We have an impartial game that works like this. We start with a pile of 10 coins. The player who starts splits the pile into two piles (he can choose sizes). The players then move in turns. At each turn a player selects a pile of size larger than one and split the pile. (To state it clearly, a pile of size one can never be used). The first player who cannot move loses the game.
Who wins this game? What is the optimal strategy?
Can this be generalised to the case with a pile with n coins?
The game ends when all $n$ coins are in $n$ separate piles. It starts with one pile, and each move increases the number of piles by $1$. Thus the game ends after $n-1$ moves, the first player wins if $n$ is even and the second player wins if $n$ is odd. Strategies are irrelevant; the result is independent of the moves the players make.