Wythoff's Nim with Finite Alterations

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.