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White win with 1.g6 fxg6 2.Kxe6
If black doesnt take g pawn and plays Kf8 or Nc2 someother move, then next move for white is g7.
I remember this puzzle from a few years back. It is tempting to try 1.Kf6, but the black king escapes:
1. Kf6?? Nc2
2. Ra8 Kh7
3. g6 fg6 (only move)
4. Ra7 Kh6
5. Ra8 a1(Q) with check and white loses the rook for the bishop and the game.
The right approach is 1.g6 which forces black to capture with the f-pawn one move earlier:
1. g6! fg6 (forced)
2. Ke6
And now the a-pawn is unprotected for one move and will be captured:
2. ……Nc2 (to win at e3)
3. Ra2 Ne3
4. Kf6 Ng4 (what else better?)
5. Kg6 Ne5 (again, what else?)
6. Kf5 Nf7
7. Ra8 Kh7
8. Ra7! Kg8
Here, Kxh6 9.Rxf7 is a losing endgame for black for sure. Continuing:
9. h7! Kh7
10.Rf7 will win easily.
I might be missing a more tenacious line for black in the above, but right now I don’t see one.
g6! with the threat of Ra1 mate looks winning, since fxg results in KxB and collection of the a pawn.
g6 winning
1.g6 fxg6
(1…Nc2 2.Ra8+ Bc8 3.Rxc8#)
(1…Kf8 2.Ra8+ Ke7 3.h7 f6+ 4.Kf4 Nb3 5.h8=Q Kd6 …)
2.Kxe6 Nb3
(2…Nc2 3.Rxa2 Nxe3 4.Kf6 Nc4 5.Kxg6 Ne5+ 6.Kxf5 Nc6 7.Kg5 Kh7 8.Rc2 Ne5 9.Rc7+ Kg8 10.Re7 Nf7+ 11.Kh5 Nxh6 12.Kxh6 …)
3.Rxa2 Kh7
4.Kf6 Nc5
(4…Kxh6 5.Rh2#)
5.Ra8 Ne6
6.Kxe6 Kxh6
7.Ke5 Kg5
8.Rb8 b5
9.Rxb5 Kg4
10.Rb1 Kf3
11.Kd4 Kf2
12.Rh1 f4
13.exf4 e3
14.Kd3 e2
15.Kd2 Kf3
16.Ke1