Fortress (chess)


Fortress (chess)

In chess, the fortress is an endgame drawing technique in which the side behind in material sets up a zone of protection around their king that cannot be penetrated by the opponent. This only works when the opponent does not have a passed pawn or cannot create one, unless that pawn can be stopped (e.g. see the opposite-colored bishops example). An elementary fortress is a theoretically drawn (i.e. a book draw) position with reduced material in which a passive defense will maintain the draw (Müller & Pajeken 2008:183).

Fortresses commonly have four characteristics:

  1. Useful pawn breakthroughs are not possible
  2. If the stronger side has pawns, they are firmly blocked
  3. The stronger side's king cannot penetrate, either because it is cut off or near the edge of the board
  4. Zugzwang positions cannot be forced, because the defender has waiting moves available (de la Villa 2008:23).

Fortresses pose a problem for computer chess: computers are unable to reason about fortress-type positions except to the extent that their endgame tablebase allows.[citation needed]

Contents


Fortress in a corner

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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  white pawn  white king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  white bishop  black king  black king  black king 1
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Bishop and wrong rook pawn: draw
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  white pawn  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  white knight  black king  white king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Knight protecting rook pawn on the seventh rank: draw

Perhaps the most common type of fortress, often seen in endgames with only a few pieces on the board, is where the defending king is able to take refuge in a corner of the board and cannot be chased away or checkmated by the superior side. These two diagrams furnish two classic examples. In both cases, Black simply shuffles his king between a8 and the available square adjacent to a8 (a7, b7, or b8, depending on the position of the white king and pawn). White has no way to dislodge Black's king, and can do no better than a draw by stalemate or some other means.

Note that the bishop and wrong rook pawn ending (i.e. where the pawn is a rook pawn whose promotion square is the color opposite to that of the bishop) at right is a draw even if the pawn is on the seventh rank or further back on the a-file. Heading for a bishop and wrong rook pawn ending is a fairly common drawing resource available to the inferior side (Müller & Pajeken 2008:183).

The knight and rook pawn position at right, however, is only a draw if White's pawn is already on the seventh rank, making this drawing resource available to the defender much less frequently. White wins if the pawn is not yet on the seventh rank and is protected by the knight from behind. With the pawn on the seventh rank, Black has a stalemate defense with his king in the corner (Müller & Pajeken 2008:189).

Serper vs. Nakamura
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8 a8 black king b8 black king c8 black king d8 black king e8 black king f8 black king g8 black king h8 black king 8
7 a7 black king b7 black king c7 black king d7 black king e7 black king f7 black king g7 black king h7 black king 7
6 a6 black king b6 black king c6 black king d6 black king e6 black king f6 black king g6 black king h6 black king 6
5 a5 black king b5 black king c5 black king d5 black king e5 black king f5 black king g5 black pawn h5 black king 5
4 a4 black king b4 black king c4 black king d4 black king e4 black pawn f4 black king g4 black king h4 black king 4
3 a3 black king b3 black king c3 black bishop d3 black king e3 white pawn f3 black king g3 black pawn h3 white pawn 3
2 a2 black king b2 black king c2 black king d2 black king e2 white king f2 white knight g2 white pawn h2 black king 2
1 a1 black king b1 black king c1 black king d1 black king e1 black king f1 black king g1 black king h1 black king 1
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White to play
Serper vs. Nakamura
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8 a8 black king b8 black king c8 black king d8 black king e8 black king f8 black king g8 black king h8 black king 8
7 a7 black king b7 black king c7 black king d7 black king e7 black king f7 black king g7 black king h7 black king 7
6 a6 black king b6 black king c6 black king d6 black king e6 black king f6 black king g6 black king h6 black king 6
5 a5 black king b5 black king c5 black king d5 black king e5 black bishop f5 black king g5 black pawn h5 black king 5
4 a4 black king b4 black king c4 black king d4 black king e4 black king f4 black king g4 black king h4 black king 4
3 a3 black king b3 black king c3 black king d3 black king e3 black king f3 black king g3 black pawn h3 white pawn 3
2 a2 black king b2 black king c2 black king d2 black king e2 black king f2 black king g2 white pawn h2 black king 2
1 a1 black king b1 black king c1 black king d1 black king e1 black king f1 black king g1 black king h1 white king 1
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Black to play. White has achieved a fortress.

