  Wythoff's Game is a variation of Nim in which players may take an equal
number of stones from each pile or make valid Nim moves. W. A. Wythoff proved
that the set of P-Positions (losing position), $C$, for Wythoff's Game is given
by $C := \left\{ (\lfloor k\phi \rfloor, \lfloor k\phi^2 \rfloor), (\lfloor
k\phi^2 \rfloor, \lfloor k\phi \rfloor) : k \in \mathbb Z_{\geq 0} \right\}$.
An open Wythoff problem remains where players make the valid Nim moves or
remove $kb$ stones from each pile, where $b$ is a fixed integer. We denote this
as the $(b,b)$ game. For example, regular Wythoff's Game is just the $(1,1)$
game. In 2009, Duch${\^e}$ne and Gravier proved an algorithm to generate the
set of P-Positions for the $(2,2)$ game by exploiting the periodic nature of
the differences of stones between the two piles modulo $4$. We observe similar
cyclic behaviour for any $b$, where $b$ is a power of $2$, modulo $b^2$, and
construct an algorithm to generate the set of P-Positions for this game. Let
$a$ be a power of $2$. We prove our algorithm works by first showing that it
holds for the first $a^2$ terms in the $(a,a)$ game. Next, we construct an
ordered multiset for the $(2a,2a)$ game from the $a^2$ terms, and an inductive
proof follows. Moreover, we conjecture that all cyclic games require $a$ to be
a power of $2$, suggesting that there is no similar structure in the
generalised $(b,b)$ game where $b$ isn't a power of $2$. Future directions for
generalising this result would likely utilise numeration systems, particularly
the PV numbers.
