Corresponding squares are an extension of the opposition concept for complex pawn endgames where simple opposition rules are not sufficient. Each critical square for one king has a corresponding square that the other king must occupy to maintain the position. This network of square relationships maps out the endgame's geometry.
The system works by numbering or lettering pairs of squares: if the white king is on square 1, the black king must be on square 1 to hold the position. If White moves to square 2, Black must reach square 2 in time. If Black cannot reach the corresponding square, White gains a decisive advantage, often through zugzwang.
Determining corresponding squares requires careful analysis of the pawn structure and available king routes. In complex positions, the network of corresponding squares can be extensive and non-intuitive. This is one of the most advanced endgame concepts, but mastering it provides deep insight into pawn endgame dynamics and the interplay between king positions.