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This paper studies a single-suit version of the card game War on a finitedeck of cards. There are varying methods of how players put the cards that theywin back into their hands, but we primarily consider randomly putting the cardsback and deterministically always putting the winning card before the losingcard. The concept of a \emph{passthrough} is defined, which refers to a playerplaying through all cards in their hand from a particular point in the game. Weconsider games in which the second player wins during their first passthrough.We introduce several combinatorial objects related to the game: game graphs,win-loss sequences, win-loss binary trees, and game posets. We show how theseobjects relate to each other. We enumerate states depending on the number ofrounds and the number of passthroughs.

Combinatorial Aspects of the Card Game War