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Rh8 ..
1) If king doesn’t move Rg8 followed by Bg6 is Checkmate
2) If King takes Rh8 Pawn is Queen at f8 Check … Followed by Pawn at Bg6 .. Checkmate
1.Rh8!!!
A>1…..Kxh8.2.f8=Q+.Kh7.3.g6#!!
B>1…..Re8.2.fxe8=Q!.any move.3.Qg8#!!
C>1…..Nd7/Rf5.2.Rg8+!.Kh7.3.Bg6#!!
D>1…..any other move.2.f8=Q#!!
1.Rh8 Kxh8 2.f8=Q+ Kh7 3.g6#
1…. Rf5 2.Rg8+ Kh7 3.Bg6#
1. Rh8 forcing black to play Kxh8
2. f8=Q+ Kh7 3.g6#
1. Rh1! Nd7
(1. …Rxg5 2. f8Q#; 1. …Kxh8 2. f8Q+ Kh7 3. g6#; 1. …Re8 2. fxe8 Nd7
3. Qg8#; Rf5 2. Rg8+ Kh7 3. Bg6#)
2. Rg8+ Kh7
3. Bg6#
Rh8
1. Rh8 Rf5!? I suggest 1. Rf6 followed by the advance of pawn g in case of Kf8.
You are right, it is still mate
1. Rh8! (1. … Kxh8 2. f8Q+ Kh7 3. g6#; 1. … Rf5 2. Rg8#) 1-0