  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.
