This page is well-known stuff, in any good introductory book on chess.
Here is rough but useful rule about endgames with only kings and pawns (from a book I had once). We will suppose that White has one extra pawn:
With 1 pawn against 0, White wins in 50% of cases.
With 2 pawns against 1, White wins in 90% of cases.
With 3 pawns against 2, White wins in 95% of cases.
Clearly exchanging a pair of pawns must be done with care! Many people might be surprised at the low figure of 50%. The basic reason is the following:
THEOREM.
If the Black King is:
i. directly in front or 2 squares in front of the pawn, and
ii. not on the back row,
then the position is a draw, with best play.
If both conditions hold, we don't need to know the position of the White King or who is to play. (But if it IS on the back row then the status of the position does depend on this information). When the Black King is 3 squares in front of the pawn, there are also cases when the position is drawn, as we shall see.
The strategy of Black to achieve the draw, when the above conditions are met is briefly this: take the square in front of the pawn if possible, if already on this square move straight back not diagonally back, when neither of these apply move sideways to the square two in front of the White King.
To discuss this endgame further, we consider two cases.
A. The pawn is a rook's pawn. This is quite simple.
If the Black King is in front of the pawn (at any distance), the position is drawn.
There is also a draw when the White King is in front of the pawn and either cannot get out of the way, or can get out of the way only at the cost of letting the Black King into the corner.
B. The pawn is on any other file.
An important position is the following: White King in front of the pawn, Black King 2 squares in front of that, e.g.
White pawn on e4, White King on e5, Black King on e7.
This position is a draw with White to play but a win with Black to play. Moving the whole position back 1 or 2 squares, so that the pawn is now on e3 or e2. the result is same. But moving the position forward 1 square, so that the Black King is on e8, the result is now a win with either side to play. This applies to all the files b to g.
Example 1. White pawn on e4, White King on d5, Black King on e7. White, to play, must play Ke5 and can then win - moving the pawn to e5 instead gives Black a draw, according to our theorem.
Example 2. White pawn on e2, White King on d2, Black King on d6. This will be a draw with either side to play.
Suppose first that White is to play. If White moves his pawn, Black moves to e5 (1 or 2 squares in front of the pawn) with an immediate draw. So White must move his King.
But Ke3, Ke5 is a draw and so is Kd3, Kd5. And moving the White King backwards is useless, since Black moves to e5 and then to e4. with a draw.
If instead Blake is to play, he should play Ke6 and again achieves a draw, the analysis being unchanged.
Note that if Black plays Ke5 or Kd5 he loses - White plays Ke3 or Kd3 in the two cases.
With the Black King on e6 (instead of d6) we will again have a draw with either side to play. If Black is to play, he plays Kd6.
(last modified 7th January 2023)