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h6 wins
h6
Looks as if 1 h6 does the trick:
1 … Rxf4 2 hxg7+ Kg8 3 g8=Q+ and now:
3 … Rxg8 4 Rh1+ Rh4 5 Rxh4+ Qh5 6 Rh5#
3 … Kh6 4 Rh1+ Rh4 5 Rxh4+ Qh5 6 Qg7#
1 … Bf8 2 Qxf6+ Bg7 3 Qxg7#
1 … Bxh6 2 Qxh6 Qf8 (or Rg8) 3 Qf6+ followed by 4 Qxg7#
1 … Qg5 2 Rxg5 and black still can’t take the queen because of 2 … Rxf4 3 hxg7+ Kg8 4 g8=Q+. Instead, 2 … Bxh6 3 Qxf6+ Bg7 4 Qxg7#. 2 … fxg5 3 hxg7+ Kg8 4 Qxd4 and black has lost a Queen and Bishop.
If 1 … f5, white can take the bishop, or play 2 Qg5 which forces the exchange of queens via 2 … Qe5 3 Qxg7+
–gb
1.h6…RxQ
2.hxg7+…Kg8
3.Bxh7+…Kxh7
4.Rh1+ mates in three.
1.h6!
1.h6! is murderous, e.g. 1…Rxf4 2.hxg7+ Kg8 3.Bxh7+! Kxh7 4.Rh1+ forces mate.
Took me a while but with the queen attacked 1.h6! was not that obvious (at least to me). A possible continuation would be 1…Rxf4 2.hxg7+ Kg8 3.Bxh7+ Kxh7 4.Rh1+ and white wins in style. Very nice!
Beelze
h6/hg/Bh7+/Rh1+
1. Ba6 not only discovers an additional attacker of the rook on d4, it also attacks the rook on c8. If Black moves the rook down the c file, the pin on the White c pawn will be maintained, but the rook on d4 will have two attackers vs. one defender. On the other hand, if Black moves the rook to d8, that does add a defender, but now the White c pawn can be counted as an additional attacker. Either way, White will net a rook.
1.h6! seems to be very strong, when 1…Rxf4 2.hxg7+ Kg8 3.Bxh7+, followed by 4.Rh1+ (or 4.g8=Q+, first) is winning for white.
h6 with threat hg7+ , bh7+ and rh1+
if black doesn’t capture queen loses too much material
1.h6 if Rxf4
2.hxg7 Kg8
3.Bxh7 Kxh7
4. Rh1 followed by mate on h8.
What about something like
1.h6 Rxf4
2.hxg7+ Kg8
3.Bxh7+ Kxh7
4.Rh1+ Kg8
5.Rh8 mate
(When I started the problem, no solutions had been posted; now I’m only repeating what others have said, but I spent some time on it, so…)
What would Black do with the simple 1. h6?
1. … Bf8
2. Qxf6 & goodnight Irene.
1. … Rxf4
2. hxg7+ Kg8
3. Bxh7+ Kxh7
4. Rh1+ Rh4
5. Rxh4+ Qh5
6. Rxh5#
1. … Rxd3
2. Rxd3 Qe5
3. hxg7+ Kg8
4. Qd2 Qb8
White should easily convert the advantage.
Incidentally, another way to finish up is 1.h6! Rxf4 2.hxg7+ Kg8 3.Bxh7+! Kxh7 4.g8(Q)+!, also forcing mate.