Awards for exemplary writing in MAA journals and magazines were handed out in August at MAA MathFest in Hartford, Connecticut. The winning articles (PDF) are now available online, along with the award citations explaining why each article merited an award.
George Pólya Awards (presented to authors of articles of expository excellence published in The College Mathematics Journal)
Jacob Siehler
The Finite Lamplighter Groups: A Guided Tour
The College Mathematics Journal, Vol. 43, no. 3, May 2012, pp. 203-211
Citation: Does the imperative of technical accuracy compel mathematical exposition to be impersonal and officious? Not if you’re Jacob Siehler. He introduces us to his pals, the lamplighter groups, in a manner that feels like an after-school chat at the corner bar.
The finite lamplighter group L<sub>n</sub> has n lamps arranged in a circle with a lighting mechanism. An operation consists of rotating the lighter and then possibly toggling one of the lamps on-off. Siehler urges us to get our hands dirty, to play around with these operations to see how they behave. He leads us to see what conjugates, centers, and commutators “look like.”
At first it seems difficult to identify the conjugate elements. Surprise! To recognize the conjugacy class of an element that rotates k steps, we identify two-colored necklaces of length gcd(n, k). The number of rotation k classes is the same as the number of length gcd(n, k) necklaces. Guided by this insight, we soon are comfortable using necklaces to describe the center and commutator subgroups. We feel like we really know these groups.
The finite lamplighter groups provide appealing examples of nonabelian groups just a touch beyond the dihedral groups. Siehler has given us a wealth of material suitable for classroom use in a beginner course. Before leaving us, he suggests a list of exercises and open questions. Read the list, and even more questions readily come to mind. Compelling mathematics raises more questions than it answers.
David Applegate, Marc LeBrun, and Neil J.A. Sloane
Carryless Arithmetic Mod 10
The College Mathematics Journal, Vol. 43, no. 3, May 2012, pp. 203-211
Citation: In the fabled Carryless Islands of the South Pacific, inhabitants use a type of arithmetic that is so close, yet so far from our ordinary integer arithmetic. The “carefree” residents of these islands have the perverse habit of adding and multiplying numbers with no carries into other digit positions.
It is within this whimsical setting that Applegate, LeBrun, and Sloane take readers on an exploration of this culture’s mathematics and along the way, discover some interesting sequences and number-theoretic results.
The authors begin by listing the first twenty carryless primes (non-units whose only factorizations are of the form u x p, where u is a unit and p is an integer and x represents carryless multiplication), and mentioning that, due to some strange factorizations of the numbers (for example, 2 = 2 x 51), it is difficult to verify directly, or even by computer, that their list is correct. Instead, they cleverly use algebra to create an algorithm that not only proves the accuracy of their list, but also generates all carryless primes.
This number system also differs from ours in that there exist numbers with an infinite number of divisors, and there is a formula for the number of k-digit carryless primes.
The article is an innovative and engaging treatment of a topic that will naturally inspire readers to ask and explore similar questions. Instructors can easily use this material in a first course on number theory or abstract algebra, and the open questions at the end will provide students with ideas for research projects.