Homemath.COarXiv:2605.29918

On Ending Partizan Subtraction Nim

math.CO2026-05v1license

Abstract

We consider Subtraction Nim, where two players have exactly the same options, but which is partizan in the sense that at the game ending, a partizan rule is applied for the decision of the winner. We consider the following example: Let SS be the set of removable numbers, which is a non-empty finite subset of positive integers greater than or equal to 22, applied for both players Left and Right. At the end of the game, Left wins if the number of remaining tokens is even, and Right wins if the number of remaining tokens is odd. We computed the outcomes for many SS, and found surprising phenomena that in most examples of SS (almost 98%98\% of some samples), the outcomes are L\mathcal{L}-positions for all large enough nn. In comparison, R\mathcal{R}-positions appear only occasionally. The main theorem explains why this phenomenon occurs. We prove that n+1n+1 and n1n-1 are L\mathcal{L}-positions when nn is an R\mathcal{R}-position. Similarly, L\mathcal{L}-positions appear whenever P\mathcal{P}-positions or N\mathcal{N}-positions appear. Only L\mathcal{L}-positions can last forever.

Cite

@article{arxiv.2605.29918,
  title  = {On Ending Partizan Subtraction Nim},
  author = {Hiyu Inoue and Shin-nosuke Kadowaki and Shun-ichi Kimura and Haruki Wada},
  journal= {arXiv preprint arXiv:2605.29918},
  year   = {2026}
}