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?

You have analyzed the game correctly.  Wikipedia lists it as "the subtraction game" under "Other variations of Nim".
