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 mth power of 1 + x + x2 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
Warren Page, Editor
A Triple Angle Formula for Tangent
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 + 10n 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
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
Warren Page, editor