Question 29
Let $n \ge 5$ be an integer. Consider $n$ distinct points in the plane, each coloured either white or black. For each positive integer $1 \le k \lt \frac{n}{2}$, a $k$-move consists in selecting $k$ points and reversing their colours. Find all values of $n$ for which, for any eligible $k$ and for any initial colouring, there exists a sequence of $k$-moves that turns all points into a same colour.