纯周期三步减法游戏
Purely Periodic Three-move Subtraction Games
- College of Information Science, School of Informatics, University of Tsukuba（筑波大学信息学研究院信息科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究三步减法游戏S={a,b,c}的纯周期性，给出纯周期的显式充分准则，确定满足准则的非可加集合的最小周期、P位置及nim值，还对可加情况单独处理并证明准则在特定条件下的必要性。
AI中文摘要：
减法游戏在一堆代币上进行，玩家轮流移除固定正整数集合S中的某个s个代币，无法行动者输。此类游戏的斯普莱格-格伦迪值序列最终呈周期性。本文研究三步集合S={a,b,c}（其中0<a<b<c）何时为纯周期，即从初始就呈周期性。在简化为原始规则集且a≥2后，本文给出纯周期性的显式充分准则；对所有满足该准则的非可加集合，确定其最小周期、所有P位置的闭式形式及所有nim值，且对可加情况c=a+b单独处理。本文推测该准则也是必要的，并在c≥2(a+b)时针对周期a+b证明了该结论。该准则是基于两步游戏{a,b}的P位置模式对角度ρ=c mod(a+b)进行的有限检验，可容许角度构成Z/(a+b)Z中的显式弧的并集。
英文摘要：
We determine the full Sprague-Grundy sequence and its least period for a class of three-move subtraction games whose sequences are periodic from the start. Write the move set as $S=\{a,b,c\}$, with $0<a<b<c$ and $\gcd(a,b,c)=1$. The case $a=1$ is known and is summarized separately. For $a\ge2$ and $c\ne a+b$, we give three explicit sufficient tests for pure periodicity, associated with the candidate periods $a+b$, $c+a$, and $c+b$. For fixed $a,b$, the tests depend only on $c\bmod(a+b)$ and are computed from the losing positions of the two-move game $\{a,b\}$. Whenever a test succeeds, we give the losing positions in closed form, reconstruct the remaining values, and prove that the least period is the smallest candidate whose test succeeds. The construction shows how the added move $c$ modifies the two-move pattern to form a repeating block. We also give a uniform formulation of the known additive case $c=a+b$. For $a\ge2$, we conjecture that the tests cover every purely periodic non-additive game. In this range, when $c\ge2(a+b)$, we prove necessity for purely periodic games with least period at most $a+b$.
