If White tries the most natural move - closing in with his queen with 1.Qf3, then 1...e6 would allow mate with 2.Qe3+ Kd5 3.Qd3, but there is no mate within the specified distance after 1...e4. The other closing- in move, 1.Kf5, would provide an instant mate after 1...e4, but fails after 1...e6+.
Back with 1.Qf3 e4, one might notice that the illegal 2.Qd7+ would do the trick: 2...Ke5 3.Rc5 is mate. That should suggest the key move 1.Qh3! (There's also a hint in the position of the pawn on c6. If it is to play a part in the solution, the black king must, in one variation at least, head backwards.)
So let's fill in the details: 1.Qh3 e4 2.Qd7+ Ke5 3.Rc5; or 1.Qh3 Ke4 2.Rc5! Kd4 3.Qe3; or 1.Qh3 e6 2.Qe3+ Kd5 3.Qd3; or 1.Qh3 Kd5 2.Qd3+ Ke6 3.Qd7 (that's where the c-pawn comes in).
Now do you see the point? White's moves are a perfect cyclic permutation of
Qd7, Rc5, Qe3 and Qd3. A fine achievement with only nine pieces.Reuse content