Paul and Luke are playing a game with stacks of black, red and green tiles. At the start of the game, there are n stacks of one black tile on the board, where n is any positive integer. Paul begins. There are only two allowed moves:
- Place a red or green tile (from a sufficiently large supply) onto an existing stack. After the move, no two pieces of the same color may be in the same stack. A stack can only consist of a maximum of three tiles.
- Remove two stacks from the game, with the top tiles of the two stacks having the same color. The height can be different.
He who can no longer draw has lost.
You decide
a) for six stacks, so $$ n=6 $$
b) for any even number $$ n>=2 $$ of stacks,
whether either player can play in such a way that she can force the win, and if so, provide such a winning strategy.
I thought about this analyzing the game. For $n=2$ we know that there are only two stacks of black, so it means Paul can directly apply the second rule of the game, so removing two stacks from the game. Does it mean, that with enough time, the same would happen also for $n=6$? This is not very clear to me.