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Vol. 83, No. 3, pp. 162-239

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ARTICLES

**A Remarkable Euler Square before Euler **

by Ko-Wei Lih

pp. 163–167

Orthogonal Latin squares have been known to predate Euler in Europe. However, it is surprising that an Euler square of order nine was already in existence prior to Euler in the Orient. It appeared in a Korean mathematical treatise written by Choe SÅk-chÅng (1646–1715). Choe’s square has several nice properties that have never been fully appreciated before. In this paper, an analysis of Choe’s remarkable square is provided and a method of its construction is supplied.

**The Graph Menagerie: Abstract Algebra and the Mad Veterinarian**

By Gene Abrams and Jessica K. Sklar

pp. 168–179

In this paper, we explore Mad Veterinarian scenarios. We show how these recreational puzzles naturally give rise to semigroups (which are sometimes groups), and we point out a beautiful, striking connection between abstract algebra and graph theory. Linear algebra also plays a role in our analysis.

Supplment: "A proof of the Graph Semigroup Group Test in 'The Graph Menagerie'" (pdf).

**The Ergodic Theory Carnival**

By Julia Barnes and Lorelei Koss

pp. 180–190

The Birkhoff ergodic theorem, proved by George David Birkhoff in 1931, allows us to investigate the long-term behavior of certain dynamical systems. In this article, we explain what it means for a function to be ergodic, and we present Birkhoff’s theorem. We construct models of activities typically found at carnivals and compare and contrast them by analyzing their ergodic theory properties. We use these carnival models to show how Birkhoff’s ergodic theorem can be used to help a photographer set up her equipment to take pictures of all children on a carousel and to aid a magician in finding a lost jewel in a sticky mess of taffy.

**Which Surfaces of Revolution Core Like a Sphere?**

By Vincent Coll and Jeff Dodd

pp. 191–199

If a cylindrical drill bit bores through a solid sphere along an axis, removing a capsule from the sphere, the object that remains is called a spherical ring. A surprising property of the sphere that is often presented in calculus courses is that any two spherical rings whose cylindrical inner boundaries have the same height also have the same volume, regardless of the radii of the spheres from which they were cut. In this article, we pose and answer the question: to what extent does this property characterize the sphere among surfaces of revolution?

**Coloring and Counting on the Tower of Hanoi Graphs**

By Danielle Arett and Suzanne Dorée

pp. 200–209

The Tower of Hanoi graphs make up a beautifully intricate and highly symmetric family of graphs that show moves in the Tower of Hanoi puzzle played on three or more pegs. Although the size and order of these graphs grow exponentially large as a function of the number of pegs, *p*, and disks, *d* (there are *pd* vertices and even more edges), their chromatic number remains remarkably simple. The interplay between the puzzles and the graphs provides fertile ground for counts, alternative counts, and still more alternative counts.

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NOTES

**When Is ***n*2 a Sum of *k* Squares?

by Todd G. Will

pp. 210–213

This note shows that with the exception of (5x 2^{k})^{2}, an integer square can be written as sums of 2, 3, and 4 positive squares if and only if it has at least one prime factor congruent to 1 mod 4. Moreover such a square *n* can be written as a sum of *k* positive squares for all *k* from 1 to *n *-14. The question of when a non-square can be written as a sum of *k* positive squares is also examined.

**How Fast Will We Lose? **

by Ron Hirshon

pp. 213–218

In a version of gambler's ruin, players start with *x* and *y* dollars respectively, and flip coins for one dollar per flip until one player runs out of money. This is a random walk with two absorbing barriers. We consider the number of ways for the first player to lose on the *n*th flip, for *n=x, n*+ 2,... We use probabilistic arguments to construct generating functions for these quantities along with explicit methods for computing them. This paper builds on the paper by Hirshon and De Simone, *Mathematics Magazine* *81* (2008) 146–152.

**More Polynomial Root Squeezing**

by Christopher Frayer

pp. 218–221

Given a polynomial with all real roots, the Polynomial Root Dragging Theorem states that moving one or more roots of the polynomial to the right will cause every critical point to move to the right, or stay fixed. But what happens to the position of a critical point when roots are dragged in opposite directions? In this note we discuss the Polynomial Root Squeezing Theorem, which states that moving two roots, *ri* and *rj*, an equal distance toward each other without passing other roots, will cause each critical point to move toward (r_{i} + r _{j} )/2, or remain fixed.

**A Counterexample to Integration by Parts**

by Alexander Kheifets and James Propp

pp. 222–225

The authors exhibit two differentiable functions *f* and *g* for which the function and are not integrable, so that the integration by parts formula does not apply.