On Remoteness Functions of Exact Slow NIM, NIM$^1_=(4,2) and NIM^1_=(5,2)

2024.

Gurvich V., Maximchuk V., Naumova M.

In three recent works: https://arxiv.org/abs/2304.06498, https://arxiv.org/abs/2311.13511, and https://arxiv.org/abs/2311.03257 a very simple explicit rule was suggested and applied to compute Smith’s remoteness function of the game Exact Slow NIMk with n piles of stones for the case n = k + 1. In this paper we extend this rule to the general case of Exact Slow NIM, NIM1 =(n, k), and apply it for n = 4, k = 2 and n = 5, k = 2. In both cases it works efficiently and reduces the remoteness function by 1 with each move in “almost all” cases. The exceptions are sparse and have regular pattern. We suggest simple formulas that cover all found exceptions

Keywords: game theory, NIM, Exact slow NIM, Smith's remoteness function