A fortress is often achieved by a sacrifice, such as of a piece for a pawn. In the game between Grigory Serper and Hikaru Nakamura,[1] in the 2004 U.S. Chess Championship, White would lose after 1. Nd1 Kc4 or 1. Nh1 Be5 or 1. Ng4 Bg7. Instead he played 1. Nxe4! Kxe4 2.Kf1! heading for h1. After another 10 moves the position in the second diagram was reached. Black has no way of forcing White's king away from the corner, so he played 12... Kf2 and after 13. h4 gxh4 the game was drawn by stalemate.

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8  black king  black rook  black king  black king  black king  black king  black king  black king 8
7  white rook  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  white pawn  white king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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The back-rank defense in rook and pawn versus rook endings: another type of fortress in a corner.

The back-rank defense in some rook and pawn versus rook endgames is another type of fortress in a corner (see diagram at left). The defender perches his king on the pawn's queening square, and keeps his rook on the back rank (on the "long side" of the king, not, e.g., on h8 in the diagram position) to guard against horizontal checks. If 1.Rg7+ in the diagram position, Black heads into the corner with 1...Kh8! Note that this defense works only against rook pawns and knight pawns (Mednis 1982:15–17).

Seirawan, 2003
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8  black king  black king  white rook  black king  black king  black king  black bishop  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  white king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Rook versus bishop, with the defending king in the "safe" corner. White to move, draw.

In the ending of a rook versus a bishop, the defender can form a fortress in the "safe" corner—the corner that is not of the color on which the bishop resides (see diagram). White must release the potential stalemate, but he cannot improve his position (Seirawan 2003:200–201).

1. Rc3 Ba2
2. Rc2 Bb3
3. Rc7 Bg8
de la Villa, 2008
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black bishop  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black pawn  black king 3
2  black king  black king  black king  black king  black king  black king  white pawn  black king 2
1  black king  black king  black king  black king  black king  white king  black king  black king 1
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Draw with bishop on either color if the white king does not leave the corner

In this position from de la Villa, White draws if his king does not leave the corner. It is also a draw if the bishop is on the other color (de la Villa 2008:25).

Rook and pawn versus queen

Van der Poel vs. Smits, 2000
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8  black king  black king  black king  black king  black king  black king  white queen  black king 8
7  black king  black king  black king  black king  black pawn  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black rook  black king  black king 6
5  black king  black king  black king  black king  white king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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The basic fortress with rook and pawn on the second rank versus a queen

In the diagram at right, Black draws by moving his rook back and forth between the d6 and f6 squares, or moves his king when checked, staying behind the rook and next to the pawn. This fortress works when all of these conditions are met:

  1. the pawn is still on its second rank
  2. the pawn is on files b through g
  3. the pawn is protecting its rook on the third rank
  4. the opposing king is beyond the defender's third rank
  5. the defending king protects its pawn (Snape 2003:91).

The white king is not able to cross the rank of the black rook and the white queen is unable to do anything useful.

1. Qd5+ Rd6
2. Qb5+ Kd8
3. Qb8+ Kd7
4. Qb5+ ½-½
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  cross  cross  cross  cross  cross  cross  cross  cross 7
6  black king  cross  cross  cross  cross  cross  cross  black king 6
5  black king  cross  black king  black king  black king  black king  cross  black king 5
4  black king  cross  black king  black king  black king  black king  cross  black king 4
3  cross  cross  black king  black king  black king  black king  cross  cross 3
2  black king  cross  cross  cross  cross  cross  cross  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Drawing places of a white pawn, from White's perspective

Positions such as these (when the defending rook and king are near the pawn and the opposing king cannot attack from behind) are drawn when: (see diagram)

  • the pawn is on the c, d, e, or f file and the second, sixth, or seventh rank
  • the pawn is anywhere on the b or g file
  • the pawn is on the a or h file and the third or seventh rank.

Otherwise, the queen wins (Grivas 2008:297).

Similar example

Whitaker vs. Ferriz, 1959
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black queen  black king  black king  black king  black king  black king  black king  black pawn 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  white rook  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  white pawn  white king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Position after 50. Re3

In this 1959 game between Whitaker and Ferriz, White sacrificed a rook for a knight in order to exchange a pair of pawns and reach this position, and announced that it was a draw because (1) the queen cannot mate alone, and (2) the black king and pawn cannot approach to help (Whitaker & Hartleb 1960). However, endgame tablebase analysis shows Black to have a forced win in 19 moves starting with 50... Qc7+ (the only winning move), again illustrating how tablebases are refining traditional endgame theory.

