Combinatorial Aspects of the Card Game War

TL;DR

This paper explores the combinatorial structure of a simplified single-suit War card game, analyzing game states, sequences, and trees to understand winning strategies and enumeration of game configurations.

Contribution

It introduces new combinatorial objects and relationships specific to War, providing a detailed enumeration framework for game states and outcomes.

Findings

Defined the concept of passthroughs in the game

Established relationships among game graphs, sequences, and trees

Enumerated game states based on rounds and passthroughs

Abstract

This paper studies a single-suit version of the card game War on a finite deck of cards. There are varying methods of how players put the cards that they win back into their hands, but we primarily consider randomly putting the cards back and deterministically always putting the winning card before the losing card. The concept of a \emph{passthrough} is defined, which refers to a player playing through all cards in their hand from a particular point in the game. We consider 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 these objects relate to each other. We enumerate states depending on the number of rounds and the number of passthroughs.