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≥2 and c≠ 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 cbmod(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≥2, we conjecture that the tests cover every purely periodic non-additive game. In this range, when c≥2(a+b), we prove necessity for purely periodic games with least period at most a+b.
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BibTeX
@misc{manabe2026purely,
title = {{Purely Periodic Three-move Subtraction Games}},
author = {Hikaru Manabe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05358},
eprint = {2609.05358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05358},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Purely Periodic Three-move Subtraction Games
AU - Hikaru Manabe
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.05358
UR - https://arxiv.org/abs/2609.05358
AB - 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≥2 and c≠ 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 cbmod(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≥2, we conjecture that the tests cover every purely periodic non-additive game. In this range, when c≥2(a+b), we prove necessity for purely periodic games with least period at most a+b.
ER -