November 2010 Contents
The Tower and Glass Marbles Problem
Richard Denman, David Hailey, and Michael Rothenberg
The Catseye Marble company tests the strength of its marbles by dropping them from various levels of their office tower, to find the highest floor from which a marble will not break. We find the smallest number of drops required and from which floor each drop should be made. We also find out how these answers change if a restriction is placed on the number of marbles allocated for testing. Investigating this puzzle motivates algorithmic thinking, and leads to an interesting recursive solution.
A Pumping Lemma for Invalid Reductions of Fractions
Michael N. Fried and Mayer Goldberg
Children often incorrectly reduce fractions by canceling common digits instead of common factors. There are cases, however, in which this incorrect method leads to correct results. Instances, such as 16/64 and 19/95, are well-known. In this paper, we consider such “weird fractions” and show how examples of them can be multiplied ad infinitum and lead to interesting questions.
Cubic Polynomials with Rational Roots and Critical Points
Shiv K. Gupta and Waclaw Szymanski
If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.
Proof Without Words: New Pythagorean-like Theorems
Claudi Alsina and Roger B. Nelsen
Finding Rational Parametric Curves of Relative Degree One or Two
A plane algebraic curve, the complete set of solutions to a polynomial equation: f(x, y) = 0, can in many cases be drawn using parametric equations: x = x(t), y = y(t). Using algebra, attempting to parametrize by means of rational functions of t, one discovers quickly that it is not the degree of f but the “relative degree”, that describes how difficult the computations become. When the relative degree is one, the parametrization technique is well-known (and quite simple). When it is two, solutions can still be directly computed using the quadratic formula. Here, we demonstrate a general method for relative degree two, focusing on specific examples.
Sprinkler Bifurcations and Stability
Jody Sorensen and Elyn Rykken
After discussing common bifurcations of a one-parameter family of single variable functions, we introduce sprinkler bifurcations, in which any number of new fixed points emanate from a single point. Based on observations of these and other bifurcations, we then prove a number of general results about the stabilities of fixed points near a bifurcation point.
Proof Without Words: Double Sum for Sine and Cosine
The Rascal Triangle
Alif Anggoro, Eddy Liu, and Angus Tulloch
A number triangle, discovered using a recurrence formula similar to that of Pascal’s triangle, yields sequence A077028 from the Online Encyclopedia of Integer Sequences.
Student Research Projects
Graphs and Zero-Divisors
M. Axtell and J. Stickles
The last ten years have seen an explosion of research in the zero-divisor graphs of commutative rings—by professional mathematicians and undergraduates. The objective is to find algebraic information within the geometry of these graphs. This topic is approachable by anyone with one or two semesters of abstract algebra. This article gives the basic definitions and provides a list of possible projects that an interested undergraduate can investigate.
On a Perplexing Polynomial Puzzle
It seems rather surprising that any given polynomial p(x) with nonnegative integer coefficients can be determined by just the two values p(1) and p(a), where a is any integer greater than p(1). This result has become known as the “perplexing polynomial puzzle”. Here, we address the natural question of what might be required to determine a polynomial with integer coefficients, if the condition that the coefficients be nonnegative is removed.
Starting with an interesting number game sometimes used by school teachers to demonstrate the factorization of integers, sum-difference numbers are defined. A positive integer n is a sum-difference number if there exist positive integers x, y, w, z such that n = xy = wz and x – y = w + z. This paper characterizes all sum-difference numbers and student exercises and projects are also suggested.
Animating Nested Taylor Polynomials to Approximate a Function
Eric F. Mazzone and Bruce R. Piper
The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or incorporated into computer labs.
Problems and Solutions
The Unimaginable Mathematics of Borges' Library of Babel by William Goldbloom Bloch
Reviewed by Dan King
Referees in 2010Additions, Corrections, Emendations, and Revisions