Floridian Solitaire: A New Variant of Bulgarian Solitaire

Abstract

Bulgarian solitaire is a well-studied, no-choice, no-loss, one-player game involving stacks of cards. More formally, it is a self-map on the set of partitions of a fixed integer $n.$ As a finite dynamical system, its long-term behavior is well understood. Every trajectory ends in a cycle. The partitions that are in a cycle are parameterized by binary vectors, and the cycles by binary necklaces. Call a partition separated if distinct part sizes differ by at least two. The vast majority of partitions belonging to a cycle are not separated. Motivated by this fact, we consider a variant where the player has choices, but is restricted to separated partitions and, if unable to make a legal move, may lose. We prove that for $n>73$, there are cycles, and hence winning initial positions. We analyze the game for small values of $n$ and describe computations which, together with our main result, show that there are cycles for $n \in \{2,6,8,11,14,16,18,21\}$ and for $n \ge 23$, but for no other $n.$