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After1.Re3, Rh3 mate is threatened If 1…Qg5 2.Rh3 wins Queen. If 1…Rg5, 2.g4 wins Rook.
As white, it would be really nice to be able to put a rook on the h-file. The obvious try is to play Re3:
1. Re3
Threatening Rh3#. Black can play Rg5 or Qg5 to interpose after Rh3#, of course white won’t mind getting the queen for the rook- especially since the d6 pawn will be tossed in, too. Best is definitely Rg5:
1. …..Rg5
Now the check from h3 is met by Rh5 very adequately. However, white has one killer move here:
2. g4!
Taking h5 from the rook. Now, Rg4 is met by the simple Qg4. Black has no good options- he can just give up the rook with either Rg4 or Rh5, or attack the white queen with the black queen from either f6, c8, or e8, but these are all a move too slow to save the rook:
2. …..Qc8
3. Rh3 Rh5
4. gh5 and white is up a rook.
1 Re3 forces 1…Rxg5
Then 2 g4 and the rook is lost after 3 Rh3+
1 g4, followed by Kg2, threatening Rh1#
g4 and if Rxg5 kg2 while if Qxg5 f3 and if Rxf3 then Kg2
I guess 1.g4 Rxg5 2.Kg2 wins for white.
1. g4 fails to Qxg5