Key-Squares in Pawn Endgames

Question

In an instructional video, I have found this position:

The instructor (Dr Muller) says about key squares that are marked with green. I am unable to get the idea behind how Dr Muller determined these squares as Key Squares. I have practiced it many times in Fritz but fail to understand the exact technique that I should employ. The instruction goes rather fast and not helpful for beginners.

Please help me to proceed as white from here.

Answer 1

If the white king reaches one of the green key squares, then white will win. There is no rule determining those squares, just concrete analysis. However, once we have the key squares in place, we can systematically search for pairs of corresponding squares.

• If the white king is on d8, then the black king must move to f8.
• If the white king is on d7, then the black king must move to f7.
• If the white king is on d6, then the black king must move to f6.
• If the white king is on c8, then the black king must move to e8.
• If the white king is on c7, then the black king must move to e7.
• If the white king is on c6, then the black king must move to e6.

If at any stage black to move has the king already placed on the square corresponding to the square of the white king, black is in zugzwang and will lose.

This example turns out to be a quite simple case of (distant) opposition and 1. Kb7! wins for white.

Answer 2

The concept of key squares, in general, is defined for pawn endgames with immobile pawns. It defines a winning strategy in two steps.

1. Occupy one of the key squares with your king (usually, using zugzwang).
2. Once this has been done, the opposing king cannot defend his pawns passively (but you should computre whether it can counterattack).

A typical example is the theory of the endgame King+pawn vs King. The result of any position with this material can easily be determined using the key squares, as explained in: https://chess.stackexchange.com/a/9239/9082 .

Your example is about opposition, and thinking in the terms of key squares doesn't seem to really clarify things here. In general, an opposition should be linked to a certain "axis", and here the plan is to use the 7th row. Here is what happens if White tries to use the opposition with the 6th row as an axis:

[FEN "8/5k2/1K4p1/5p1p/5P1P/6P1/8/8 w - - 0 1"]

1. Kb7! Ke6 2. Kc6! Ke7 3. Kd5? Kd7 4.Ke5 Ke7 {and Black holds the opposition, while the white king has not reached the key squares. Draw}

The correct plan is this one:

[FEN "8/5k2/1K4p1/5p1p/5P1P/6P1/8/8 w - - 0 1"]

1. Kb7! Ke6 2. Kc6!  Ke7 3. Kc7!? Ke6 4. Kd8 Kd5 {The tentative of a counter-attack comes too late} 5. Ke7 Ke4 6. Kf6 Kf3 7. Kxg6 Kg4 8. Kf6 Kxg3 9.Kg5 {White wins}

You can find similar examples the book of Mark Dvoretsky "Endgame Manual".