Start by looking at the captures you have the power to make and ask what stops them from working. Here you would like to play QxQ but are stopped by Black’s knight on f5. You can’t take the knight, but maybe you can experiment with ways of overworking it since you have other threats pending against Black’s king. Building a mate threat often is a matter of finding a square next to the enemy king that already is under attack by one of your pieces and then adding another attacker to it. To use the principle to overwork an enemy piece, try planting that second attacker on a square that the enemy piece protects. Here you see that you already have a rook aimed at h7, so you add another with Rh4—a square Black’s knight defends. Black can't sit still for this; he has to take one of White’s rooks to prevent mate next move with Rxh7#. His knight is his only piece that attacks either rook, so his best move is to play NxRg7 (we will consider his other options in a moment). Now you have QxQ, winning a queen for a rook.

That’s the sequence as it goes with best play by Black. But a move like 1. Rh4 does not force your opponent’s reply in the strong sense we see with checks or (sometimes) captures. Black has a couple of other ways he could respond to it, and they all need to be seen and considered:

(a) Black could play 1. …h7-h5. No problem; for of course you also see that you have the kernel of a discovery—indeed, a discovered check—with your rook and queen. Thus 2. Rxg6+, and there is nothing Black can do to stop you from mating: 2. …QxQ, Rxh5+; 3. Nh6 (a futile interposition), RhxN#.

(b) Black could play h7-h6. This seems better because Black's pawn has protection from its knight. So now your discovered check doesn’t work; Rxg6+ again is met with 2. …QxQ, and this time it won't work to give check with your rook on the h-file because Black replies NxR. You would have to simply recapture RxQ—and then Black has NxRh4. But you can do better in reply to 1. …h7-h6: play 2. Qf7, and Black cannot escape mate (e.g., if 2. …NxRg7, 3. Rxh6#). The best Black can do is throw checks at White’s king until he is out of them.

(c) Speaking of checks, Black can reply to 1. Rh4 with Ng3+. It doesn’t work. You come back with 2. Kh2. This looks a little scary because you are creating the kernel of a discovered check with Black’s knight and queen. But Black can't take advantage. If he tries Nf5+, you still play 3. QxQ, NxQ; 4. Rxh7#.

The main line of analysis in this position is another example of how an overworked piece can be exploited two ways. If it protects a piece and a square needed to mate, the first thought may be to take the piece; then again, it sometimes works better to play the mating attack, allow it to fail, and then take the enemy piece left vulnerable by those defensive efforts.