White has a battery of queen and rook (and a pawn, too) on the g-file, but let's assume he can’t quite find a way to mate with it. So examine the Black king more closely and notice that it is completely immobilized. When a king has no flight squares the merest check may be fatal. Look for such checks, and especially for open lines leading toward the king on which you might put, say, a bishop; look for these things and see that Bd5+ almost mates. The trouble is that Black’s rook on d8 guards the mating square. What to do? Use our current logic: when a rook on the enemy’s back rank is frustrating your offensive plans up the board, consider putting one of your heavy pieces—a rook or queen—onto that same rank to threaten his king and overwork his rook. The path for such a piece can be diagonal as well as vertical. Here White does it with Qe8+. Black is forced to play RxQ. Now the coast is clear for White to play Bd5+; he mates a move later (after Black exhausts a futile interposition with his rook on e6).

At first the queen’s move to e8 may seem counterintuitive because you imagine that it is doing important work hemming in Black’s king. But most of that work is being done by the pawn on g7, and the rest of the job—the sealing of the f7 square—is done by your bishop once it moves to d5.

Now as a quick exercise, consider the consequences of starting here with 1. Bd5+. Well, that works, too: 1. …RxB; 2. Qe8+, Nf8; 3. QxN, Kh7; 4. Qh8#.