The first thing to notice is Black's idea of stalemate with ...h5. After, for example, 1.Kc7 c5 2.bxc5 h5, White cannot avoid a draw. So what about 1.Kc7 c5 2.b5? That's no good because the black pawn queens with check giving a comfortable draw. 1.Kb7 c5 2.b5 also fails, because the king gets in the way of the b-pawn. The first move must therefore be 1.Ka7 or 1.Ka8.
Let's try the obvious one: 1.Ka7 Kh5! (1...c5 2.b5 or 1...h5 2.Ka6 c5 3.b5 win for White) 2.Ka6 Kh4 3.Ka5 Kh5 4.Ka4 Kh4 5.Kb3! (not 5.Ka3 c5! and the pawn queens with check) and now 5...Kh5 6.Kc2 Kh4 7.Kd3 wins comfortably for White. But Black plays 5...c5! 6.b5 c4+ 7.Kc3 h5 8.Kd4 c3 with a draw.
Now look at the solution:
1.Ka8!! Kh5 2.Ka7 Kh4 3.Ka6 Kh5 4.Ka5 Kh4 5.Ka4 Kh5 6.Kb3! winning after either 6...Kh4 7.Kc2 h5 8.Kd3 c5 9.Kxe3 c4 10.Kd2 c3+ 11.Kc1 c2 12.e4! or 6...c5 7.b5 c4+ 8.Kc3 Kh4 9.b6 h5 10.Kd4! when White mates after both pawns queen.
1.Ka8!! loses a move in order to gain one. Very neat.Reuse content