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The Paintball Party

by James Propp

Year of Award: 2018

Publication Information: Math Horizons, Volume 25, Number 2, November 2017, Pages 18-21.

Summary: Geometry, combinatorics, and finite fields are used to compose teams for seven games of paintball. Eight players are to be divided into two teams of four so each pair of players is on the same team exactly three times. Vertices of a cube in ℝ3 viewed as elements of GF(2)3 represent the players. Teams are formed based on the planes of GF(2)3. Symmetry is employed to ensure that each pair of players is on the same team exactly three times. Readers are encouraged to explore further by considering other numbers of players, and different values for the number of times each pair is on the same team. Humor and lively writing help illustrate an excellent example of the power of symmetry. The article is well-written at a level accessible to students who have not yet encountered abstract algebra, but ramps up nicely to cover open problems at the end, having something for everyone.

Response from the Author:

I'm honored to receive an award from the MAA, and it's a special treat to have the MAA acknowledge this particular piece of writing. The life of a mathematician-with-children offers many conflicts between the personal and the professional; it's delightful that in the case of the paintball party saga, my mathematical knowledge enabled me to be a more effective parent, and the resulting anecdote, in turn, contributed to my efforts at mathematical outreach. Editor Dave Richeson deserves special thanks; he was the one who, perusing my Mathematical Enchantments blog, singled out the paintball party essay as being especially suited to Math Horizons, and he did most of the work in whittling the original, longer essay down to size.

About the Author:

James Propp is a Professor at the University of Massachusetts, Lowell. He did his PhD work in ergodic theory (at U.C. Berkeley) but is best known for his contributions to combinatorics and probability and his mentoring of young mathematicians through supervised research. He serves on the advisory council of the Museum of Mathematics and is an Ambassador for the Global Math Project. He blogs at, posts videos at, and tweets at @JimPropp.