3. (CZS) A cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is given. The point $X$ is moving at a constant speed along the square $A B C D$ in the direction from $A$ to $B$. The point $Y$ is moving with the same constant speed along the square $B C C^{\prime} B^{\prime}$ in the direction from $B^{\prime}$ to $C^{\prime}$. Initially, $X$ and $Y$ start out from $A$ and $B^{\prime}$ respectively. Find the locus of all the midpoints of $X Y$.
3. By inspecting the four different stages of this periodic motion we easily obtain that the locus of the midpoints of $X Y$ is the edges of $M N C Q$, where $M, N$, and $Q$ are the centers of $A B B^{\prime} A^{\prime}, B C C^{\prime} B^{\prime}$, and $A B C D$, respectively: