White to move! How should white proceed?
a) 1...hxg6, 2. Qxg6+ Qf6, 3. Qxf6+ Kd5, 4. Qc6#
b) 1...Kf7, 2. Qxf5+ Qf6, 3. Qxf6#
c) 1...Qf6, 2. Rxf6+ Ke7 (2...Ke5, 3. Qxf5#), 3. Qf7#
1.Rg6 should do it. Mate in 4 is the longest defense I can see here, but only by sacrificing material to block checks. 1.Rg6 hxg6 2.Qg6 Qf6 (Kd5 3.Qc6#) 3.Qxf6 with mate on the next move. Every other move by black on move 1 is a mate in 3.
1.Bxb2 Rxg3 2. Bc4 Kd7 3.c6