Slow $k$-Nim

TL;DR

This paper analyzes two variants of the slow k-Nim game, computing their Sprague-Grundy functions for small parameters and characterizing P-positions in specific cases, advancing understanding of these combinatorial games.

Contribution

It provides explicit Sprague-Grundy functions for small cases and characterizes P-positions for certain parameter ranges in slow k-Nim variants.

Findings

Sprague-Grundy functions computed for n=k=2 and n=k+1=3.

P-positions characterized for n ≤ k+2 and n=k+3 ≤ 6.

Enhanced understanding of game structure for small parameters.

Abstract

Given $n$ piles of tokens and a positive integer $k \leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from each of these piles. For the normal and mis\`ere version of each game we compute the Sprague-Grundy function for the cases $n = k = 2$ and $n = k+1 = 3$. For game Nim$^1_{n, \leq k}$ we also characterize its P-positions for the cases $n \leq k+2$ and $n = k+3 \leq 6$.