2013 MAA Writing Award Winners: American Mathematical Monthly

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.

Paul R. HalmosLester R. Ford Awards (presented to authors of articles of expository excellence published in The American Mathematical Monthly)

Robert T. Jantzen and Klaus Volpert
On the Mathematics of Income Inequality: Splitting the Gini Index in Two
The American Mathematical Monthly, Vol. 119, no. 10, December 2012, pp. 824-837

Citation: This article starts with the stunning fact that a certain hedge fund manager earned more in 2010 than 50,000 math professors combined. Followed by a graph showing us the percent of annual national income received by the top 1% of wage earners since 1913, this paper easily entrances the reader into exploring various measurements and characteristics of income inequality.

Using concrete examples, the authors give clear definitions of Lorenz curves and the Gini index. This index is a number that gives an indication of income inequality in an economy, but its computation requires more information than would normally be available from government sources.

The authors use the 2009 U.S. quintile income data to show the challenge of modeling, in a valid and meaningful way, this economic situation with the incomplete data. Going back and forth between economic examples and mathematical models, we are led to a Lorenz curve that both fits the data plus uses meaningful parameters.

Most of this is done using just knowledge of functions and their integrals, yet there is enough mathematical rigor and economic data that readers have the pleasure of investigating a mathematical concept in depth and feeling on top of current events.

Dimitris Koukoulopoulos and Johann Thiel
Arrangements of Stars on the American Flag
The American Mathematical Monthly, Vol. 119, no. 6, June-July 2012, pp. 443-450

Citation: What do the American flag, the multiplication table, and the average number of prime factors of an integer have in common? Koukoulopoulos and Thiel illuminate these connections in their fascinating article.

Reasonable star patterns on the American flag correspond to special factorizations; the density of such factorizations is less than the density of values in a multiplication table; Paul Erdös showed this density asymptotically approaches zero by considering the average number of prime factors of an integer. The conclusions are robust, applying to a broad class of star patterns.

Along the way, the authors discuss multiple refinements of results on the average number of prime factors of an integer. The authors combine the history of star patterns on the flag with significant number theory propositions, a happy marriage of an easily understood concrete model with commendable mathematical generality.

This paper exemplifies the Monthly’s goal to “inform, stimulate, challenge, enlighten, and even entertain” its readers.

Lionel Levine and Katherine E. Stange
How to Make the Most of a Shared Meal: Plan the Last Bite First
The American Mathematical Monthly, Vol. 119, no. 7, August-September 2012, pp. 550-565

Citation: This paper provides an entertaining and remarkably transparent discussion of the “Ethiopian Dinner Game”: Two players take turns eating morsels from a common platter. The players may have different utility values (i.e. tastiness) for the morsels but each knows the utility values of the other. What strategy should a player use assuming that they are not cooperating and that both players wish to maximize the sum of the utility values of the morsels they get to eat?

The subject of the paper is a strategy for each of the players discovered by Kohler and Chandrasekaran, which is based on what the authors of this paper call the “crossout strategy”: When it is your turn, eat your opponent’s least favorite remaining morsel on your own last move. Working backwards prescribes the choices of each player.

The authors give a new, rigorous visual proof that this pair of strategies forms a Nash equilibrium; that is, neither player can benefit by changing their strategy unilaterally. The authors give new proofs that the crossover strategy pair is a subgame perfect equilibrium and Pareto efficiently computable. Along the way, the authors provide a lucid and enjoyable introduction to perfect information game theory.

Daniel Kalman and Mark McKinzie
Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function
The American Mathematical Monthly, Vol. 119, no. 1, January 2012, pp. 42-51

Citation: Despite the great fame, many proofs, and well-known history of Euler’s result from 1735 that the sum of the reciprocal squares is π^2/6, this paper shows that all is not yet known about this result. Moreover, in this paper the authors wrap their presentation of an unfamiliar but very intuitive proof of the result in a fascinating historical mystery.

The proof combines very natural ideas from calculus and generating functions to arrive at what the authors call “the roadblock”. The roadblock can be removed but only via two identities—both of them due to Euler. The puzzle then remains: “Did Euler know this proof?”

An engaging historical section acknowledges L. Lewin’s Polylogarithms and Associated Functions from 1981 as the source of the argument to come. The authors then proceed with the calculus derivation that leads to the roadblock. They gracefully employ a Dilog identity of Euler’s, which he might possibly have known in 1730, and his famous identity e^πi = -1 to sum the series.

The meat of the paper is then a very careful and lucid section that makes the Eulerian style argument above completely rigorous. The paper finishes with the historical puzzle.

This article brings a beautiful but unfamiliar proof of Euler’s result to the public eye while also illustrating how powerful an intuitive proof combined with the necessary rigorous argument can be. Finally the authors succeed in bringing the mathematical mind of Euler to life when they speculate about whether he knew this proof, and if he did know it why it was not published.