Chess is a game of geometrical patterns, which should be assimilated to train our intuition to find right moves. This is especially true for the endgame, where we usually have to blitz our moves relying more on our knowledge/intuition than on calculation. In pawn endgames, one of the most important patterns is the rule of the square, which tells us whether a pawn can reach promotion or it will stopped by the opponent's king.
In the position below, White won't win if he tries to capture Black's pawn with 1.Kc3? Kg5 2.Kd3 Kf5! diagonal opposition, 3.Ke3 Ke5 opposition. White cannot reach Black's pawn, and his extra pawn counts nothing. We should look for another solution.
Instead, the pawn break 1.b4! cxb4 2.c5 (diagram) creates a passed pawn.
White wants to be sure that one of his pawns will make it to promotion before sacrificing his b pawn. The rule of the square tells us to look at the corners of an imaginary square: c5 (the pawn), c8, f5, f8. Black's king needs 2 moves to enter the square, therefore he is lost.
There are many exercises that are longer and more complicated than this one, but they are all based on this basic rule.
Let's discuss now a study created by Richard Réti (1921). Although White moves, his position looks completely hopeless. His king is 3 moves away from the square of the h pawn (h5,h1,d5,d1), while Black kings is within the square of White's c6 pawn (c6,c8,a6,a8). However, compared to the previous example, White has an additional element that allow him to save the game; the c6 pawn, which acts as a diversion. White has to play flexible moves threatening to enter the square of the h5 pawn and to support the promotion of his own pawn.
1.Kg7 h4 2.Kf6. Black can try 2...h3 (diagram), after which White's king is too far to stop the pawn.
However, after 3.Ke6! h2 4.c7 h1=Q 5.c8=Q+ (diagram), the game is draw because White promotes with check, and Black does not have the time to play the tricky Qh3+.
It may seem that Black would have done better playing 2...Kb6 stopping White's pawn. However, White saves the game with 3.Ke5! (diagram), another flexible moves with two intentions. If Black captures the c6 pawn, White will enter the h4 pawn square, drawing.
Instead, after 3...h3 4.Kd6 h2 5.c7 h1=Q 6.c8=Q (diagram) the game is a draw because Black has no way to win White's queen with tricky checks.
It really looks like a miraculous endgame!
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