Awards for exemplary writing in MAA journals and magazines were handed out in August at MAA MathFest in Portland, Oregon. The full text (PDF) of all the winning articles is now available online.
Citation: How many ways can n people each choose two gloves from a pile of n distinct pairs of gloves, so that nobody gets a matching pair? In this article, authors Sally Cockburn and Joshua Lesperance consider a fun and surprisingly challenging twist on this familiar combinatorics problem, replacing gloves with socks. They ask, “How many ways can n people each choose two socks from a pile of n distinct pairs of socks, with no one getting a matching pair?”
The sock problem extends the glove problem by removing the crucial assumption that right- and left-handed gloves are distinguishable. In the glove problem, the matched pairs are effectively sorted in two piles: left-handed gloves in one and right-handed gloves in another, and since each person will take one left-handed and one right-handed glove, the ways to derange gloves are limited. But we do not distinguish between left and right socks, and this allows for a wider range of possibilities. As the authors illustrate, the seemingly innocuous switch from gloves to socks significantly complicates the problem.
The authors begin their discussion with counting derangements, permutations in which every object gets moved. From there, they develop solutions to the more complicated sock problem, starting with a recursive formula. Next they come up with a non-recursive solution, and as the authors develop their ideas, the reader becomes thoroughly engaged by the connections of this problem with other mathematical results.
The reader is led on a lively tour of a variety of discrete mathematical tools: partitions, cyclic permutations, recurrence relations, ordinary and exponential generating functions. At the end the authors deliver a final pleasing touch: using complex analysis to show that the fraction of all sock distributions that are deranged in this sense converges to 1/e.
Susan H. Marshall and Donald R. Smith
Feedback, Control, and Distribution of Prime Numbers
Mathematics Magazine, Vol. 86, no. 3, June 2013, pp. 189-203
Citation: In this article, Susan Marshall and Donald Smith describe an unusual application of a technique of mathematical modeling, feedback and control, to a classical mystery of number theory, the distribution of primes.
In a famous result due to Gauss, the density of primes is (approximately) inversely proportional to the natural logarithm. The differential equation below reasonably models the density of primes. Here f(x) represents the density of primes:
f’(x) = f(x) f(√x)/2x
Although this is a known application in differential equation literature, it appears to be largely forgotten in number theory. In the process of deriving this model, the authors give the reader a lively introduction to the theory of feedback and control, complete with a cast of characters representing different feedback phenomena in the face of perturbations. We have the “cool, calm, and collected” responder, the “whimsical” responder, and finally the “panicky and overreacting” responder. The authors note that the distribution of prime numbers has an element of randomness, yet it also stays on track, much like a feedback and control system, with either a “whimsical” or a “cool, calm, and collected” response.
The authors demonstrate how one verifies not only that the differential equation (above) predicts the correct density function, but also that the model is robust. That is, while the true density of primes at times deviates from 1/ln x, they show that 1/ln x is the ideal path of the true density. For a “perturbed solution” to the differential equation, as x increases we see that f(x) approaches 1/ln x. This represents stability. The mathematics is presented as a beautifully simple (manageable) change of variables. With stability comes the conclusion that the model predicts the prime number theorem. After further computation, the authors show that Littlewood’s Theorem is also predicted by the model. For a complete lesson in modeling, the authors also describe the limitations of their model; it is successful as far as gross behavior goes, but most likely fails at the fine scale, as it is incompatible with the Riemann Hypothesis.
In this engrossing article, descriptions and arguments are interspersed with history, which serves to round out a satisfying tour through both prime density and mathematical modeling.