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 $n.$

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“member with in-degree 0? • Are there modifications of the lifting rules or additional lifting rules of the same nature that give more edges or merge components in the modified Q? Tool and computational resource disclosure ChatGPT was used to proofread the manuscript for correct grammar, spelling, and punctuation. 31”

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2026-08-23T05:01:28.216528Z

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