All's Fair in Love and WAR: Combinatorics in a card game

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Marc Brodie


A popular, yet very basic, card game is War. Played by children and adults alike, War is a game thought to have very little strategy involved in the game's outcome. Unlike card games such as blackjack and poker, in War, it seems the luck of the deck alone will determine the fate of the players. Under this commonly held assumption, a player has no idea when starting a game his or her chances of winning. An observant player, however, may recognize patterns in a game. For example is it possible that a game may not end? Is it plausible that an entire game be played without a match occurring? Some of these questions were asked and answered by Angela Chappell in her 1998 senior honors thesis. To find her answers, however, she had to establish a set of assumptions for the way the game was to be played. To illustrate, she generated her data using the convention that, if player A and B were playing War, and on the first hand player A's card was greater than player B's card, then A's card would go back to A's hand first before B's card. My project, then, is to change several of Ms. Chappell's assumptions and replay the games. In this way, I hope to further knowledge about the conjectures and theses she made in her research. In terms of the mathematical application, this project will explore a specialized use of permutations and optimally define them in group theory. As permutations have seemingly infinite potential, the ability to categorize and conjecture would add to a better understanding of their behavior.