Counting Cards: Combinatorics, Group Theory, and Probability in War

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Marc Brodie


My project involved taking a look at the card game War, searching for patterns which develop in the game, and then looking for Mathematical explanations for those patterns. The project began when my advisor, Professor Marc Brodie, was playing War with his children and began to notice patterns and ask questions about them. The questions I set out to answer were: What is the probability of playing a game of war in which a loop develops? If we know the size of the deck we are using, can we determine what loop lengths are possible? How are cards cycling between players within a loop? What patterns of winning occur within a loop? What effect does changing the number of suits in the deck or the number of players have on the cycling of cards within a loop and the loop length? We found at least partial answers to all of these questions and more using Mathematica programs of simulated games along with basic theory from Combinatorics, Group Theory, and Probability.

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