Partizan Serial Nim
Abstract
A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to determine the outcome (i.e., which player has a winning strategy) of a given position in combinatorial games. Nim is a well-known and fundamental ruleset in combinatorial game theory. This paper proposes a novel partizan variant of Nim called Partizan-Serial-Nim, defined as follows: there are n piles of stones indexed by 1, 2, …, n; the two players have permutations σL and σR of (1, 2, …, n), respectively; a move is to remove any positive number of stones from the non-empty pile with the minimum value in the player's permutation; the player who cannot make a move loses. This ruleset is a generalization of Serial-Nim and Partizan-End-Nim. We give an algorithm to compute the outcome of a given position in Partizan-Serial-Nim in O(n2) time, provided that each arithmetic and comparison operation is performed in O(1) time. Also, for the case where all non-empty piles have the same number m of stones, we prove that the outcome does not depend on m for m ≥ 2 and present an algorithm to compute the outcome in O(n) time. Further, we prove that the atomic weight of every position in Partizan-Serial-Nim is an integer.
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BibTeX
@misc{hashimoto2026partizan,
title = {{Partizan Serial Nim}},
author = {Kengo Hashimoto},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.19876},
eprint = {2608.19876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19876},
}
RIS
TY - UNPB
TI - Partizan Serial Nim
AU - Kengo Hashimoto
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.19876
UR - https://arxiv.org/abs/2608.19876
AB - A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to determine the outcome (i.e., which player has a winning strategy) of a given position in combinatorial games. Nim is a well-known and fundamental ruleset in combinatorial game theory. This paper proposes a novel partizan variant of Nim called Partizan-Serial-Nim, defined as follows: there are n piles of stones indexed by 1, 2, …, n; the two players have permutations σ^L and σ^R of (1, 2, …, n), respectively; a move is to remove any positive number of stones from the non-empty pile with the minimum value in the player's permutation; the player who cannot make a move loses. This ruleset is a generalization of Serial-Nim and Partizan-End-Nim. We give an algorithm to compute the outcome of a given position in Partizan-Serial-Nim in O(n^2) time, provided that each arithmetic and comparison operation is performed in O(1) time. Also, for the case where all non-empty piles have the same number m of stones, we prove that the outcome does not depend on m for m ≥ 2 and present an algorithm to compute the outcome in O(n) time. Further, we prove that the atomic weight of every position in Partizan-Serial-Nim is an integer.
ER -