Question 85
We play a game of musical chairs with $n$ chairs numbered 1 to $n$. You attach $n$ slips of paper, numbered 1 to $n$, to the chairs in such a way that the number on a slip does not match the number on the chair.
A player sits on each chair. When you clap, each player looks at the number on the slip attached to their current chair and goes to sit on the chair with that number. Prove that for every $m$ that is not a prime power with $1\lt m \leq n$, it is possible to attach the slips in such a way that after $m$ claps, for the first time, everyone is back on their own chair.
A player sits on each chair. When you clap, each player looks at the number on the slip attached to their current chair and goes to sit on the chair with that number. Prove that for every $m$ that is not a prime power with $1\lt m \leq n$, it is possible to attach the slips in such a way that after $m$ claps, for the first time, everyone is back on their own chair.