Corresponding squares

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This article uses algebraic notation to describe chess moves.

Corresponding squares in chess are squares of reciprocal (or mutual) zugzwang. They occur most often in king and pawn endgames, especially with triangulation, opposition, and mined squares (see Zugzwang). A square that White can move to corresponds to a square that Black can move to. If one player moves to such a square, the opponent moves to the corresponding square to put the opponent in zugzwang (Dvoretsky 2006:15-20).

[edit] A simple example

Numbered squares are corresponding squares in king and pawn versus king. Squares marked "x" are key squares.

One of the simplest and most important uses of corresponding squares is in this king and pawn versus king endgame. Assume that the black king is in front of the pawn and the white king is behind or to the side of the pawn. The black king is trying to block the white pawn and the white king is supporting its pawn. If the white king gets to any of the key squares, he wins. Suppose the black king moves to the square labeled "1" near him (square c8). Then if the white king moves to the corresponding square (also labeled "1", square c6), he wins. Conversely, if the white king moves to the "1" square then the black king must move to the corresponding square to draw. Thus if both kings are on the "1" squares, the position is a reciprocal zugzwang. Note that the second player moving to one of the corresponding squares has the advantage. Being on a square when the opponent is not on the corresponding square is a disadvantage.

The squares labeled "2" are similar corresponding squares. If the white king is on the square labeled "3", he is threatening to move to either the "1" square or the "2" square. If the white king is on the d5 square (the middle one labeled "3", then the black king must be in a position to move to either his "1" square or his "2" square in order to hold the draw, therefore he must be on one of his "3" squares. This makes the defense for Black clear: shift between the squares labeled "3" until the white king moves to his "1" or "2" square, and then go to the corresponding square, gaining the opposition. If the black king moves to the "1" or "2" squares under any other circumstances, the white king moves to the corresponding square, takes the opposition, the black king moves, and White advances the pawn and will promote it and win, with a basic checkmate.

The c5 and e5 squares can also be label "3" squares, since if the white king is on one of them, the black king must be on one of his "3" squares.

[edit] A second example

Rösch-Mast 1995, White to move, but drawn with either side to move

This is another example that is fairly simple. The key squares (see king and pawn versus king endgame) are e1, e2, e3, and f3. If the black king gets to any of those squares, Black wins. The job of the white king is to keep the black king off those squares. One might think that Black has the advantage, since he has the opposition. White can defend the two key squares of e3 and f3 by oscillating. White's defense is simple if he observes the corresponding squares:

• 1. Kf2! (keeping the black king off e3 and f3)
• 1... Kd3
• 2. Kf3! moving to the corresponding square
• 2... Kd2
• 3. Kf2! Kd1
• 4. Kf1!

Each time the black king moves to a numbered square, the white king moves to the corresponding square (Müller & Lamprecht 2000:191).

[edit] An example with separated key squares

(e1 is a "5" for White) White to move wins, Black to move draws. A study by Nikolay Grigoriev, 1924.

In this position, the squares marked with "x" are key squares and the e1 square is a "5" for White. If White occupies any of the key squares, he wins. With separated key squares, the shortest path connecting them is significant. If White is to move in this position, he wins by seizing a key square by moving to e2 or f2. If Black is to move, he draws by moving to his "5" square. Black maintains the draw by always moving to the square corresponding to the one occupied by the white king (Müller & Lamprecht 2000:188-89).

[edit] An example with triangulation

The white king is on one of his "1" squares, the black king is on his "1" square. Key squares are e2, e3, and d4, marked with "x" except for e3. The corresponding squares help show White's winning process. Study by Grigoriev.

In this position, e2, e3, and d4 are key squares. If the white king can reach any of them, White wins. Black can't move out of the square of White's d-pawn (see king and pawn versus king endgame). The square c3 is adjacent to d4 and the "1" square the White king is on, so it is numbered "2". Therefore e3 is "2" for Black. White threatens to move to c3, so this is labeled "3". Since Black must be able to move to "1" and "2", f4 is his corresponding "3" square. If the White king is on b2 or b3, he is threatening to move to "2" or to "3", so those are also "1" squares for him. White has more corresponding squares, so he can out-maneuver Black to win (Müller & Lamprecht 2000:189).

• 1. Kc2 Kf4
• 2. Kb3 Kf3
• 3. Kb2 Kf4 The black king must leave his "1" square, and has no corresponding "1" square to which to move.
• 4. Kc2! Kf3 The black king is on "3", so he can't move to "3". White has used triangulation.
• 5. Kd2 Back to the starting position, but with Black to move.
• 5... Kf4 Black is on his "1" square, so can't move to a "1" square.
• 6. Ke2!

White occupies a key square and can support the advance of his pawn until he is able to win the black pawn, e.g.: 6... Kf5 7. Ke3 Ke5 8. d4+ Kd5 9. Kd3 Kd6 10. Ke4 Ke6 11. d5+ Kd6 12. Kd4 Kd7 13. Kc5.

[edit] Lasker-Reichhelm position

Lasker vs. Reichhelm, 1901

White to move wins, Black to move draws. "X" indicates key squares, some of the corresponding squares are marked

One of the most famous and complicated positions solved with the method of corresponding squares is this position from a game between Emanuel Lasker and Gustavus Charles Reichhelm in 1901.

• 1. Kb1 Kb7
• 2. Kc1 Kc7
• 3. Kd1 Kd8
• 4. Kc2 Kc8
• 5. Kd2 Kd7
• 6. Kc3 Kc7
• 7. Kd3 Kb6
• 8. Ke3

and White wins by penetrating on the kingside. Each of White's first seven moves are the only one that wins (Müller & Lamprecht 2000:193-94).

[edit] See also

• Triangulation
• Opposition
• Zugzwang
• King and pawn versus king endgame

[edit] References

• Dvoretsky, Mark (2006), Dvoretsky's Endgame Manual (second ed.), Russell Enterprises, ISBN 1-888690-28-3 

• Müller, Karsten & Lamprecht, Frank (2000), Secrets of Pawn Endings, Everyman Chess, ISBN 1-85744-255-5