Blocking Wythoff Nim

TL;DR

This paper investigates a variant of Wythoff Nim with a blocking rule, determining winning strategies for small blocking parameters and conjecturing asymptotic behavior of P-positions for larger parameters based on computational evidence.

Contribution

It provides a complete solution for the game with blocking parameters k=2 and k=3, and proposes conjectures for larger k supported by computer simulations.

Findings

Winning strategies for k=2 and k=3 are fully characterized.

Conjectures on the asymptotic distribution of P-positions for 4 ≤ k ≤ 20.

Computer simulations suggest specific patterns in the game's P-positions.

Abstract

The 2-player impartial game of Wythoff Nim is played on two piles of tokens. A move consists in removing any number of tokens from precisely one of the piles or the same number of tokens from both piles. The winner is the player who removes the last token. We study this game with a blocking maneuver, that is, for each move, before the next player moves the previous player may declare at most a predetermined number, $k - 1 \ge 0$, of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and does not have any further impact on the game. We resolve the winning strategy of this game for $k = 2$ and $k = 3$ and, supported by computer simulations, state conjectures of the asymptotic `behavior' of the $P$-positions for the respective games when $4 \le k \le 20$.