Corresponding squares
Corresponding squares form a map for blocked pawn endings: each square the attacking king might use is paired with the square the defending king must occupy in reply, and missing a pair lets the attacker break through.
The opposition is a ready-made answer for kings facing each other across one square. Positions with many fixed pawns need something more detailed, because the kings can approach several targets by different routes and the right reply depends on exactly where the other king stands.
The map is built from the end. First mark the positions where a king is in zugzwang: the defender, on move, would have to let the attacker in. Those give the first pairs. Then step backward: if the attacker can reach a square from which two entry pairs are threatened, the defender needs a square from which both partners are one move away. Repeat until the pairs cover the area where the kings walk.
In play, the map turns calculation into lookup. The defender only asks whether the king can reach the partner of the attacker's square. The attacker looks for a square whose partner the defender cannot reach in time, often by taking a longer route or a triangle so that the pairing comes out with the defender on move.
The Lasker and Reichhelm study below is the classic test, because direct opposition does not help there at all. Work out the pairs once for a structure and the rest of the play becomes almost mechanical.
It sits close to Distant opposition. Distant opposition is the long-range form of the kings' standoff: the kings share a line with an odd number of empty squares between them, and the side not on move can keep that count odd until the kings meet.
If "Corresponding squares" has cost you half a point, a first lesson is a good place to find the pattern behind it: George Li goes through your games and builds a plan.
Related terms
Opposition is a standoff between the two kings on one file or rank with a single empty square between them; whoever does not have to move holds it, because the other king must give up ground.
Distant opposition is the long-range form of the kings' standoff: the kings share a line with an odd number of empty squares between them, and the side not on move can keep that count odd until the kings meet.
Zugzwang is the bind of having to move when standing still would be best: the player on move has only moves that weaken the position, and since passing is not allowed, the duty to move itself costs something.
A mutual zugzwang is a position in which whichever player has to move is worse off, so each side would gladly pass, and the result depends entirely on whose turn it is.
Key squares are the target squares for the attacking king in a pawn ending: once it reaches one while its pawn is safe, the pawn promotes by force, no matter whose move it is.
Triangulation is a way for a king to hand the move to the opponent: it makes a three-move loop over neighboring squares and returns to its starting square, while the enemy king, with less room, cannot copy the loop.
Guides that go further
What to decide first, the questions to ask, what a first lesson should show and the red flags to avoid.
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A game review, a read on your level and a written study plan from George Li.
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A first lesson starts with your games: George reviews them and builds a study plan around what he finds.