the game is structured like this:
- two players
- the players alternate moves
- 4 heaps $h_1, h_2, h_3, h_4$ with sizes $n_1, n_2, n_3, n_4$
- at each move, the player can either remove one or two elements from any of the heaps (meaning that if the player takes two elements, those two can either be from the same heap or from two different heaps, as long as the total number of removed elements per turn is 2).
- the game is played misère-ish, so it stops when the player (who loses the game) is left with 3 heaps of size $0$ and one heap with size greater than $0$.
Is this game a variant of the Index-k Nim (also called $Nim_k$)? The problem is that in the $Nim_k$ (and in every multi-heap Nim) game I can either remove an arbitrary or bounded amount of elements $r$ in each of the (at most) $k$ heaps per turn, while in this game the maximum number of elements that can be removed in total is bounded.
Is it a valid Nim or not? Can you suggest a valid winning strategy?
(thank-you)