Contents and Summaries for December 2010 Mathematics Magazine
For subscribers, read recent issues online (Requires MAA Membership)The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements
by Nicholas Pippenger
Using a classic puzzle concerning a cube of resistors as a point of departure, we use its generalization to a hypercube of resistors as an excuse to survey a number of results concerning generating functions, asymptotic expansions, and combinatorial enumeration. We conclude by giving, apparently for the first time, the complete solution to the problem of the hypercube of resistors.
Ambiguous Groups and Cayley Graphs—A Problem in Distinguishing Opposites
By Richard Goldstone, John McCabe, and Kathryn Weld
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For any finite group, if we are given the complete Cayley graph for the group, and if the undirected edges are colored in a natural way, does this partial information determine the multiplication table for the group? It turns out that that the answer to this inverse problem is usually yes, but not always. We call a group whose multiplication table cannot be determined from its complete colored Cayley graph an ambiguous group. A simple example of such a group is the quaternion group. We are able classify all ambiguous groups. We show that the complete Cayley graph with colored edges does determine the isomorphism class for the group. Along the way we revisit contributions made to the development of group theory by the eminent mathematicians Cayley, Hamilton, Dirichlet, and Baer.
Probability in Look Up and Scream
By Christopher N. Swanson
In the game Look Up and Scream, players stand in a circle, close their eyes, and on the count of three, open their eyes, with each player looking directly at another player. If two players look directly at each other, they scream and are out of the game. In this paper, the author derives a formula for the probability that there are y pairs of yells when n people play a round of the game. Using this formula, the author derives formulas for the mean and variance of the number of pairs of yells and demonstrates how to calculate the mean rounds a game will last when starting with n players. The author also presents alternative derivations for the mean and variance of the number of pairs of yells.
Solving the Noneuclidean Uniform Circular Motion Problem by Newton’s Impact Method
By Robert L. Lamphere
We compute the centripetal force exerted on a particle moving uniformly on the circumference of a noneuclidean circle using Newton’s impact method.
Sums of Evenly Spaced Binomial Coefficients
by Arthur T. Benjamin, Bob Chen, and Kimberly Kindred
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We provide a combinatorial proof of a formula for the sum of evenly spaced binomial coefficients, . This identity, along with a generalization, are proved by counting weighted walks on a graph.
How Long Until a Random Sequence Decreases?
by Jacob A. Siehler
Increasing runs of numbers are a naturally attractive feature in any randomly-generated sequence. Surprisingly, the average length of such runs is easy to compute and does not depend on the distribution of the random numbers, at least in the case of continuous random variables. We prove this, along with similar results for runs in sequences generated by rolling dice.
Monotonicity of Sequences Approximating
by Eugene Gover
Apparently, it has not previously been observed that as , a sequence of the form with can first decrease for more than any arbitrarily specified number of terms before increasing monotonically towards the limiting value, . We prove that when , values for x and can always be found so that this type of reversal in the growth of terms of the sequence is realized, and outside this range, convergence is strictly monotonic starting from the first term of the sequence.
by Jerzy Kocik
Finding appearances of the golden ratio in various nooks and crannies of mathematics brings delight, often surprise. This note presents, in the form of a puzzle, a configuration of circles that is replete with the golden ratio. But that is only the surface. One tool to analyze such figures is the “master matrix equation” that rules circle (and n-sphere) configurations. This equation generalizes the famous circle theorem of Descartes (known also as Soddy’s kissing circle theorem).