IMO 1962 - Problem 3

Problem: A cube $ABCDA'B'C'D'$ is given. The point $X$ is moving at a constant speed along the square $ABCD$ in the direction from $A$ to $B$. The point $Y$ is moving with the same constant speed along the square $BCC'B'$ in the direction from $B'$ to $C'$. Initially, $X$ and $Y$ start out from $A$ and $B'$ respectively. Find the locus of all the midpoints of $XY$.
Solution: By inspecting the four different stages of this periodic motion we easily obtain that the locus of the midpoints of $XY$ is the edges of $MNCQ$, where $M,N$ and $Q$ are the centers of $ABB'A'$, $BCC'B'$ and $ABCD$, respectively.