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1.Rh8+
The first shot is easy!
1.Rh8+!!Kf7.
(If 1…Kxh8.2.Qh6+.Kg8.3.Qxg7#)
2.Qf4+.Ke7
(If 2…Kg6.3.Rh6+!gxh6.4.Qf6+!Kh5. 5.g4#!!.Here if 4….Kh7.5.Qg7#!)
(If 2.Ke6.3.Rxe8+ mating quickly)
3.Qe5+.Kd7.
(If 3…Kf7.4.Qxe8#!)
4.Rxe8. white is a R up and mate is around the corner!
Harry
1. Rh8+, Kf7 (otherwise, mate)
2. Qf4+, Kg6
3. Qg4+, Kf7
4. Qf5+, Ke7
5. Qe5+, Kd7
6. QxR+ or RxR (wins, can’t be bothered to analyze further!)
1.Rh8+ Kf7
2.Qf4+ Ke6
(2… Kg6 3.Rh6+ gxh6 4.Qf6+ Kh5 5.g4#)
3.Rxe8+ Kd5
[3…Kd7 4.Qf7+ Kc6 (4…Kd6 5.Re6+ Kd5 6.Rb6#)5.Qe6+ Kb5 6.Rb8+ Ka5 (6…Ka4 7.Qa2+ Ba3 8.Qxa3#) 7.Ra8+ Kb5 8.Qa6#]
4.Qe5+ Kc4
5.Qe6+ Kb5
6.Rb8+ Ka5
7.Ra8+ Kb5
8.Qa6#
Well, I couldn’t see the final mate right from the start, but I could see that on 1.Rh8+, black could not take the rook because the pin on g7 would allow Qh6+ followed by mate on the next move, but would have to play Kf7 instead- after that I think it is still mate, but in many more moves:
1. Rh8 Kf7 (Kh8 2.Qh6+-)
2. Qf4 Ke6
Of course, Ke7 still allows 3.Qe5+ and black loses the rook at e8 with check on the next move and is still going to get mated at some point, though I haven’t bothered to look at that line to its conclusion since it isn’t fundamentally different than the main line below. And on Kg6, white has a quick mate starting with 3.Rh6: [2. …Kg6 3.Rh6! gh6 4.Qf6 with mate on the next move]. Continuing:
3. Re8 Kd5
Or [3. …Kd7 4.Qf7 Kc6 5.Qe6 Kb5 6.Rb8 Ka5 (or 6. …Ka4 7.Qa2+-) 7.Ra8 Kb5 8.Qa6#]. Continuing:
4. Qe5 Kc4 (Kc6 5.Qe6+-)
5. Qe6 Kb5
6. Rb8
And mate follows as in the line in the parenthetical note after black’s 3rd move above.
Bxg7!!! Will do the winning line and mate in 4
In reply to the suggestion 1.Bxg7, it has a point, which is that Black can’t capture the Bishop. But Black doesn’t have to. For instance Black can play 1…Bd2 and a lot of Black pieces are hanging or under attack. Also, the Bishop indirectly defends against a potential Qg5+ or Qh6+ (by X-ray)
1.Bxg7?! Bd2
2. Rh8+ …. If 2.Qe2 simply Kxg7 because on 3.Qg4+ Kxh6 simply snapping off the Rook
…. Kxg7 White has no time to play RxR (Rxe8) because his Queen is under attack. And the Black Bishop being on the same diagonal as the Queen prevents Qg5+ or Qh6+
– Craigaroo