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American Mathematical Monthly -May 2004

MONTHLY, May 2004

Geometry of Chaotic and Stable Discussions
by Donald G. Saari
Why is it that, no matter how polished and complete a proposal may be, when it is presented to a group for approval, there always seems to be a majority who wants to "improve it?" The mathematical modeling provides an immediate and intuitively complete answer for this behavior, and it probably will cause the reader to worry about recent personal decisions and discussions. To address some natural questions associated with the model, we need to use a variety of interesting mathematic approaches that range from elementary geometry to game theory to orbits of symmetry groups to singularity theory.


How Many Squares Are There Mr. Franklin?: Constructing and Enumerating Franklin Squares
by Maya Mohsin Ahmed
Benjamin Franklin constructed three famous squares that have several interesting properties: the entries of every row and column add to a common sum called the magic sum, the entries of the bent diagonals add to the magic sum, and so forth. These squares have fascinated both expert and amateur mathematicians for centuries and such squares came to be known as Franklin squares. People have tried to understand the method Franklin used to construct his squares and many theories have been developed along these lines. In this paper, we present a new method of constructing the three famous squares, and all other Franklin squares. We also provide formulas for counting the number of Franklin squares with a given magic sum. Our approach uses a very general algebraic-geometric method that exploits the notion of Hilbert bases of polyhedral cones. The method described in this paper is both powerful and useful, and constructing and enumerating Franklin squares is only an example of how it can be applied.


Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions
by Barbara S. Edwards and Michael B. Ward,
It is no surprise that students struggle with some mathematical definitions. Most professors assume the content of the definitions is the cause of all the difficulty. Recent research in mathematics education suggests a more fundamental problem: students do not understand the special nature and role of definitions in mathematics. This paper contains a discussion of that research and lists some classroom activities that might help to address students' misunderstanding.


Problems and Solutions


Dissecting Cuboids into Cuboids
by Jeroen Spandaw

An Elementary Proof of Euler’s Formula for ζ(2m)
by Hirofumi Tsumura

A Short Proof of the Explicit Formula for Bernoulli Numbers
by Grzegorz Rzadkowski

Moving a Rectangle around a Corner—Geometrically
by Raymond T. Boute

An Old Friend Revisited
by Christoph Leuenberger


A Course in Approximation Theory.
by Ward Cheney and Will Light
Reviewed by G.E. Fasshauer

Telegraphic Reviews

Editor’s Endnotes