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I encountered the following puzzle on Lichess. I instinctively knew I had to play Kc4 and then it would be about counting possible moves since neither king can move with out losing a pawn but I was lost.
How should I approach this position as white?
[fen "8/5pp1/3k3p/p1pP4/P7/3K1P2/6PP/8 w - - 0 1"]
As you rightly point out Kc4 is forced for white and then whichever king has to move first loses.
It is worth quickly checking if white can abandon the d5 pawn and go for the a5 with the intention of queening the a4 pawn. You do this by counting to see who queens first.
For white it goes Kb5, Kxa5, Kb4 (to threaten the c pawn), a5, a6, a7, a8=Q - 7 moves.
For black it goes Kxd5, c4, Kd4, c3, c2, c1=Q - 6 moves and black will win the white queen after checking on the b and a files.
White can avoid this by going Kb6 instead of Kb4 but then black can miss out the move Kd4 and queen two moves before white instead of just one. Again lost for white. So the kings are locked in place.
We can see that here:
[fen "8/5pp1/3k3p/p1pP4/P7/3K1P2/6PP/8 w - - 0 1"]
1. Kc4 h5 2. Kb5 Kxd5 3. Kxa5 c4 4. Kb4 (4. Kb6 c3 5. a5 c2 6. a6 c1=Q 7. a7 Qb2+) 4...Kd4 5. a5 c3 6. a6 c2 7. a7 c1=Q 8. a8=Q Qb2+ 9. Ka4 Qa1+ 10. Kb4 Qxa8
Then it becomes a case of counting "reserve moves". Basically the number of reserve moves each side has in this position is the number pawn moves before losing something. Black has two problems here. First they have to move first and second the symmetry in the position suggests that the number of reserve moves for each side is the same.
Obviously, unless you have a tactical possibility, then you only ever want to move a pawn one square forward and never two. That would be a blunder and just lose a precious reserve move. Also, if you can move an opposed pawn past the halfway line between your pawn and your opponent's pawn you can maybe gain a move / deprive your opponent of a move.
Checking that is relatively easy. Here is what it could look like:
[fen "8/5pp1/3k3p/p1pP4/P7/3K1P2/6PP/8 w - - 0 1"]
1. Kc4 h5 2. f4 g6 3. g3 f6 4.h3 g5 5. f5 g4 6. h4
and black, as predicted, has run out of reserve moves first and so loses.
