### ARTICLES

**Znam’s Problem**

Lawrence Brenton and Ana Teodorescu Vasiliu

Problems in unit fraction arithmetic originating in dynastic Egypt continue to intrigue modern day mathematicians. As formulated by Znam in 1972, one such problem in the theory of systems of congruences has turned out to have surprising applications both within number theory and in geometry.

**The Josephus Problem: Once More Around**

Peter Schumer

If a group of people numbered 1 to 40 is placed around a circle and every seventh person remaining is eliminated in succession, where would you stand to be the last chosen? This was the problem facing Josephus Flavius nearly 2000 years ago. Variations on the original problem with different numbers of people, n, and different step sizes, q, are collectively called the Josephus problem. In this paper we consider for given n how to choose q so that all the odd-numbered people are removed first. Other variants are also discussed including the following: given n and a particular position p, how to choose q so that position p is the last to be chosen.

**A Brief History of Factoring and Primality Testing B.C. (Before Computers)**

Richard Anthony Mollin

The notions of factoring and primality testing are rooted in antiquity, but have become increasingly important in our information-based society. The reason is that they have produced techniques used in the modern transmission of data. Thus, Cryptography Â– the study of sending messages in secret Â– has given high profile to number theory in general and factoring/primality testing in particular. What is often lost in the modern-day shuffle of information are the pioneers who ushered in the computer age and their ideas, some of which are still used as the under-pinnings of powerful algorithms for factoring and primality testing today. This article pays homage to these individuals, and their work in these areas B.C. Â– Before Computers.

### NOTES

**The Perfect Shape for a Rotating Rigid Body**

Frank Morgan

What shape minimizes the gravitational plus kinetic energy of a rotating rigid body of given volume and angular momentum? It's probably not what you think.

**One Sequence; Many Interesting Ideas in Analysis**

Russell A. Gordon, Charles Kicey, and Sudhir Goel

The sequence *x_n*, where *x_n* = 1/(*n*+1) + 1/(*n*+2) + . . . + 1/(2*n*), provides a nice illustration of the fact that bounded monotone sequences converge. It is elementary to prove that this sequence is increasing and bounded above by 1. Somewhat surprisingly, it turns out that this sequence also illustrates Riemann sums and rearrangements of conditionally convergent series. Furthermore, the sequence can be generalized without losing any of its interesting features. In particular, the generalized sequence makes it possible to find exact values of sums for an entire class of rearrangements of a conditionally convergent series.

**When Does a Sum of Positive Integers Equal Their Product?**

Michael W. Ecker

In 1997, an amateur math enthusiast sent Dr. Mike Ecker a neat little formula about tangents and triangles. To wit: The sum of the tangents equals the product of the tangents. Can these three values all be integers? More generally, for any positive integer *n* > 1, we can always find *n* positive integers whose product and sum are the same. Sometimes there is just one such solution for a given *n*. Usually there are more. What is the nature of such solutions, and what properties do these *n*-tuples enjoy?

**The Scarcity of Regular Polygons on the Integer Lattice**

Daniel J. O’Loughlin

There is an inherent irrationality to all regular polygons (with the exception of squares). An investigation of formulae for the area of polygons shows that twice the area of any polygon in the coordinate plane whose vertices all lie on integer lattice points must be an integer. In addition, any high school student can see that the area of a regular polygon must involve the central angle of the polygon. In this article we show that these two facts together rule out any regular polygons on the integer lattice, except (of course) squares. The result is well known, but the proof is new, and the approach via topics from vector Calculus (including Green's Theorem) makes a discussion of this interesting result appropriate for many undergraduate courses. Moreover, all of the proofs can be understood by an advanced undergraduate.

**Medical Tests and Convergence**

Stephen H. Friedberg

We use elementary real analysis and difference equations to examine a scheme to help a patient learn if she has a disease without actually getting bad news. The scheme involves her doctor as well as other doctors reporting only good news depending on the results of a sequence of coin tosses.

**Uniquely Determined Unknowns in Systems in Linear Equations**

Kenneth Hardy, Blair K. Spearman, and Kenneth S. Williams

Sometimes a system of linear equation is such that some of the unknowns are uniquely determined, while others are not. In this Note the following three questions are answered:

Question 1. *What is a necessary and sufficient condition for an unknown to be uniquely determined by a consistent linear system?*

Question 2. *How many of the unknowns are uniquely determined by a linear system?*

Question 3. *If an unknown is uniquely determined by a linear system, is there an explicit formula for it?*

**Counterintuitive Aspects of Plane Curvature**

Russell A. Gordon and Colin Ferguson

A study of the curvature of a plane curve of the form y = f(x) leads to some counter-intuitive results. For instance, the curvature of a function whose graph is concave up may not approach 0 as x approaches infinity, and the curvature of a function with a vertical asymptote at x = c may not approach 0 as x approaches c. In addition, scaling a function affects its curvature qualitatively as well as quantitatively.

### POEM

**Doing Math** Donna L. Davis

A poem

### PROOFS WITHOUT WORDS

**The Pigeonhole Principle**

Ran Libeskind-Hadas

Seven pigeons in six boxes.

**A Sum and Product of Three Tangents**

Roger B. Nelsen

A visual demonstration is given for the observation that if A, B, and C denote angles in an acute triangle, then tan A + tan B + tan C = tanAtanBtanC.

### LETTERS TO THE EDITOR

### Letter to the Editor

Stan Wagon

The article by Dresden in the October 2001 issue presents a solution to the problem of the periodicity of the sequence of rightmost nonzero digits of n!. This problem has appeared in several places. I first learned about it from

*Crux Mathematicorum* 19 (1993), 260-261 and 20 (1994) 45, where it was presented as an unused Olympiad problem. A discussion of the problem that focuses on the issue of a fast algorithm to get, say, the right-most nonzero digit of the factorial of a googol appears as Problem 90 in my book

*Which Way Did the Bicycle Go?* (with D. Velleman and J. Konhauser, MAA, 1996). And in Exercise 4.40 of

*Concrete Mathematics* by Graham, Knuth, and Patashnik (Addison-Wesley, 1989) one finds a formula (see also Problem 90.2 of my book) for this digit in base p, when p is prime.

**Letter to the Editor**

Stephen A. Fulling

A. A. Kosinski provided some interesting and valuable information about Gabriel Cramer in "Cramer's Rule is Due to Cramer" (this *Magazine*, October 2001). However, the note did not address the most controversial and bitter question: Is the man's name prounounced CRAY-mer or krah-MARE? The MARE party believes that the CRAY people are displaying typical Anglo-Saxon uncultured illiteracy. The CRAYs say that the MAREs are displaying pretentious ignorance, like those Z-phobes who insist on referring to the composers Boulez and Glazunov as boo-LAY and Glottsanoff. Each party accuses the other of confusing the eighteenth century Cramer with the twentieth century Swedish statistician H. Cramer, whose name (allegedly) is really pronounced the other way. Can Professor Kosinski, or anybody else authoritatively settle this issue? It is encrusted with about as much pseudoinformation as Noble's relationship to Mittag-Leffler.

### PROBLEMS

edited by Elgin H. Johnston

**REVIEWS**

edited by Paul J. Campbell

### NEWS AND LETTERS