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May 2003 Contents

**Do Dogs Know Calculus?**

Timothy J. Pennings

In all calculus books appears the problem of minimizing the time to get to a point on the other side of a river, running part of the way and swimming the rest. Isomorphic to this, if you are a dog, is the problem of minimizing the time to get to a ball that your master has thrown into a lake. The author has made measurements of how his dog retrieves the ball and finds that he indeed seems to choose the optimal path.

**Coin ToGa: A Coin-Tossing Game**

Osvaldo Marrero and Paul C. Pasles

Take a revolver with exactly one bullet in it, spin the chamber, put it to your head and pull the trigger. You either win or lose. In either case, the revolver passes to the next player, who reloads (if necessary) and plays the same game. Repeat around the circle of *n* players until only one remains. What is the probability that it's you, or any other player? To avoid unnecessary grisliness, the authors consider coins instead of guns and probabilities of failure other than 1/6.

**A Class of Exponential Matrices**

M. A. Khan

Even if you have never felt the need for a variable matrix M(x) with the property that *M*(*x* + *y*) = *M*(*x*)*M*(*y*), you might find it nice to have one, just in case. Here is how to find one, or as many as you like.

**Clarifying Compositions with Cobwebs**

Nial Neger and Michael Frame

The composition of two functions arises naturally all over the place, so it would be nice to have a way of seeing it graphically. The authors provide a method. It would also be nice to have a way of smelling the composition, since the more senses involved in mathematics the better, but the authors do not go into that.

**Tangent Planes of a Quadratic Function**

Panagiotis T. Krasopoulos

Two tangent lines to a vertical parabola intersect at a point halfway between the points of tangency in the horizontal direction. This property generalizes to three dimensions.

**Taking the Sting out of Wasp Nests: A Dialogue on Modeling in Mathematical Biology**

Jennifer C. Klein and Thomas Q. Sibley

Wasps in hot climates build elongated nests, while in colder areas they tend to be circular. Mathematics cannot explain that, but there are questions about numbers of cells that can be answered.

**Variations on a Theme from Pascal's Triangle**

Thomas J. Osler

Pascal's triangle, ever fruitful! Here we have everything from the coefficients of powers *x* in the *m*th power of 1 + *x* + *x*^{2} to the number of ways (57) of writing 13 as a sum of positive integers that are less than 6.

**Fallacies, Flaws, and Flimflam**

Edward Barbeau, Editor

**Classroom Capsules**

Warren Page, Editor

**A Triple Angle Formula for Tangent**

Yûichirû Kakihara

A formula for tan(3Θ) with consequences such as that 153 (a triangular number) is the sum of the squares of the tangents of 5 + 10*n* degrees, *n* = 0, 1, … , 8.

**The Murder Mystery Method for Determining Whether a Vector Field is Conservative**

Tevian Dray and Corinne A. Manogue

When is a vector field the gradient of a potential function? To find out, we can interrogate - I mean *integrate* - the witnesses.

**Visual Proof of Two Integrals**

Thomas J. Osler

Complicated integrals made simple.

**Area Realtions on the Skewed Chessboard**

Larry Hoehn

Take a chessboard and twist it, violently if that is the way you feel. What happens to the area of the black and white squares? They change, of course, and here are some nice relations.

**On the Monotonicity of (1 + 1/***n*)^{n} and (1 + 1/*n*)^{n+1}

Peter R. Mercer

How to show that the left-hand function increases with n while the right-hand one decreases, and some consequences.

**Problems and Solutions**

Benjamin G. Klein, Irl C. Bivnes, and L. R. King, editors

**Media Highlights**

Warren Page, editor