Wythoff's Nim with Finite Alterations

TL;DR

This paper studies a modified version of Wythoff's Nim where certain positions are designated as losing or winning, and shows that the set of losing positions closely resembles the original game as the game size increases.

Contribution

It introduces a finite alteration to Wythoff's Nim and proves that the P-positions of the altered game asymptotically resemble those of the original game.

Findings

The P-positions of the altered game are similar to those of the original Wythoff's Nim.

The overlap of P-positions approaches 100% as pile sizes grow.

The set of P-positions in the altered game converges to a translated version of the original set.

Abstract

Wythoff's Nim is a variant of 2-pile Nim in which players are allowed to take any positive number of stones from pile 1, or any positive number of stones from pile 2, or the same positive number from both piles. The player who makes the last move wins. It is well-known that the P-positions (losing positions) are precisely those where the two piles have sizes $\{\lfloor \phi n \rfloor, \lfloor \phi^2n \rfloor \}$ for some integer $n\geq 0$, and $\phi = (1+\sqrt{5})/2 = 1.6180\cdots$. In this paper we consider an altered form of Wythoff's Nim where an arbitrary finite set of positions are designated to be P or N positions. The values of the remaining positions are computed in the normal fashion for the game. We prove that the set of P-positions of the altered game closely resembles that of a translated normal Wythoff game. In fact the fraction of overlap of the sets of P-positions of these two games approaches $1$ as the pile sizes being considered go to infinity.