We’re going to analyze the following position. White to move.
First off, we can note that white has the advantage due to white’s superior king position. And it’s white’s move as well. While white has the advantage, it’s not necessarily clear that white can win. But it is evident that white is playing to win in this position. Black, on the other hand can only aspire to a draw as the best possible outcome.
In order to win, white must queen a pawn. We can observe immediately that white will not be able to create a passed pawn with pawn moves only. White needs the king to create a passed pawn.
In order for white to create a passed pawn, white most likely has to capture ore or more of the black pawns to be able to queen one of the pawns. Note that white doesn’t have any means to force black’s pawns to advance. White cannot assume that the pawns will advance. In fact, in a general sense, it is not logical for black to advance the pawns because any pawn moves will place those pawns closer to the white king, thus helping white to advance white’s plans. Moving the black pawns will also reduce black’s flexibility. For example, it’s more likely that the pawns will become blocked and unable to move.
Under the assumption that black’s pawns don’t move, white wants to get the king to f7 to attack the g7 pawn, force that pawn to advance, then block and capture black’s pawns.
So it’s clear that the white king needs to move closer to black’s pawns. Black on the other hand needs to decide what to do with the king. Black can try to go counter clock-wise around the white king and attack the white pawns. But that approach seems risk for two reasons. First, should white succeed in capturing black’s pawns, black’s king will be unable to interfere with the advance of white’s pawns. Secondly, since white’s king is closer, white is in a better position to defend the pawns. The alternative is for black king to “shadow” the white king to the f7 square. Black actually might not need to commit to either strategy for the next few moves, but we’ll see.
For white to get to f7, the key diagonal is the a2-g8 diagonal. Tracing backward from f7, the white king must come from d6. To get to e6, the white king might come from d5 or e5.
However, per above, both sides have potential plans. The best move for white is the move that advances white’s plan the most, while also interfering with black’s plan the most. The fewest number of moves for white’s king to reach f7 is six moves. But white has many paths to take to get there. The moves white should select should be moves that most interfere with black’s plans.
Now it’s quite clear that the best move for white is 1. Kc2!
The white king is on the way to f7. Now black must decide which way to go. Should black go counter clock-wise around the white king with Ka1? Or should black shadow the white king with Ka3? We will assume that black shadows the white king with 1. … Ka3. This is actually quite logical, because black can still consider playing around the white king or can shadow the white king. 1. … Ka1? is less flexible since it commits to only one strategy rather than keeping both options on the table.
The game continues with 2. Kc3 Ka4 3. Kc4
And it’s pointless for black to continue to shadow the white king since the white king will always be closer to the black pawns. Black must look for a way to create counter-play rather than playing passively. So black goes counter-clockwise to also reach the king side as quickly as possible. Play continues with 3. … Ka3 4. Kd5 Kb3 5. Ke6 Kc4 6. Kf7 g5 7. g4 Kd5 8. Kg7 Ke5 9. Kxh7 Kf4 10. h3! Kg3 11. Kg6 Kf4 12. Kf6 and the pawn falls and white will win.