This is problem 3 in The Ultimate Chess Puzzle Book: 1001 Original Chess Puzzles to Challenge All Standards of Player, from Beginner to Grandmaster, by John Emms. Black to move.
Black mates with 1. …, Qxc1+ 2. Rxc1, Rd1+ 3. Rxd1, Rxd1#.
Suppose that it’s white to move. The following moves potentially stop the above mate: Qb3, Qc3, Qd3, Qe3, Qf3, Qg3, Qf5, Qg4, Qh6, Qh5, Qh4, Rxd8+, Re1, Rf1, Rg1, g3, g4, and Kg1.
But that doesn’t mean these are all good moves…
Qb3, Qc3, Qd3, Qf5, and Qh6 lose the queen.
Several moves lose a piece for white. For example, 1. Qg3, Qxc1+ 2. Rxc1, Rd1+ 3. Qe1, Rxe1+ 4. Rxe1, g6, and the bishop on h7 is useless. We can rule out Qg3 and Qh4.
Re1, Rf1, and Rg1 lose the rook on c8.
g3 and g4 are questionable since they weaken squares around the king’s position, open up the long diagonal, and weaken the pawn structure.
Kg1 is suspect as it also exposes the king further and gains nothing in return.
The remaining moves are Qe3, Qf3, Qg4, Qh5, and Rxd8+. How do we decide between these moves?
Consider the bishop on h7. After black plays g6, the bishop is helpless. White will want compensation for the lose of the bishop. This can come in the form of material gains, such as gaining a pawn or two in exchange for the bishop, or positional gains, such as weaknesses around black’s king, dominance of the f-file, or a weakened pawn structure.
We also need to consider the weak a- and e-pawns, the continued threats via Qxc1+. and the pressure down the long diagonal from black’s bishop, especially on the g2 square.
In other words, white has a lot to worry about in this position.
Based on all of these considerations, Qe3, Qf3, or Rxd8+ seem natural.
Qf3 looks suspicious due to black’s bishop.
There is going to be a rook exchange at some point, and so I might prefer to play Rxd8+ immediately and keep my c1 rook. For example, 1 Rxd8+, Rxd8 2. Qe3.