White to move
A good way to begin is to make the first move with the Queen, trying to limit Black's mobility as much as possible. Thus: 1.Qc6 Kd4 2.Kd2 Already Black has only one available square.
Black to play has only one legal move
2...Ke5 3.Ke3 Kf5 4.Qd6 Kg5
Should Black play 4...Kg4 , then 5.Qg6+.
If 5...Kh5 , then 6.Kf4 and mate next move.
After 6.Qg6: the Black King is confined to four squares
King moves, and Queen mates.
Or 7...Kh4 8.Qh6 mate.
In this ending, as in the case of the Rook, the Black King must be forced to the edge of the board; only the Queen being so much more powerful than the Rook, the process is far easier and shorter. These are the three elementary endings and in all of these the principle is the same. In each case the co-operation of the King is needed. In order to force a mate without the aid of the King, at least two Rooks are needed.