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It took me a little more than 10 seconds to see the general idea, and a minute or so to verify all the lines.
1. Qe8!
Due to the threat of 2. Rh3#, Black must keep his Queen on the h3-c8 diagonal. But no matter what Black does, White will either force the Queen off the diagonal or win decisive material (with mate quickly following)
If 1. … Qd7 2. Qxd7 Rxd7 3 Rh3#
If 1. … Rf6 2. Qxe6 Rxe6 3 Rh3#
If 1. … Re2 2. Qxe2 Rxe2 3 Rh3# (2… Qc1,Qh4#)
If 1. … Qf5 2. Rg5 Qxg5 3 fxg5+ Kh5 4. Qxf6
Q-e8
How about 1. Qe8 to deflect the queen from guarding h3?
Qe1 is black’s kiss of death
1.Qe8 is a pretty move!
1.Rg8!! followed by Re8 will put pressure on the weak pawn on e4.
1. Qd8 Qf5 (else 2. Rh3#)
2. Qxf7
1-0
taking the queen leads to Rh3#
not taking it means black is a rook down, and the same mate threat harassment persists (along with Qg7+ and other mates).
Qe8 with the idea of Ra3 mating immediately suggests itself. For example if Re7 then Qxe7 or if Qf5 then Qxf7. etc.
Queen e8
After 1- Qe8; Qf5 (black’s best reply as far as I can determine), Qxf7 is just not sufficient for white (2… Qxf7 3-Rh3+ Qh5 4-Rxh5+; Kxh5, the threats are defused and material is almost equal).
So here’s my line:
1- Qe8; Qf5
2- Qg8! and the black queen is now overburdened, having to defend against two different mate threats (Rh3 and Qg5)
I’ll admit, it took me quite a bit longer than 10 seconds to get there.