AI 中文总结
该研究提出保加利亚纸牌游戏的新变体佛罗里达纸牌游戏，证明n>73时存在循环及获胜初始位置，明确特定n值下该游戏存在循环的结论。
AI 中文摘要
保加利亚纸牌游戏是一种被广泛研究的无选择、无损失的单人游戏，涉及纸牌堆，更正式地说，它是固定整数n的分拆集合上的自映射，作为有限动力系统，其长期行为已被充分理解：每条轨迹最终都会进入一个循环，处于循环中的分拆由二元向量参数化，循环则由二元项链参数化。若分拆的不同部分大小至少相差2，则称其为分离分拆，绝大多数处于循环中的分拆都不是分离分拆。受此事实启发，我们考虑一种变体：玩家有选择，但仅限于分离分拆，若无法做出合法走法则可能输。我们证明，当n>73时，存在循环，因此存在获胜初始位置；我们分析了小n值下的该游戏，并描述了相关计算，结合我们的主要结果可知，n∈{2,6,8,11,14,16,18,21}以及n≥23时存在循环，其他n值则不存在。
英文摘要
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 $n.$
Comments32 pages, 5 figures
