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Clear the h-file with force:
1. Be6
Threatens Qf7#. Black is forced to either take the bishop, or guard f7. Continuing:
1. …..Qc7 (alternatives later)
2. Bf7!
I think Rh8 is a mate, too, but I got bogged down in every line in which white doesn’t capture at f7 on the third move, and it became clear that the path is shorter if white takes immediately. The other move that I had in mind is Qh4 (actually is the move I thought of from the start for the follow to Be6, but after black defends with Bg7, I couldn’t find a clear cut victory. Continuing:
2. ….Qf7 (Kg7 3.Qf6#)
3. Rh8 Kh8 (Kg7 4.Rdh1 Qf4 5.R1h7)
4. Qf7 with mate to follow in some moves depending on how much material black throws away.
Back at move 1, it doesn’t matter what black guards f7 with, though I can’t find a quick mate with 1. …Rd7 since the queen can come to f8 to guard h6:
1. Be6 Rd7
2. Bf7 Rf7
3. Rh8 Kh8 (mate otherwise)
4. Qf7 Bh6 (mate otherwise)
5. Rh1 Qf8
6. Rh6 Qh6
7. gh6 Rg8
And, I am sure there is a mate line in here for white, but it is going to be several moves reaching it, with lots of side variations.
Finally, at move 1, black cannot take the bishop at e6:
1. Be6 fe6
2. Rh8 Kh8 (or get mated)
3. Qf7 with mate coming after black is through giving up material.
1. Be6 wins. If … fe6:, then 2. Rh8+ Kh8: 3. Qf7 and mates.
Really hard captchas today. Maybe my eyes are tired.
mb Be6 and then Bf7 and Rh8…smt should be there
The h-file and the black king position speaks strongly for a bishop move. An enforcing bishop move is Be6+, so my first idea here is:
1. Be6+ fxe6
2. Qf6 Bg7 (only playable move)
Now the main question would be: How does white gain tempo to get Rd1 into h-file?
3. Qxe6+! (not Qxg6) Kf8 (only)
4. Rh7! Rd7
5. Rh1!
I don’t see the defence against Rh8+ Bxh8 Rh8#.
Or:
1. Be6+ Rd7
2. Qf6 Bg7
3. Qxg6 fxe6
4. dxe6
with the plan Qh7+ etc.?