VN-TST 1990 Problem 6

There are $n\geq 3$ pupils standing in a circle, and always facing the teacher that stands at the centre of the circle. Each time the teacher whistles, two arbitrary pupils that stand next to each other switch their seats, while the others stands still. Find the least number $M$ such that after $M$ times of whistling, by appropriate switchings, the pupils stand in such a way that any two pupils, initially standing beside each other, will finally also stand beside each other; call these two pupils $A$ and $B$ , and if $A$ initially stands on the left side of $B$ then $A$ will finally stand on the right side of $B$ .