Trying to find the name of this Nim variant Consider this basic example of subtraction-based Nim before I get to my full question:
Let $V$ represent all valid states of a Nim pile (the number of stones remaining):
$V = 0,1,2,3,4,5,6,7,8,9,10$
Let $B$ be the bound on the maximum number of stones I can remove from the Nim pile in a single move (minimum is always at least 1):
$B = 3$
Optimal strategy in a two-player game then is to always ensure that at the end of your turn, the number of stones in the pile is a number found in $V$, and that it is congruent to $0$ modulo $(B+1)$. During my first move I remove 2 stones because $8$ modulo $4$ is $0$.
My opponent removes anywhere from 1-3 stones, but it doesn't matter because I can then reduce the pile to $4$ because $4$ modulo $4$ is $0$. Once my opponent moves, I can take the rest of the pile and win.
This is straightforward, but my question is about a more advanced version of this, specifically when $V$ does not include all the numbers in a range. Some end-states of the pile are not valid, which implies that I cannot find safe positions by applying the modulo $(B+1)$ rule.
Does this particular variant of Nim have a name that I can look up for further research? Is there another way to model the pile?

It is simply Nim. The change of rule doesn't introduce anything new.
I assume that we have a fixed set $S$ of numbers and the number of stones to remain in a pile after each move must be from $S$ (and in initial position, the number of stones in each pile is a number from $S$). In particular, i am assuming that the numbers allowed doesn't vary from pile to pile.
For an example suppose that the set $S=\{0,1,2,4,7,8\}$. Keep in mind that we are in a world of cardinal numbers here. So what you did here is just a change of symbols. I shall use the following notation: usual cardinal number ----> (number in your game).
0---->(0)
1---->(1)
2---->(2)
3---->(4)
4---->(7)
5---->(8)
Now simply do the "congruent to 0 modulo $(B+1)$ move" on usual cardinal number and you are done.
PS: Even when the set $S$ varies from pile to pile, the situation is similar. Just interpret the number of items in each pile according to the set $S$ for that pile in the way described above and using those numbers play nim.
Thank you
