The problem by Yevgeny Bogdanov combines elegance of construction with difficulty of solution. It is White to play and mate in three, and even if you find the answer you could well miss the composer's point.

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.