Number of reachable cycles in Beggar-My-Neighbor

Determine the total number of distinct cycles (periodic loops of game states) that are reachable in Beggar-My-Neighbor played with a standard 52-card deck under the 1911 Encyclopaedia Britannica rules, across all possible starting deals, where a cycle is defined as a finite sequence of tricks that returns the game to a previously encountered full game state and then repeats indefinitely.

Background

The authors construct and verify a non-terminating game of Beggar-My-Neighbor, identifying a cyclical structure of 62 tricks that is reached by 30 distinct starting hands. This resolves the long-standing question of whether a non-terminating game exists but raises the broader question of how many such cycles exist in the full state space of the game.

Their methodology combines forward and backward play, the identification of independent “pieces,” and a template-based search, culminating in a balanced non-terminating game and a family of starting deals converging to the same cycle. The existence of multiple starting deals for one cycle suggests a richer landscape of cycles whose total number remains unknown.