Consider the following game:
There are a number of piles of stones. On each turn a player can remove as many stones he likes (at least 1) from up to $N$ piles (at least 1). It is allowed to remove a different amount of stones from each of the up to $N$ piles. The first player that can't move loses.
For $N=1$ we have the original game of nim with its well known optimal strategy.
For $N \geq$ number of piles the obvious optimal strategy is to remove all the stones.
What is the optimal strategy for $N=2,3,..$?
Extra question: What is the sprague-grundy value for these games?