The winning plan is place the bishop on the a1-h8 diagonal. To avoid any h pawn advance. Keep an eye on your only pawn. And make waiting moves with the bishop on the long diagonal while you slowly get your king to take on g4.
Also the wrong bishop is only for the rook pawn. With any other pawn you can make a waiting move to get the oponents king out and then queen.
This is a "Metallica" endgame. Reach a target position (Black's king on f7, Black's bishop on f6), and nothing else matters.
White's king can be on f5 or anywhere else. It can even be White to move (or Black, no zugzwang is involved at this stage, even though zugzwang will be used later to invade and win the g4 pawn).
The justification for the target position is threefold : (1) The bishop on the long diagonal prevents any disruption based on the h-pawn ; (2) on f6 the bishop protects the g5-pawn and (3) it is protected by Black's king on f7. White must give ground and looses.
How to reach the target position ? There are a few pressing considerations in the initial position before you can think of any long-term goal. The exact path to the target positon depends of course on White's replies, but here is my suggested variation (where White combines three defensive attempts).
1.Kf5 Bc3 (don't let White's king into the f6 square)
2.Ke6 Kg7 (or the f7 square)
3.Kf5 Bd2 (defend the g5 pawn)
4.Ke6 Ba5 (charge !)
5.Kf5 Bd8 (just in time)
6.Ke6 Bf6 and now White must give Black access to the f7 square.
White's attempts to corner Black's king with Ke6-e7-e6 or with Kf6-f7-f6 fail. White's counterplay against the g5 pawn is accounted for. I can't see any other defensive attempt. The fight is over. Black invades and wins the g4 pawn.
As long as the bishop is on the long diagonal (ideally f6) we will be able to outflank White in a King vs King endgame since we have infinite waiting moves with the bishop. The king will walk in and slowly collect.
@A-Cielbleu @JARANDujo @kenzaburo
Thanks for your replies, appreciated.
I realized only later on that with so little amount of pieces on the board the cool Lichess tablebases can be used.
I've updated the study content with tablebases results for this position.
Interesting to see is that it shows two "only moves" for black in one variation.
Stockfish finds the win in seconds, you just have to enable infinite analysis (in the hamburger menu).
The idea that might lead you to find the correct moves here is the play against the locked g pawns. Without the bishop and extra h pawn for white, play would revolve around the key squares for outflanking the g4 pawn (by reaching d4 e4 or f4, black should win the g4 pawn) - so we can assume that a black king on those squares in our game is also winning.
If you magic the black king onto e6, there is no way for white to stop us from achieving d4, e4, or f4 and winning the g pawn - so our goal to win is there - reach e6 with the black king. This goal is short enough to do a straight calculation.
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