Slow k-nim

© 2020, Colgate University. All rights reserved. Given n piles of tokens and a positive integer k ≤ n, we study two impartial combinatorial games, Nim1n,≤kand Nim1n,=k. In the first (resp. second) game, each move consists of choosing at least 1 and at most k (resp. exactly k) non-empty piles and removing one token from each of them. We study the normal and misére versions of both games. For Nim1n,=k we give explicit formulas of its Sprague-Grundy function for the cases 2 = k ≤ n ≤ 4; for Nim1n,≤k we provide such formulas for 2 = k ≤ n ≤ 3 and characterize the P-positions for the cases n ≤ k + 2 and n = k + 3 ≤ 6.