Purely Periodic Three-move Subtraction Games

Abstract

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 $a+b$.