The PMO Magician has a special party game. There are n chairs, labelled 1 to n. There are n sheets of paper, labelled 1 to n.[*] On each chair, she attaches exactly one sheet whose number does not match the number on the chair.
[*] She then asks n party guests to sit on the chairs so that each chair has exactly one occupant.
[*] Whenever she claps her hands, each guest looks at the number on the sheet attached to their current chair, and moves to the chair labelled with that number.Show that if 1<m≤n, where m is not a prime power, it is always possible for the PMO Magician to choose which sheet to attach to each chair so that everyone returns to their original seats after exactly m claps. number theorycombinatoricspermutationsPMO