The Number Pad Game
Alex Fink and Richard Guy
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
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
The geometric problem of finding the number of normals to the parabola y = x2 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 Neilís (semi-cubical) parabola.
Surprising connections Between Partitions and Divisors
Thomas J. Osler, Abdulkadir Hassen, and Tirupathi R. Chandrupatla
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
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
A Direct Proof that Row Rank Equals Column Rank
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
We investigate the differential equation yn + 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
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. College Mathematics Journal Homepage
Read the September issue of the College Mathematics Journal online. (This requires MAA Membership.)