This is a very good exercise in tempo-counting, which is something I have to write down to get.

We can see that W's goal is to clean up the queenside and get to b7 before B can take his kingside pawn and get to c8. B has to lose 3 tempi (b5, bxa4 and b3). W has to lose two (bxa3 and the initial capture on g5). It will take B a minimum of 6 K tempi to capture a pawn on the g file and get to c8. Two more if the pawn is on the h file. It will take W a minimum of 7 K tempi to get from the f file to the b7 after cleaning up. 8 if he starts from g5. Arithmetically, hxg5 results in W needing 7+2 = 9 tempi and B 6+3=9 tempi. Similarly, Kxg5 results in W needing 8+2 = 10 tempi and B 8+3=11. Naively, either should therefore result in a W win (though Kg5 gives more margin)

Let's look at hxg5 and see if the arithmetic adds up

1. hxg5 b5 2. g6 bxa4 3. g7 Kf7 (can't ignore it any more) 4. Ke5 a3 5. bxa3 Kxg7 6. Kd6 Kf7 7. Kxc6 Ke7 8.Kb5 Kd7 9. Kxa5 Kc8 B got to c8 in 9 tempi. W is two tempi away from b7, having lost two pawn tempi

So lets see if we can do better by not moving the pawn,

1. hxg5 b5 2. Ke4 bxa4 3.Kd4 Kf5 4.Kc5 Kxg5 5.Kxc6 a3 6.bxa3 Kf6 7. Kb5 Ke7 8. Kxa5 Kd8. 9. Kb6 Kc8. We are one short to grab b7. How did that happen? W should've got there in 9 according to the arithmetic, but W lost a tempo with the (forced) K-d4-c5-c6 path.

Compare to Yancey's line above

1. Kxg5 Kf7 2. Kf5 b5 3.Ke5 bxa4 4. Kd6 Kg6 5.Kxc6 Kh5 6.Kb5 a3 7.bxa3 Kxh4 8.Kxa5 Kg5 9. Kb6 Kf6 10. Kc7 Ke6. W is on schedule, but now B has lost a further tempo with the Kh5-h4 path

Can black do better? If he tries to deny W the d6 square, perhaps?

1. Kxg5 Kf7 2. Kf5 b5 3.Ke5 Ke7 4.h5 bxa4 5. h6 Kf7 6.Kd6 Kg6 7. Kxc6 Kxh6 8. Kb5 b3 9.axb3 Kg7 10. Kxa5 Kf8 11. Kb6 Ke8 12. Kb7 Kd8. W loses one tempo due to the pawn move, but Ke7-f7 loses two tempi for B.

Now if I could do this mentally, I'd actually be able to play these on the board :-)

The problem for white is going to be black exchanging down on the queen side with b5, leaving white with, at best a rook's pawn on the a-file or creating his own passed pawn. First principles here would suggest white needs to drive the black king as far away as possible from the queen side corner at a8, so this means white should capture at g5 with the king, not the pawn- it threatens to queen the h-pawn if the black king doesn't respond:

1.Kxg5

White is now threatening to play 2.Kg6 followed by a queen on move 6. Black doesn't have time to creat a passed pawn on the queenside- for example

1.Kg5 b5 2.Kg6 ba4 3.h5 a3 4.ba3 c5 5.h6 c4 6.h7 c3 7.h8(Q) wins.

So, after 1.Kg5, black must try to stop the h-pawn:

1.Kg5 Kf6 2.Kf5

There are probably lots of winning moves for white at move 2, but 2.Kf5 seems simplest to me- white is setting the king on the journey to the queenside- the pawn on h4 is just fine- the black king can't leave it and must go capture it, taking him far from the a8 corner. However, I can see one last trick in this position.....

2......b5! 3.Ke5 ba4 4.Kd6 Kg6 5.Kc6 Kh5 6.Kb5 a3 7.ba3 Kh4 8.Ka5 wins

And I am pretty sure that 3.axb5 is a mistake that costs the win- I don't see how white prevents black from exchanging off both white pawns after that point.