Finiteness in the Card Game of War

The Game of War is an internationally popular children’s card game with a very basic set of rules. At the start, the pack is shuffled and divided into two equal parts; then each player reveals their top card. The player with the highest value card then collects both cards and returns them to the bottom of their hand. This process continues until one player loses all cards. Many who play this game in childhood lose patience before the end as the back and forth nature of card transfer often allows a player to almost reach the point of victory only to see their hand reduced dramatically in a couple of minutes. In effect children playing are conducting basic mathematical experiments and observing the development of chaotic dynamics. It is often wrongly assumed that this game is deterministic and the result is set once the cards have been dealt. However this is not so: the rules of the game do not stipulate in which order the winner of each play returns the cards to the bottom of their hand own card first and then rival’s or vice versa. We shall at first consider a model with an arbitrary (even) number of cards and only one