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MONTHLY, May 2004

**Geometry of Chaotic and Stable Discussions**

by Donald G. Saari

dsaari@uci.edu

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

ahmed@math.ucdavis.edu

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

edwards@math.orst.edu, wardm@wou.edu

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**

**Notes**

**Dissecting Cuboids into Cuboids**

by Jeroen Spandaw

j.g.spandaw@xs4all.nl

**An Elementary Proof of Euler’s Formula for ζ(2***m*)

by Hirofumi Tsumura

tsumura@tmca.ac.jp

**A Short Proof of the Explicit Formula for Bernoulli Numbers**

by Grzegorz Rzadkowski

rzadkowski@uksw.edu.pl

**Moving a Rectangle around a CornerÂ—Geometrically **

by Raymond T. Boute

boute@intec.Ugent.be

**An Old Friend Revisited**

by Christoph Leuenberger

christoph.leuenberger@eif.ch

**Reviews**

**A Course in Approximation Theory.**

by Ward Cheney and Will Light

Reviewed by G.E. Fasshauer

fasshauer@math.iit.edu

**Telegraphic Reviews**

**Editor’s Endnotes**