bibkey: morisawa2026mpileperiodicity
authors: Takayuki Morisawa
year: 2026
title: "Winning strategy and periodicity of m-pile divisor Nim"
doi: 10.1007/s00182-026-00991-5
url: https://doi.org/10.1007/s00182-026-00991-5
claim: "Eventual periodicity for fixed-board m-pile divisor Nim sequences."
strata_touched:
  - D5/S3/Combinatorics/Games/DivisorNimFixedBoard
  - D5/S3/Combinatorics/Games/DivisorNimGcdTwo
license: citation-only
triage: anchor

Fixed-board Divisor Nim periodicity

Verified locator

DOI and publisher URL: https://doi.org/10.1007/s00182-026-00991-5

Scope

Morisawa supplies the broad eventual-periodicity background. This citation does not assert that the paper states the comparison spacing built from the minimum fixed heap and the least periods of every direct fixed successor, or the gcd-two multiplier identity. The native formalization uses the literal finite multiset game and its full Sprague–Grundy sequence.

The finite-memory argument uses bounded SG blocks sampled at a common forcing phase. Equality of two such output states propagates along the exact recurrence.

The two-pile predecessor, Odd case of 2-pile divisor Nim, DOI 10.1007/s00182-025-00948-0, has a narrower domain. Nohagi’s Sprague-Grundy Value for Common Divisor Nim, arXiv:2312.05466, concerns a different move rule. Neither is used as a proof of the arbitrary-multiset comparison-spacing result.