Black to move and draw! Again, this is NOT a one move answer. What is the drawing plan for black?
In short, 1...Kb3 allows the black king to approach the pawns from behind, and any attempt to foil this puts the white king on the wrong side of the b1-h7 diagonal, hindering the white king from getting to the proper squares to keep the black king out of the corner.
I am not going to try to construct a fully detailed explanation of this forced draw- best for the curious to just play through the Nalimov Tablebase lines- but I will try to give a bit of insight into the beginning and show why it is the only drawing move.
It should be obvious that the bishop alone can't hold up the pawns. I will just give you this, my working line, that demonstrated this concept to me pretty early on:
1...Bg6 2.h4 Kb5 3.h5 Bc2 4.g5 Kc6 5.Ke5 Kd7 6.Kf6 Ke8 7.Kg7 and it should be clear this is winning for white- the black king cannot aid in the defense.
Now, you have to ask yourself- shouldn't black move the king first to try to save time? I tried relentlessly to make this work, but was far too dogmatic in my approach. See this working line:
1...Kb5 2.h4 Kc6 3.Ke5
You see the problem, right? It is just like the first line, but with the white pawns not as far advanced, but still clearly winning (though it took me a lot of time and effort to accept there was no draw for black, though I did try various maneuvers with the bishop after move 1).
It just looked to me that there was no way to draw this- the white king can always keep the black king from approaching while the pawns themselves are too much for the bishop. Or so I believed. Here is the right start:
I never once considered this. I know why I didn't think of it- it just seems ridiculous- if1...Kb5 is wrong because it can't get to the pawns, then Kb3 has to be even more wrong because it takes the black king "further away". However, my feeling is an illusion, and a powerful one at that. Consider this beginning line:
1...Kb3 2.h4 Kc2 3.g5 Kd2 4.h5 Ke2
And can you see the problem white has? No? Let's go a bit further in the line:
5.g6 Kf3 6.g7 Bh7 (Ba2 good too)
Study this a bit. The white king will not be able to support the g-pawn's promotion, nor will it be able to protect the h-pawn. Now, the reason this is different in this line than in the first two lines I showed is that the white king got out in front of the pawns and prevented the bishop from establishing itself on h7 while white's own pawns hinder the white king's access to the square g7. So, now you have to ask yourself why can't the white king just cut off the black king first, then support his pawns from behind or from ahead depending on how black replies. Take a look at this line:
1...Kb3 2.Ke3 Kc4 3.Kf4 Kd5 4.Kg5 Ke6 5.Kh6 Kf7 and white will not win.
Do you see that? Do you see how the squares along the b1 to h7 diagonal were never open to the white king because of the bishop. It is the first move 1...Kb3 that sets this motif in motion. The last thing I want to show is what happens if the white king, instead of trying to stop the black from reaching the pawns from behind instead tries to just dash for the g7 square immediately:
1...Kb3 2.Ke5 Kc4 3.h4 Kd3 4.g5 Ke3 5.h5 Kf3 6.Kf6 Kg4
Can you see why this is drawn? If white pushes 7.h6, then black puts the king on h5 to draw, and if white pushes 7.g6, then black just takes on h5 and uses the bishop to cover g8 from a2- also drawn.
This is an amazing postion. No shame in losing here- I don't know if any human could have solved this over the board- the idea is just so non-obvious as to be seemingly impossible to actually be the right solution.
This was a real game but the player moved Kb5 and lost.
I gave up on this one. I don't think I even thought about the right first move a single time in 2 days. I can see now why it holds, but sheesh!