Applying Burnside’s Lemma to a One-dimensional Escher Problem
TomaÂž Pisanski, Doris Schattschneider, and Brigitte Servatius
The graphic artist M. C. Escher counted the number of patterns resulting from repeatedly translating a square block of side length two, which has its four unit subsquares filled with copies of an asymmetric motif. We simplify this two-dimensional Escher problem from the plane to a strip, but at the same time generalize it to various block sizes.
Dropping Lowest Grades
Daniel Kane and Jonathan Kane
This paper considers the problem of identifying r grades to drop from a list of k grades in order to maximize the resulting average grade. Many examples are given showing that when the k grades are not all worth the same number of points, the optimal solution can be non-intuitive and tricky to identify. A brute-force algorithm for finding the best grades to drop would be to calculate the average grade for each subset of k - r grades of the k grades. This algorithm is inefficient and impractical to use. The authors present a very efficient algorithm which works well in practice.
Subsequent to publication we have learned of two related references:
Eppstein, David & Hirschberg, Daniel S. Choosing subsets with maximum weighted average. J. Algorithms 24 (1997).
McGrail, Robert W. & McGrail, Tracey Baldwin A grading dilemma or the abyss between sorting and the knapsack problem. Journal of Computer Sciences in Colleges (2004).
Compound Platonic Polyhedra in Origami
Place a cube in each vertex of a dodecahedron allowing adjacent cubes to intersect at one of their vertices. David Mitchell invented an origami model called "Twenty-Cubes"that approximates this polyhedral compound. We give a mathematically accurate description of Twenty-Cubes and consider the more general scenario.
The Bernoulli Trials 2004
Christopher G. Small and Ian Vanderburgh
"TRUE or FALSE? Undergraduate mathematics contests can be exciting. TRUE! (Obviously.) This note presents an unusual format for a mathematics contest, along with the problems, solutions and results from the 2004 edition."
The Comparison Test--Not Just for Nonnegative Series
Michele Longo and Vincenzo Valori
Is it possible to generalize the Comparison Test to generic real series? At first glance, many (certainly the authors) could argue something like "If it were true then it certainly would be written in some of the books standing on the shelves in my room." As a matter of fact, all the books on the authors’ shelves state the test only for nonnegative series. However, a straightforward generalization of this test is possible and useful to study the convergence of series for which standard tests do not apply as we show in this note.
From the Cauchy-Riemann Equations to the Fundamental Theorem of Algebra
Alan C. Lazer
We give a proof of the Fundamental Theorem of Algebra which is based on the Cauchy-Riemann equation.
Path Representation of a Free Throw Shooter’s Progress
Christopher L. Boucher
A recent Putnam exam asked for the expected number of successful shots from a free throw shooter whose success on a given shot depends in a simple way on the history of the process. By extending an idea from an earlier article in Mathematics Magazine, we are able to address this and the considerably more difficult question of the probability that the shooter ever finds herself having made k more shots than she has missed.
Fun, Fun, Functions
Brian D. Beasley
As mathematics teachers, we hope to give our students good vibrations about the various topics that we cover. For example, if we show that 409 is prime, then we want everyone to get excited rather than shut down. Wouldn’t it be nice if we could convey that sense of fun, fun, fun that we feel about math? God only knows, we do try.