The rule of the square

Step 1

Draw a square from the pawn to its promotion square. If the defending king, with the move, can step inside it, the king catches the pawn.

Step 2

Black to move. Can the king catch the pawn?

1. Yes: it is inside the square
2. No: the pawn is too fast

Step 3

And now, with the king two files further away?

1. No: it cannot reach the square in one step
2. Yes, just in time

Step 4

With the pawn's side to move, the pawn steps first, and the square shrinks by one.