Endgames with just kings and pawns are fundamental to an understanding of the endgame as a whole. When there are other pieces still on the board, there is always the possibility that those pieces may be exchanged, leading to a pawn endgame. For that reason, not only is it important to know how to win a pawn endgame, it is also important to know when is a good or bad idea to initiate exchanges leading to a pawn endgame.
Sometimes pawn endgames are easy - if the enemy king is too far away to catch your pawn, it can run all the way to the end of the board. A quick way to tell if the enemy king can catch your pawn is to use a method known as 'the square'. Picture an imaginary square running from the pawn to the end of the board, and an equal number of squares to the side. If the enemy king can step into the square, then he can catch the pawn. If it's not clear, take a look at the board below:
If you have a single pawn against a bare king, whether or not you can get the pawn to the end of the board will depend on whether or not you can get the enemy king out of the way. Simply pushing the pawn can often lead to stalemate, as in the example below:
The correct technique is to use your king to force the enemy king to give ground, by gaining the opposition. The opposition is when two kings stand face to face with one square between them. The king whose turn it is to move must break the opposition and move away, allowing his rival to advance. Here is the same example as above, with white using the opposition to drive the black king out of the pawn's path:
Although the side with the extra pawn has good chances of winning using the technique above, there is an exception: If the pawn is a rook pawn - an a-pawn or h-pawn, which start in front of the rooks at the edge of the board - then there is no way to force the enemy king out from his blockade, and the game is usually drawn.
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