Modular Nim (Q673073)

We give a survey on modular Nim or Kotzig's Nim is a 2-player perfect information game without chance moves, invented by Anton Kotzig in 1940. A finite move set \(M=\{a_1,\dots,a_m\}\in(\mathbb{Z}^+)^m\) (of distinct elements \(a_i\)) is given, as well as a modulus \(n\in\mathbb{Z}^+\). We imagine the numbers \(0,1,\dots,n-1\) written on equally spaced points lying consecutively around a circle in clockwise direction, with 0 initially labeled. Player I begins by labeling (also called covering or going to) some point \(a_i\) on the circle, provided \(a_i\in M\). Player II and player I then alternately label points \(a+a_j\bmod n\), where \(a\) is the most recently labeled point, \(a_j\in M\), and no point may be labeled twice. In normal play, the player first unable to move loses and his opponent wins. The outcome is reversed for misère play. Note that in normal play, a player may lose either because the circle has just been tiled (every point on the circle has been labeled), or because it has been locally obstructed, although there are still unlabeled points on it.