Brain teaser Strategic choice... $X$ and $Y$ are playing a game. There are $11$ coins on the table and each player must pick up at least $1$ coin, but not more than $5$. The person picking up the last coin loses. $X$ starts. How many should he pick up to start to ensure a win no matter what strategy $Y$ employs?

A player facing $n$ coins can force a win iff there exists $1\le k \le 5$ such that a player facing $n-k$ coins cannot escape a loss.
Clearly, $n=1$ is a lost position.
Therefore $n=2, 3, 4, 5, 6$ are won positions.
Therefore $n=7$ is a lost position.
Therefore $n=8, 9,10,11,12$ are won positions.
One readily sees that this pattern continues and a position $n$ is lost if and only if $n\equiv 1\pmod 6$. Therefore $n=11$ is a won position - and the (only) winning move consists in going to the lost position $7$.
