# Towers game strategy

Given the following game, what is the strategy to win?

Given $N$ towers of different heights. Two players play against each other. Each player (in his turn) divides each of the towers which are greater than 1 into two separate towers. The player who wins is the one that after his (last) turn all towers are of height 1.

Any help would be appreciated.

Are you familiar with Nim? As an impartial game, this is Nim in disguise. You need to figure out which positions are $N$ (wins for the next player) and $P$ (wins for the previous player). You only care about the largest stack because each player has to play in all stacks greater than $1$. A largest stack of size $1$ is $P$ by the rules. $2$ plays to $1$ and wins, so is $N$. $3$ has to leave a $2$, so is $P$. $4$ can move to $3$ and win, so is $N$. Keep going a while and try to find a pattern.
• Can you figure out who wins for $5,6,7,8$? – Ross Millikan Dec 9 '14 at 14:19