### January 2006 Contents

### ARTICLES

**Graeco-Latin Squares and a Mistaken Conjecture of Euler**

*Dominic Klyve and Leo Stemkoski*

2-15

A Graeco-Latin square of order *n* is an *n*×*n* array whose entries are the *n*^{2} ordered pairs of numbers from 1 to *n*, and in each row and each column the first elements of the ordered pairs are all different, as are the second elements. This article traces the history of the results that came out of work on a false conjecture of Euler, namely, that there are no Graeco-Latin squares of an order congruent to 2 modulo 4. The story covers more than two centuries, involves more than twenty researchers from five countries, and the proofs used

**Do Dogs Know Related Rates Rather than Optimization?**

*Pierre Perruchet and Jorge Gallego*

16-19

Although dogs seemingly follow the optimal path where they get to a ball thrown into the water, they certainly do not know the minimization function proposed in the calculus books. Trading the optimization problem for a related rates problem leads to a mathematically identical solution, which, it is argued here, is a more plausible model for the strategy of dogs

**Do Dogs Know Calculus of Variations?**

*Leonid A. Dickey*

20-23

As the title says, this article considers the dog-on-the-beach problem from the perspective of the calculus of variations, making connections with the brachistochrone problem and Snell's law.

**Archimedes Quadrature of the Parabola: A Mechanical View**

*Thomas J. Oster*

24-28

In his famous quadrature of the parabola, Archimedes found the area of the region bounded by a parabola and a chord. His method was to fill the region with infinitely many triangles each of whose area he could calculate. In his solution, he stated, without proof, three preliminary propositions about parabolas that were known in his time, but are not widely known today. It is the purpose of this short paper to prove the ideas presented in these obscure propositions so that a complete presentation of Archimedes' solution can be given. Our proofs are novel in that they are "mechanical"; that is, they use simple ideas from elementary physics rather than geometry. We use the fact that a particle, not acted on by friction, in motion near the surface of the earth, has a parabolic trajectory. The proofs give this way are very simple.

**David Gale: Restless Pioneer**

*Walter Meyer*

29-38

David Gale was one of the mathematicians responsible for the modern form of the theory of duality in linear programming and the associated proof of the minimax theorem in the theory of games. He is a member of the National Academy of Sciences and is Professor Emeritus of Mathematics and Operations Research at the University of California at Berkeley. He is cited by John Nash as being partly responsible for the simplicity of the proof of the theorem for which Nash was awarded the Nobel Prize. In addition to doing research in pure geometry, game theory, and mathematical economics, Gale wrote a recreational math column in the Mathematical Intelligencer for some years. In this interview, he shares some thoughts on his education and career, his research in game theory and mathematical economics, and his relationship with Nash.

### Fallacies, Flaws, and Flimflam

*Ed Barbeau, editor*

39-42

### Classroom Capsules

*Michael Kinyon, editor*

43-53

**Pizza Combinatorics Revisited**

*Griffin Weber and Glen Weber*

43-44

A pizza company advertises that the number of different selections of four of their pizzas in a box is more than six million. This note shows that the number is actually more than twenty billion (20,695,218,670 to be exact).
**Using Random Tilings to Derive a Fibonacci Sequence**

*Keith Neu and Paul Deiermann*

44-47

Some congruences are proved using a technique that Benjamin and Quinn developed to prove some Fibonacci identities.

**The Sample Correlation Coefficient from a Linear Algebra Perspective**

*C. Ray Rosentrater*

47-50

This note brings statistics and linear algebra together using inner products.

**Pythagoras By the Cross Ratio**

*Rebecca M. Conley and John H. Jaroma*

50-52

The Pythagorean theorem is given that uses the cross ratio of complex numbers.

**Parity and Primality of Catalan Numbers**

*Thomas Kosny and Mohammad Salmassi*

52-53 The following two facts about Catalan numbers are established: (a) The *n*th Catalan number is odd if and only if *n* is of the form 2^{m} -1, that is, *n* is a Mersenne number. (b) The only prime Catalan numbers are *C*_{2} and *C*_{3}.

**Student Research Project**

*Brigitte Servatuis, editor*

**Integer Points on a Hyperboloid of One Sheet**

*Margaret Beattie and Chester Weatherby*

54-58

This research project involves the study of the equation *x*^{2} + *y*^{2} = *z*^{2} + *k* for various integer values of *k*n. A procedure for generating a tree of all the primitive integer solution, if there is one. This is done for the case *k* = 2.

### Problems and Solutions

### Media Highlights