Figure[Black to move]

White has a loose rook on g5. Black’s queen has no checks, and the mating threat Qd5 doesn’t work because White's rook guards the square (as well as g2). But there are other ways one can threaten mate. Study the constraints on White’s king. It is blocked by its own pawns. Qe7 thus attacks the loose rook and threatens mate on the back rank with Qe1. But this is another of those cases where you have to ask whether White might save the target and defuse the mate threat at the same time. The rook can’t do this by moving, but White does have the simple resource of h2-h4, guarding the rook and giving the king an escape square. Yet as we saw most recently in the previous problem, this sort of possibility should not discourage you. Ask what check would then be possible, and with what consequences. Black could play Qe1+, which would force Kh2. And then what check could Black offer? Qxh4—checking the king and again attacking the rook, and this time winning it. It's all about persistence with checks.