The Power Of Two Squares

HERE is a handy way to judge many king-and-pawn versus king endings.
If the pawn (I will call it a white pawn for convenience) is in its own half of the board, White wins if the white king can occupy any point two squares ahead of the pawn and either on its file on an adjacent file (assuming the black king is not able to simply capture the pawn).
That is quite a mouthful, but here is a simple example drawn from Amateur To IM - Proven Ideas And Training Methods by Jonathan Hawkins (Mongoose Press 2012).

White to play - win or draw?

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1.Kd2
1.Kf2 also wins.
1...Ke7 2.Ke3 Ke6 3.Ke4
The white king reaches a point two squares ahead of its pawn. We know from the rule above that 3.Kf4 and 3.Kd4 also win.
3...Kd6 4.Kf5 Ke7 5.Ke5 Kd7 6.Kf6 Kc6 7.e4 Kd7 8.e5 Ke8 9.Ke6
Not 9.e6?? Kf8 10.e7+ Ke8 11.Ke6 - stalemate.
9...Kd8 10.Kf7 1-0

But note that if it were Black to move in the above diagram, the same rule shows the position is drawn:
1...Ke7 2.Kd2 Ke6 3.Ke3 Ke5
The white king cannot occupy d4, e4 or f4, and so the position is drawn.

If the pawn is in the enemy half of the board, White wins if the white king can occupy any one of two sets of squares ahead of the pawn.

The white pawn has crossed the halfway line

With a white pawn on e5, for example, the two sets of winning squares for White are d7, e7 and f7 (two squares ahead of the pawn, just as if the pawn were in its own half the board) and d6, e6 and f6.
If the white king can get to any of those six squares, White wins.
Here is an example:

White to play and win

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1.Kf6 Kf8
Black gets the opposition, but it does him no good as White takes the opposition away with his next move.
2.e6 Ke8 3.e7 Kd7 4.Kf7 1-0

If it were Black to move in the above diagram, Black would draw with 1...Kf7 or 1...Ke7.

EXCEPTIONS
It seems every rule in chess has its exceptions, and these pawn-and-king versus king endings are, well, no exception.
With a rook's pawn, there are just two key squares.
If the black king is in the vicinity, ie can reach the pawn before the pawn reaches the eighth rank, White only wins if the white king can get to the seventh or eighth rank on the adjacent knight's file.

Exceptional rook's pawns

So with an h pawn, the two winning squares are g7 and g8, and with an a pawn the two winning squares are b7 and b8.