Here is a variant of the Nim game which I could not find out the winning strategy, the game rule is like this:
The games starts with 16 stones arranged as follow:
o (first pile)
ooo (second pile)
ooooo (third pile)
ooooooo (fourth pile)
1.Two players take turns to take away stones.
2.Each time you can at most take away all stones in the same pile , and you must at least take away one stone.
3.The player who takes away the last stone loses.
4.If one pile is split into two piles, you have to take away them in at least two turns.
For instance the third and fourth stones in the third pile was taken: (x represents the taken stones)
Then you are not allowed to take away all stones in the third pile in one turn. You have to take away all of them in at least two turns since it has been split into two piles:
First turn: oo xxo
Second turn: xxxx o
And for pattern like oxxoxoo, you have to take away all of them in at least 3 turns, etc.
The first three rules are similar to the classic NIM game(I know the winning strategy in the classic one), however the last rule makes it not applicable. My question is who has the winning strategy in this game, and what is the winning strategy?