The kernel for Black again is the bishop behind the rook, and again the bishop’s path leads toward the position of White’s king but not yet quite at White’s king. How can the king be drawn onto the bishop’s diagonal? With the same sacrificial idea considered in the previous positions: Qf2+, requiring White to play KxQ. Now what? Again, with the queen sacrificed and White’s king vulnerable, Black thinks in terms of mate; this means using the rook to add to the pressure on the king’s position by making a check of its own or cutting off some flight squares. Black could try a double check with Rd2, but trace the consequences: the king returns to the back rank with Kf1, and if Black follows him there with Rd1, the rook gets taken by White’s knight.

So what else might Black do with his rook? The key here is to be mindful of all the Black pieces bearing on the situation. After his queen is off the board, Black still will have another bishop at g4 cutting off the square above White’s king (f3) and to the side (e2); Black also has a pawn at f4 that cuts off e3 and g3 (well-advanced pawns can serve very valuable purposes in closing off squares; do not overlook them). This means that once White’s king is drawn onto f2 it will have no escape squares to the north, east, or west; the third rank and the e2 square might as well be filled with White pawns. So the really useful thing Black’s rook can do is cut off the king’s escape to the south by occupying the back rank: Rd1+ unmasks check by the bishop and leaves the king nowhere to go. White can interpose his bishop with Be3, but this just delays mate by a move (Black plays BxB#, with his bishop getting cover from the f4 pawn).

Here’s another way to have seen the idea. As Black you notice that your queen and bishop both are aimed at f2, adjacent to White’s king. You might naturally consider Rd1+; this requires a reply from White (NxR) to save his king; and in the meantime the bishop has now been unmasked and is ready to support mate with Qf2. But then you see it doesn’t quite work because once White’s knight is on d1 it controls f2. So you play with the move order and see that if you start with Qf2 it works after all.