CMJ, November 2010
ARTICLES
The Tower and Glass Marbles Problem
By: Rick Denman, David Hailey, and Mike Rothenberg
denman@southwestern.edu, dhailey@sstz.org, mikerothenberg@gmail.com
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 an Invalid Reduction of Fractions
By: Michael N. Fried and Mayer Goldberg
mfried@bgu.ac.il, gmayer@little-lisper.org
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
By: Waclaw Szymanski and Shiv Gupta
WSzymanski@wcupa.edu, SGupta@wcupa.edu
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.
Finding Rational Parametric Curves of Relative Degree One or Two
By: Dave Boyles
boyles@uwplatt.edu
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
By: Judy Sorenson and Elyn K. Rykken
sorensj1@augsburg.edu, elrykken@muhlenberg.edu
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.
The Rascal Triangle
By: Eddy Liu, Angus Tulloch, and Alif Anggoro
cheeseplayer@hotmail.com
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 PROJECT
Graphs and Zero-Divisors
By: Mike Axtell and J. Stickles
maxtell@stthomas.edu, jstickles@millikin.edu
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.
CLASSROOM CAPSULES
On a Perplexing Polynomial Puzzle
By: Bettina Richmond
tom.richmond@wku.edu
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.
Sum-Difference Numbers
By: Yixun Shi
yshi@bloomu.edu
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
By: Eric Mazzone and Bruce Piper
efmazzone@gmail.com, piperb@rpi.edu
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
BOOK REVIEW
The Unimaginable Mathematics of Borges' Library of Babel by William
Goldbloom Bloch, reviewed by Dan King
MEDIA HIGHLIGHTS
REFEREES IN 2010
ADDITIONS, CORRECTIONS, EMENDATIONS AND
REVISIONS