February 2011 Contents
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Markov Chains for Collaboration
Robert Mena and Will Murray
Consider a system of n players in which each initially starts on a different team. At each time step, we select an individual winner and an individual loser randomly and the loser joins the winner's team. The resulting Markov chain and stochastic matrix clearly have one absorbing state, in which all players are on the same team, but the combinatorics along the way are surprisingly elegant. The expected number of time steps until each team is eliminated is a ratio of binomial coefficients. When a team is eliminated, the probabilities that the players are configured in various partitions of n into t teams are given by multinomial coefficients. The expected value of the time to absorbtion is (n - 1)2 steps. The results depend on elementary combinatorics, linear algebra, and the theory of Markov chains.
Crosscut Convex Quadrilaterals
A convex quadrilateral ABCD is “crosscut” by joining vertices A, B, C, and D to the midpoints of segments CD, DA, AB, and BC, respectively. Some relations among the areas of the resulting pieces are explored, both visually and analytically.
From Fourier Series to Rapidly Convergent Series for Zeta(3)
Ernst E. Scheufens
Exact values of the Riemann zeta function ζ(s) for even values of s can be determined from Fourier series for periodic versions of even power functions, but there is not an analogous method for determining exact values of ζ(s) for odd values of s. After giving a brief historical overview of ζ (s) for integer values of s greater than one, and showing how we can determine ζ(2) and ζ (4) from Fourier series, we consider the Fourier series for a continuous and piecewise differentiable odd periodic function from which we can find a series with logarithmic terms for ζ (3). Using power series for logarithmic functions on this series, a rapidly convergent series for ζ (3) is obtained. Using a partial sum of this series we can compute ζ (3) with an error which is much smaller than the error obtained by using a similar partial sum of the infinite series defining ζ (3).
Three Approaches to a Sequence Problem
In this note we examine a well-studied problem concerning the terms of a certain linear recurrence modulo prime numbers. We present three solutions to this problem and examine the similarities and differences between them. In particular, despite using primality in different ways, all three proofs yield the same generalization of the original problem.
Isoperimetric Sets of Integers
Steven J. Miller, Frank Morgan, Edward Newkirk, Lori Pedersen, Deividas Seferis
The celebrated isoperimetric theorem says that the circle provides the least-perimeter way to enclose a given area. In this note we discuss a generalization which moves the isoperimetric problem from the world of geometry to number theory and combinatorics. We show that the classical isoperimetric relationship between perimeter P and area A, namely P = cA1/2, holds asymptotically in the set of nonnegative integers.
Choosing Rarity: An Exercise in Stopping Times
Ewa Kubicka, Grzegorz Kubicki
You enter an online game, hoping to win a prize. You know that thirteen envelopes of two colors will appear one by one on your screen, and you know that there will be five of one color and eight of the other. Each of the five envelopes, those with the “rare” color, contains $100, while the others are empty. However, you do not know which color is which, and you can only select one envelope; once you click, the game is over. How should you play to maximize the probability of winning? We provide an optimal strategy for this game and a generalization.
Roger C. Alperin, Vladimir Drobot
What is the smallest number of inch marks on a ruler that allow us to measure all integral distances? This question motivates our survey of Golomb rulers, perfect rulers, and minimal rulers—different types of rulers for allowing the most measurements with the smallest number of marks.
Pascal's Hexagon Theorem Implies the Butterfly Theorem
Pascal's Hexagon Theorem is used to prove the Butterfly Theorem for conics, a well known result in Euclidean geometry. In the course of the proof, some basic concepts in projective geometry are introduced.