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I guess black wins with Kc5.
A challenging endgame. A simple move count revealed the problem with the most “obvious” of the beginnings- the creation of the black passed pawn on the queen side. Black can queen first, but only on the half move before white does- a situation that is likely drawn on basic principles, but I will show that explicitly below:
1. ………..a4?
2. ba4! ba4 (b4 is no better on time, and leaves white with an unnecessary pawn)
3. d4
And one can probably now tell why black can’t win this- white will pick up a tempo with the check from d5 and queen on the half move after black does. Continuing:
3. …………a3 (nothing better now)
4. d5 Kc5
I think black draws with Kc7 here, too- white has to play Ke7 before continuing with d6+ and will still queen on the half move after black regardless. Now, is black’s position a bit better with the king closer (on c6 after d6+) to the new white queen and king? I can’t see how, and I have looked at it for a while now. Continuing:
5. d6 a2
6. d7 a1(Q)
7. d8(Q)
And this should be utterly drawn with proper play. Any attempt to exchange off the queens is bad for black since it is his pawns that are weakest in the king ending, and if black attempts to win the white h-pawn with the sequence Qe1+ followed by Qh4+, then white has a pretty simple draw by repetition:
7. …………..Qe1
8. Kf6 Qh4
9. g5 Qh3
10.Qa5 Kd4 (the only plausible route to escape the checks)
11.Qb4 Kd5! (of course, Kd3/e3 is suicide)
12.Qb5
Lots of draws here for white, this move is the most straightforward to my own eye since it lays a simple trap:
12. ………..Ke4
Of course, Kd4 just repeats moves, and Kd6 is dangerous for black because it opens the possibilities for white to check from e6 forcing the queens off the board with the black king no closer than the c-file- a pawn ending that is surely lost. Continuing:
13.Qc4 Kf3
14.Qd3 Kg2
The reason I selected this line is that it is only one that demonstrates the outcome in cases other than white just drawing by repetition. Here, 14. …..Kg2 (or Kg4 for that matter) put white to the question- can he draw/win with 15.Qxh3? The answer is, probably not- after the exchange at h3, white can win the h-pawn, but black will win the g-pawn on the next half move protecting his remaining pawn and win the game. At that point, white will have to forgo winning the h-pawn and protect the g-pawn from f6, but this will allow black’s king to take up residence on h5 putting white in zugzwang. So white must persist in checking the black king, or at least forcing the exchange of queens on the 2nd rank rather than the third. Continuing:
15.Qe2 Kg3 (no better is Qg1/h1)
16.Qe3 Kh4 (or offer repeats after Kg2/h2)
17.Qf2 Kh5 (or offer repeating moves via Kg4/Qg3 etc.)
18.Qe2
Probably drawn with other moves like Qf4, but those are risky because black is given the initiative, and there is no reason at all to do that- and I can’t see the position deeply enough to be sure it is drawn for white. Continuing:
18. ………..Qg4
19. Qh2 Qh4
20. Qe2 and the position will repeat 3 times eventually.
As Bill points out above, though lacking detail, black should win with 1. …..Kc5 since that steals a tempo from white due to where white’s king is forced to go in the most accurate line. I will cover this in the next comment.
On a first glance, a move like Kc5 might seem no different than the line I showed above- it simply allows white to play Ke5 followed by the tempo-winning move of d4+ rather than the check from d5 in the line I showed above with the same situation- black queens on the half move before white does. However, that is not the case!
1. ………….Kc5!
Obviously, black is threatening Kd4 followed by Kxd3. White’s options are limited- he can win the black h-pawn by playing 2.Kf6/3.Kg7/4.Kxh7, but then it is still six more moves to queen a king side pawn, while black still has the four move sequence immediately available with 2. …a4 to win. Also, since black still controls d6 at this point, white must wait for Kd4 in order to move on the queen side black pawns, but black then black can just play 2. …a4 if white plays a waiting move like g5 or h4 since the threat of d4+ no longer matters. So white must maintain the balance with 2.Ke5:
2. Ke5 a4 (more accurate than Kb4, but clearly winning now either way)
3. ba4
Time wise, nothing is different with d4+:
3. …………ba4!
4. d4 Kb5 (Kb6 is ok, too, though clumsier)
5. d5 a3
6. d6 a2
7. d7 a1(Q)+
And right there is why 1. ….Kc5 wins- it forces the white king onto e5 in the most tenacious defense, but the defense is an illusion that cost white a critical tempo.
Yes:
This is what I saw.