April 2012 Contents
Our first two colorful articles show the benefits of playing with toys and games. Ben Coleman and Kevin Hartshorn describe the mathematics of the game of SET, and Peter Hilton and Jean Pedersen use a large Magz kit to find some remarkable polyhedra inside "Pascal's Tetrahedron." The Notes include a treatment of the cycles formed by Fibonacci sequences, when they are reduced mod p.—Walter Stromquist, Editor
Game, Set, MathBen Coleman and Kevin Hartshorn
We describe the card game SET, and discuss interesting mathematical properties of the game that illustrate ideas from group theory, linear algebra, discrete geometry, and computational complexity. We then suggest a criteria to identify when two card collections are similar to one another and appeal to Pólya’s Theorem to determine the number of structurally distinct collections. For example, we find there are 41,407 collections of 12 cards, the layout most commonly seen in gameplay.
Mathematics, Models, and Magz, Part I: Patterns in Pascal’s Triangle and TetrahedronPeter Hilton and Jean Pedersen
Illustrations by Sylvie Donmoyer and Photographs by Chris Pedersen
This paper describes how the authors used a set of magnetic toys to discover analogues in 3 dimensions of well-known theorems about binomial coefficients. In particular, they looked at the Star of David theorem involving the six nearest neighbors to a binomial coefficient . If one labels the vertices of the bounding hexagon with the numbers 1, 2, 3, 4, 5, 6, consecutively (in either direction), then the product of the coefficients with even labels is the same as the product as the coefficients with odd labels. Furthermore the two figures formed by connecting the odd and even vertices are both equilateral triangles arranged so that a sixty-degree rotation exchanges the triangles. There is a generalized Star of David theorem concerning a semi-regular hexagon with similar results. The paper describes analogous results for trinomial coefficients involving, sometimes but not always, tetrahedral instead of triangles.
Picturing IrrationalitySteven J. Miller and David Montague
In the 1950s, Tennenbaum gave a wonderful geometric proof of the irrationality of the square root of two. We show how to generalize his arguments to prove the irrationality of other numbers, and invite the reader to explore how far these arguments can go.
Gauss’s Lemma and the Irrationality of Roots, RevisitedDavid Gilat
An idea of T. Estermann (1975) for demonstrating the irrationality of is extended to obtain a conceptually simple proof of Gauss’s Lemma, according to which real roots of monic polynomials with integer coefficients are either integers or irrational. The standard proof of the lemma is also reviewed.
Minimizing Areas and Volumes and a Generalized AM–GM InequalityWalden Freedman
Solving optimization problems via Lagrange Multipliers leads us to a generalized AM-GM inequality. We give several related optimization problems, suitable as projects for calculus students, with answers provided at the end.
Proof Without Words: Is IrrationalGrant Cairns
A Generalization of the Identity cosErik Packard and Markus Reitenbach
Using Euler’s Theorem and the Geometric Sum Formula, we prove trigonometric identities for alternating sums of sines and cosines.
A Class of Matrices with Zero DeterminantAndré L. Yandl and Carl Swenson
Let a1, a2, . . . , an , b1, b2, . . . , bn be real numbers and the nxn matrix C be defined with entries , where k is a positive integer. If n > k + 1, then det(C) = 0, and if n= k + 1, then det(C) is a product involving two Vandermonde determinants.
Splitting Fields and Periods of Fibonacci Sequences Modulo PrimesSanjai Gupta, Parousia Rockstroh, and Francis Edward Su
We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.
A Short Proof of the Chain Rule for Differentiable Mappings in RnRaymond Mortini
Based on the notion of M-differentiability, we present a short proof of the differentiability of composite functions in the finite dimensional setting.
The Surprising Predictability of Long RunsMark F. Schilling
When data arise from a situation that can be modeled as a collection of n independent Bernoulli trials with success probability p, a simple rule of thumb predicts the approximate length that the longest run of successes will have, often with remarkable accuracy. The distribution of this longest run is well approximated by an extreme value distribution. In some cases, we can practically guarantee the length that the longest run will have. Applications to coin and die tossing, roulette, state lotteries and the digits of π are given.