We consider the following combinatorial game (with two players
alternatively playing optimally). Posititions are given by heaps containing $b\geq 0$ black and $w\geq 0$ white stones and are
encoded by $(b,w)$ in $\mathbb N^2$ where $\mathbb N=\lbrace 0,1,\ldots\rbrace$.
Players are allowed to remove either a unique black stone or
a set of two stones containing at least one white stone.
(Moves starting at $(b,w)$ correspond therefore to steps
in $\{(-1,0),(-1,-1),(0,-2)\rbrace$ which do not
leave the first quadrant.)
Final positions with no move left are $(0,0)$ and $(0,1)$
and are either winning or losing independently (there are
therefore $4=2^2$ possibilities).
There seems to be no easy obvious pattern in the set of winning positions for these four games but my computer seems to be convinced of the following fact (for each possible game):
Fix $(\alpha,\beta)$ in $\mathbb N^2$ and consider for every
$(a,b)$ in $\mathbb N^2$ the characteristic sequence $W_{(\alpha,\beta)}(a,b)$ in $\lbrace 0,1\rbrace^{\mathbb N}$ encoding
winning positions in the arithmetic progression $(\alpha,\beta)+\mathbb N(a,b)$. All sequences $W_{(\alpha,\beta)}(a,b)$ with $a,b\geq 1$ are
seemingly ultimately periodic and moreover the union $\cup_{a,b\geq 1}W_{(\alpha,\beta)}(a,b)$ of sequences
corresponding to all arithmetic progressions (with integral steps in the open first quadrant) starting at $(\alpha,\beta)$ seems to be a finite (and generally fairly small) set.
Should I throw away my computer?
Upgrade: The characteristic sequence for
winning positions $(0,3)+\mathbb N(1,1)$
(with respect to the rule that the last player with moves win)
has a short non-periodic prefix since it is seemingly given by $110\overline{101}$.
Best Answer
Consider the case where $(0,0)$ and $(0,1)$ are winning (i.e. P) positions. There's a biphasic structure.
Let $\operatorname{off}_w(b) = 3 \lfloor \frac b2 \rfloor + (b \bmod 1)$. Then if $w < \operatorname{off}_w(b)$ you look at the periodic table $$\begin{matrix} P & P & N & N & N & N \\ N & N & N & P & P & N\end{matrix}$$ with row indexed by $b \bmod 2$ and column by $w \bmod 6$; otherwise you look at the periodic table $$\begin{matrix}P & P & N & N \\ N & N & P & N \\ N & N & P & P \\ P & N & N & N\end{matrix}$$ with row indexed by $b \bmod 4$ and column indexed by $(w - \operatorname{off}_w(b)) \bmod 4$.
For nimbers, the tables are $$\begin{matrix}0 & 0 & 3 & 1 & 1 & 3 \\ 1 & 1 & 2 & 0 & 0 & 2\end{matrix}$$ and $$\begin{matrix}0 & 0 & 1 & 1 \\ 1 & 2 & 0 & 3 \\ 1 & 1 & 0 & 0 \\ 0 & 2 & 1 & 3\end{matrix}$$
I haven't worked through the others to the same level of detail, but I anticipate that they will have similar structures.