### September 2005 Contents

### ARTICLES

**Jan Hudde and the Quotient Rule before Newton and Leibniz**

*Daniel J. Curtin*

262-272

This article describes some of the work of Jan Hudde who anticipated some results of calculus. Prior to a career as a Burgomaster of Amsterdam, Hudde engaged in mathematics. His method of finding maxima and minima is especially interesting.

**Curious consequences of a Miscopied Quadratic**

*Jeffrey L. Poet and Donald L. Vestal, Jr.*

273-277

The starting point of this article is a search for pairs of quadratic polynomials *x*^{2} + *bx* ± *c* with the property that they both factor over the integers. The search leads quickly to some number theory in the form of primitive Pythagorean triples, and this paper develops the connection between these two topics.

**The Platonic Solids from their Rotation Groups**

*Larry Grovei*

278-283

The five Platonic solids are constructed (as graphs) from their rotational symmetry groups. The constructions are based on an idea of Bertram Kostant and are quite simple; conjugacy classes in the group are the vertices of the graphs and products determine adjacency.

**On Primes, Density Measures, and Statistical Independence**

*Yung-Pin Chen*

284-288

A result known as the Borel-Cantelli lemma is about probabilities of sequences of events. This article presents an example in which it appears that the hypotheses of the lemma are satisfied but the conclusion is not. The explanation of why not combines elements of probability theory, number theory, and analysis.

**Visibles Revisited**

*Mark Bridger and Andrei Zelevinsky*

289-300

Within the set of points in the plane with integer coordinates, one point is said to be visible from another if no other point in the set lies between them. This study of visibility draws in topics from a wide variety of mathematical areas, including geometry, number theory, probability, and combinatorics.

**"Mathematical Games" and Beyond: Part II of an Interview with Martin Gardner**

*Don Albers*

301-214

This is a continuation of an interview (begun in the May issue) with Martin Gardner, in which he discusses his "Mathematical Games" columns in *Scientific American *and some of his other writings. He goes on to share his views on a variety of aspects of science and mathematics.

### Fallacies, Flaws, and Flimflam

*Ed Barbeau, editor*

315-317

### Classroom Capsules

*Michael Kinyon, editor*

318-329

**Self-Integrating Polynomials**

*Jeffrey A. Graham*

318-320

The polynomials studied here have the property that . The main results concern their density on a given interval.
**A Variant of the Patition Function**

*John F. Loase, David Lansing, Carrie Hryczaniuk, and Jamie Cahoon*

320-321

This note concerns the number of ways that a positive integer can be expressed as a sum of primes. Some elementary results on this variant of the partition function and some questions for further study are raised.

**Exactly When is (***a* + *b*)^{n} ≡ *a*^{n} + *b*^{n}(mod*n*)?

*Pratibha Ghatage and Brian Scott*

322

Although primes satisfy the congruence in the title, they are not the only numbers that do. It is shown that a necessary and sufficient condition is that Fermat's little theorem hold for *n*.

**How Do You Slice the Bread?**

*James Colin Hill, Gail Nord, Eric Malm, and John Niord*

323-326

The bread in question is like a standard loaf with a rounded top, and the cut to be made is to be diagonal, thereby giving two pieces that are roughly triangular in shape. Two problems are discussed: how to make the cut so that the two parts have an equal amount of bread, and how to do it so the amounts of crust are the same.

**Limits of Functions of Two Variables**

*Ollie Nanyes*

326-329

An example is given of a function that is discontinuous at the origin, but for which the limit along every curve of the form *y* = *x*^{n} is 0. The function is a modification of the standard examples of this type given in textbooks, but has this much stronger path property.

### Problems and Solutions

### Media Highlights