Question 17
There are $n$ markers, each with one side white and the other side black, aligned in a row so that their white sides are all up in the beginning. In each step, if possible, we choose a marker with the white side up (but not the one of outermost markers), remove it, and reverse the closest marker to the left and the closest marker to the right of it. Prove that one can achieve the state with only two markers remaining if and only if $(n-1)$ is not divisible by $3$. (Short-list, IMO-2005)