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1. Bxh6 Nxh6 2. Rxh6 Kxh6 3. Qh4+ Kg7 4. Qh7#
1. Bxh6 Nxh6 2. Rxh6 Rh8 3. Rxh8 ( 3. Nh5+ Kxh6) wins a pawn
1. Rxh6 Nxh6 2. Bxh6+ Kxh6 [Kh8 3. Bxf8 Rxf8 5. Qh3+ etc] 3. Qh4+ Kg7 4. Qh7#
1. Rxh6 Rh8 2. Nh5+ Kg8 3.Rxg6+ fxg6 4. Qxg6+ Kf8 5. Qf6+ Kg8 6. Qxe6+ Kh7 7. Qxf5+ Kg8 8. Qxc8+ etc.
1. Rxh6 Rh8 2. Nh5+ Kg8 3.Rxg6+ Kf8 wins two pawns
So 1. Rxh6 looks better.
1.Rxh6 Rh8 2.Nh5+ Kg8 3.Rxg6+ fxg6 4.Qxg6+ Kf8 5.Qf6+ Ke8 6.Qxh8+ Kd7 7.Qh7+ Kc6 8.Qxa7 and White is a couple of pawns up.
1.Qxf5 fxf5 (1…gxf5 2.Bxh6+ Kh8 3.Bxf8#) 2.Rg3+! Kh8 3.Bxh6 Rg8 4.Bg7+ Rxg7 5.Rh3+ Rh7 6.Rxh7#
What? I meant 1.Qxf5 gxf5 (1…exf5 2.Bxh6+ Kh8 3.Bxf8#) 2.Rg3+! Kh8 3.Bxh6 Rg8 4.Bg7+ Rxg7 5.Rh3+ Rh7 6.Rxh7#
1. Qxf5 exf5 2. Bxh6+ Kh8 3. Bxf8#
1. Qxf5 gxf5 2. Rg3+ Kh8 3. Bxh6 Rg8 4. Bg7+ Rxg7 5. Rh3+ Rh7 6. Rxh7#