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Another repeat, I think:
1. Bg6! Kg7
Every thing else is a shorter mate. On Kg8 or Kh8, white plays Qh6 and will mate quickly, and if black takes at g6 white plays Rg6 and nothing can prevent Qh6#.
2. Bh7!!
The hardest move to find. One wants to attack h6 with the rook, but white doesn’t have time to do this if he tries 2.Rh3 since black has time to play Rh8 to guard it. With 2.Bh7, white clears the bishop out of the way with check, and forces the black king to occupy h7 and hindering his own defense:
2. …..Kh7 (Kh8 3.Qh6 mates in 1)
3. Rh3 and nothing can prevent white from playing Qh6 and mating black since the f8 escape square will be covered by the white queen from h6.
Well, if this was a real game, I’ll be a chicken a take a draw:
1. Bg6+ Kg7
2. Bh5+ Kh7
etc
black will lose if
1. Bg6+ fxg6
2. Rxg6
and 3. Qh6#
———–
but since this is a puzzle, there must be a better solution 🙂
1. fxe6
if 1… exf6
2. Rg6 wins
if 1… Rg8
2. Qd3+ Kh8
3. Qf3 Rxg3
4. Qxh6+ Kg8
5. Bxf7#
if instead of 3… Rxg3,
3… Kh7
4. Qxf6 Rxg3
(4…. a2, 5. Qxf7+ Kh8 6. Qf6+ Kh7 7. Bg6+ wins)
5. hxg3 Rg8
(5… a2 6. Qxf7+ Kh8 7. Qf6+ Kg8 8. Bf7+ Kh7 9. Bg6+ Kg8 10. Qf7+ wins)
6. Qxf7+ Kh8
7. Qf6+ Kh7
8. e7 wins
e.g., 8… a2, 9. Qf7+ Kh8 10. Qxg8+ Kxg8 11. e8(Q)+ Kg7 12. Qf7+ Kh8 13. Bg6
—–
Lines that don’t work:
if 4. exf7 instead of 4. Qxf6 above:
3… Kh7
4. exf7 Rxg3
5. hxg3 a2
6. Nxa2 Qxa2
7. Qxf6 Qc1+
8. Kh2 Qg5
9. Qf3 Rf8
10. g4
seems to be good for black
——-
in the first line, if white doesn’t repeat position with 2. Bh5+, it appears very risky:
1. Bg6+ Kg7
2. Bxf7+ Kxf7
3. fxe6+ Ke8
4. Qd3 a2
5. Qg6+ Kd8
and black seems to be winning
1. Bxf7, threatening 2. Rg6 is a little slow:
1. Bxf7 a2
2. Rg6 a1(Q)+
Bg6 Kg7
Bh7 Kh7
Rh3
there was an error in the previous analysis. 4. Qxh6+ is not a legal move after 3… Rxg3. So:
1. fxe6
if 1… exf6
2. Rg6 wins
if 1… Rg8
2. Qd3+ Kh8
3. Qd4 Rxg3
4. hxg3 Kg7
5. Qg4+ Kh8
6. Bxf7 wins
or 5… Kf8
6. exf7 Ke7
7. Qe2+ Kd6 (else 8. Qe8+)
8. Nb5+ Kc5
8. Nxa3 Qxa3
9. Qe7+ K any
10. Qxa3 Rxa3
11. f8(Q)
wins
8…. Kc6 (instead of Kc5)
9. Nd4+ Kd6
10. Nf5+ Kc7/c5
11. Qe7+
black’s a-pawn is still dangerous, but the king is vulnerable.
if the king goes to b5 or c4, then Be2+
there is probably still some error here, who knows.
1. Bxf7 (Followed by Rg6+-)
1…Rg8 2. Bg6+ (with the idea of 2…Rxg6? 3. Rxg6 +- threatening Qh6#).
2…Kg7 3. fxe6 (with the idea of 4. Bf7+ Kh7 5. Qd3+ Kh8 6. Qd4 Rxg3 7. hxg3 Rg8 8. Qf6+ Rg7 9. Qh6+ Rh7 10. Qf8#).
3…Kf8 4. Qxh6+ Ke7 5. Qh7+ Kd6 (5…Kxe6 6. Re3+ Kd6 7. Qe7+ Kc6 8. Re6#)
6.Qd7+ Kc5 7. Qc7 Bc6 (7…Kd4 8. Rd3#; 7…Kb4 8. Rf4+ d4 9. Rxd4#)
8. Qe7+ Kd4 7. Rd3+ Ke5 8. Qc7+ Kxe6 9. Re3#.
and yes you are right if you found the flow: 8…Kc4! but I guess white should be able to catch the king soon. I don’t have an engine at the moment. I should make my mind sharper to see til the end. :))
Bg6+ Kg7
Bh7+ Kxh7
Rh3 h5
Rxh5+ Kg7
Qh6+ Kg8
Qh7#
1.Bg6+ Kg7 (Not fxg6 2.Rxg6 followed by 3.Qxh6#)
2.Bh7+ Kxh7
3.Rh3 h5
4.Rxh5+ Kg8
5.Qh6 Qxc3
6.Qh8#