There are $n$ ants randomly distributed on a ring facing clockwise or anticlockwise. Each ant is carrying a flag numbered $1,\ldots,n$. They all start moving around the ring at the same speed (in the direction they were facing).
When they meet they swap flags and turn around and continue walking in the direction they came from (still at the same speed). Effectively a perfect bounce of the ants.
Find a configuration where the initial position is never recovered, or prove that the initial position will always be recovered.