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1. Rxh8+ Kxh82. Rc8+ Kh73. Bxg6+ Kh64. Rh8#
1. Rxh8+ kxh8 2. Rc8+ kh7 3. Bxg6+ kh6 4. Rh8#
Rxh8+ Kxh8Rc8+ Kh7Bxg6+ Kh6Rh8#
1. hxg6+ with the threat Rh1+!
1. Rxh8+ Kxh82. Rxc8+ Re83. Rxe8+ Rf84. Rxf8+ Kh75. Bxg6+ Kh66. Rh8#Black cannot avoid the forcingof the king to h7 and checkmate follows.
Rxh8+...and the rest just follows
1. R:h8+ K:h82. Rc8+ Kh7 (interpositions of black rooks delay, but do not prevent the mate)3. B:g6+ (hg also works) Kh64. Rh8#
1. Rxh8+ Kxh82. Rc8+ Kh73. Bxf6+ Kh64. Rh8#
1.Rxh8+ Kxh8 2.Rc8+ Kh7 3.hxg6+ Kh6 4.Rh8++
Isn't this a simple mate in 6?1. Rxh8+!! Kxh82. Rc8+ Re83. Rxe8+ Rf84. Rxf8+ Kh75. Bxg6+ Kh66. Rh8+ checkmate!
1. Rxh8+ Kxh8 2. Rc8+ Kh7 3. Bxg6+ (or hxg6+) Kh6 4. Rh8# Of course, if Black sacrifices both his rooks (2. ... Rd8 and 3, ... Rf8) then the checkmate will appear after 6 moves, instead of 4.
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