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1. Nxg6+!
>A-1….hxg6.2.Rh3+.
>>A1-2….Kg7.3.Qxg6#
>>A2-2….Qh7.3.Rxh7+.Kxh7/g8.4.Qxg6+.Kh8. 5.Qh7#
>B-1…Qxg6.2.Bxg6.
>>B1-2….hxg6.3.Rxf8+.Kg7.4.h4!.Kxf8.5.Qf3+.Kg7.6.Qxa3! and white should win easily with Q vs B+R, Q controlling black passed pawns and white’s passed pawn and black king still in a precarious position with a gaping hole in f6.
>>B2-2….Rxf3.3.Qxf3!.hxg6.4.h4! with a easy win!
>C-1…Ng7.2.Nf7++.Kh8.3.Nxg8. and wins easily with Q for a N!
The best I could find is –
1.Nxg6+ Qxg6
(1…hxg6 2.Rh3+ Qh7 (2…Kg7 3.Qxg6#) 2.Rxh7 Kxh7 3.Qxf6+ Kh8 4.Qh7#)
(1…Kg7 2. Nd7+ Kh8 (2…Kh6 3.Rh3#) 3.Nxg8 with advantage)
2.Rxf8+ Kg7 (2…Qg8 3.Rxg8#)
3.Bxg6 hxg6! (now black has a threat of Rb1#)
4.Rg8+ Kxg8
5.Qxg6+
Now white has advantage in material and should win from here.
From last Spring according to my notes. The first move literally jumps out at me, though I have to admit it isn’t clear cut, and I question I could find it easily if I didn’t know there was a tactic here to find:
1. Ng6 Qg6
If black takes with the h-pawn, it is mate since the pawn that will come to g6 is double attacked: [1. …hg6? 2.Rh3! Qh7 3.Rh7! Kh7 4.Qg6 with mate on the next move]. I will cover 1. ..Kg7 later. Continuing:
2. Bg6
I think white wins with 2.Rf8, too, but I liked this move better last Spring, and see no reason to alter that decision. Continuing:
2. ……Rf3 (what better?)
3. Qf3 hg6
4. h4
Required at some point to take care of white’s back rank. In any case, I will just refer to the previous posting of this problem from April 12, 2014 on this blog. Not a trivial win from this point, but not difficult either since black has several weak pawns and a lousy bishop.
Now, at move 1, black could have played 1. …Kg7, and I think this leads to mate:
1. Ng6 Kg7
2. Nf8!
White can win fairly easily with Ne7 here, but that ending is sort of like the one above- 2.Nf8 is just a lot more forcing:
2. ……Kh8 (Kh6 3.Rh3#)
3. Ng6! Qg6
Or [3. …hg6 4.Qh4! Qh7 (or 4. …Kg7 5.Qf6 Kh7/h6 6.Rh3#) 5.Qd8! Qg8 6.Rh3 Kg7 7.Qf6#]; or [3. …Kg7 4.Ne7! with mate on the next move regardless of where the king goes.] Continuing:
4. Bg6 hg6
I am not going to look at it, but this has to be mate if black doesn’t take the bishop. With the bishop still covering b1, white doesn’t even have to worry about Rb1. Continuing:
5. Qh4 Kg8 (Kg7 6.Qe7)
6. Qd8 Kh7/g7
7. Qe7 with mate on the next move.
1. N:g6+ for if 1…hg 2. Rh3+ mates quickly.
So black must play 1…Q:g6 allowing 2. R:f8+ Kg7 3. B:g6 hg 4. Rg8+ K:g8 5. Q:g6+ and 6. h4 with a big advantage.
One interesting line is
1.Nxg6+ Kg7
2.Nxf8+ Kh8
3.Ng6+ Kg7
4.Ne7+ Kh8
5.Qxg8#