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1. Bxh7+ Kxh7
(If 1. . . Kf7 g6-g7 mate)
2. Qh5+ Kg8
3. g6-g7 and Black cannot prevent
4. Qh7 mate
If 1. . . Kh8
2. Qh5 g7-g6
3. Bxg6+ Kg8
4. Qh7 mate
If Black plays 2. . Qf7 it does not change anything
3. Bg6+ Kg8
4. Qh7 mate
Why not Qh5? It’s mate in 3
1. Bh7+ Kh8 2. Bg6!, 1. Bh7+ Kh7 2. Qh5 Kg8 3. g6
One correction:
1.Bxh7+
If 1. . . Kh8
2.g6-g7 and Black cannot prevent mate
3. Qh5
4. Bg8+
There are other possible moves (N (d)-e5) which can delay the mate, but not stop it.
better
1 bh7+ kh8
2 Bg6 and mates or win heaps
if 2 g6 then black has a move
ie Rf7 and black lives
Most of the combinations are mate but the first move has to be 1.Bxh7+!.Then,if black tries 1…Kf7,then 2.g6 is mate.If black tries to capture the bishop,(with 1…Kxh7)then 2.Qh5+ Kg8 3.g6!+- and black can’t stop mate which is
4.Qh7 mate.If black tries the defence 1…Kh8,then 2.Bg6!(also 2.g6! will win because if 2…Rf7 then 3.Qh5 and black cannot prevent 4.Bg8+! followed by Qh7 mate.)and black cannot stop 3.Qh5 followed by 4.Qh7 mate.