPSULES**

**A Direct Proof that Row Rank Equals Column Rank**

Nicolas Loehr

300-301

We give a direct proof of the result in the title that gives more geometric intuition for the result than other proofs do. The key is seeing what happens to the row rank and column rank when a row that is dependent on the other rows is deleted.

**An Elementary Proof of an Oscillation Theorem for Differential Equations**

Robert Gethner

301-303

We investigate the differential equation *y*^{n} + *q(t)y* = 0 and what can be learned about the zeros of its solutions using graphical arguments from elementary calculus.

**STUDENT RESEARCH PROJECT**

**From Cyclic Sums to Projective Planes**

Roger Zarnowski

304-308

Given a cycle of numbers (like beads on a necklace), if we add some consecutive set of numbers, the result is called a cyclic sum. In this project, we present some results and problems on this interesting topic.