Example with more pawns

Salov vs. Korchnoi
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black pawn  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black pawn 6
5  black king  black king  black king  black king  black king  white rook  black king  black king 5
4  black king  black king  black king  black king  black king  black king  white pawn  black king 4
3  black king  black king  black king  black king  black king  white pawn  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black queen  black king  white king  black king 1
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Even with equal pawns, Korchnoi could not break through Salov's fifth-rank defense

From the diagram at right, in Salov vs. Korchnoi, Wijk aan Zee 1997,[2] White was able to hold a draw with a rook versus a queen, even with the sides having an equal number of pawns. He kept his rook on the fifth rank blocking in Black's king, and was careful not to lose his rook to a fork or allow a queen sacrifice for the rook in circumstances where that would win for Black. The players agreed to a draw after:

48.Kg2 Kg6 49.Rh5 Qe2+ 50.Kg3 Qf1 51.Kf4 Qe1 52.Rd5 Qc1+ 53.Kg3 Qc7+ 54.Kg2 Qf4 55.Rh5 Kf6 56.Rd5 Ke6 57.Rh5 Qd2+ 58.Kg3 f6 59.Rf5 Qc1 60.Rh5 Qg1+ 61.Kf4 Qe1 62.Rb5 Qc1+ 63.Kg3 Qg1+ 64.Kf4 Qh2+ 65.Ke3 Kf7 66.Rh5 Qg1+ 67.Kf4 Kg6 68.Rd5 Qh2+ 69.Ke3 Kf7 70.Rh5 Qg1+ 71.Kf4 Ke6 72.Rb5 Qh2+ 73.Ke3 Kd6 74.Rf5 Qb2 75.Rh5 Ke6 76.Kf4 Qc3 77.Kg3 Qc7+ 78.Kg2 Qf7 79.Rb5 Qe8 80.Rf5 Qg6 81.Rb5 ½–½

Opposite-colored bishops

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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  white bishop  black pawn  black king  black king 6
5  black king  black king  black bishop  black king  black pawn  black king  black pawn  black king 5
4  black king  black king  black king  black king  white king  black king  black king  black pawn 4
3  black king  black king  black king  black king  black king  black king  black king  white pawn 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Drawing fortress with bishops on opposite colors, Black to move.

In endings with bishops of opposite colors (i.e. where one player has a bishop that moves on light squares, while the other player's bishop moves on dark squares), it is often possible to establish a fortress, and thus hold a draw, when one player is one, two, or occasionally even three pawns behind. A typical example is seen in the diagram at right. White, although three pawns behind, has established a drawing fortress, since Black has no way to contest White's stranglehold over the light squares. White simply keeps his bishop on the h3 to c8 diagonal (Dvoretsky 2006:92).

Example from game

Lautier vs. Rublevsky, 2003
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black pawn  white king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  white bishop  black king 5
4  black king  black king  black king  black king  black king  white pawn  black bishop  white pawn 4
3  black king  black king  black king  black king  black king  black king  white pawn  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Black to move gives up a pawn to set up a fortress.

In an endgame with opposite-colored bishops, positional factors may be more important than material. In this position, Black sacrifices a pawn (leaving him three pawns down) to reach a fortress.

1... Kf5!
2. Kxf7 Bh5+
3. Kg7 Bd1
4.Be7 ½-½

After 4... Be2 5. Kh6 Bd1 6. h5 Black just waits by playing 6... Be2 (Müller & Pajeken 2008:191).

Queen versus two minor pieces

Here are drawing fortresses with two minor pieces versus a queen (Dvoretsky 2006:289). Usually the defending side will not be able to get to one of these positions.

Bishop and knight

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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black bishop  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  white queen  black king  black king  black king  black king 6
5  black king  black king  white king  black knight  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Drawing fortress (in either corner which the bishop controls).

The bishop and knight fortress is another type of fortress in a corner. If necessary, the king can move to one of the squares adjacent to the corner, and the bishop can retreat to the corner. This gives the inferior side enough tempo moves to avoid zugzwang. For example: (Müller & Lamprecht 2001:339–41)

1. Kb5 Ka7
2. Qd8 Ba8
3. Ka5 Bb7.

Two bishops

Lolli, 1763
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black bishop  black bishop  black king  white queen  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  white king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Drawing fortress (in any corner), the Lolli position

In the two bishop versus queen ending, the queen wins if the Lolli position is not reached, but some of them take up to seventy-one moves, so the fifty-move rule comes into play. From the diagram:

1. Qe7+ Kc8
2. Qe6+ Kb7
3. Qd6 Ba7
4. Qe7+ Kb6!
5. Qd8+ Kb7!
6. Ka5 Bc5!

and White cannot prevent ... Bb6, which gets back to the Lolli position (Müller & Lamprecht 2001:339–41).

Two knights

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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black knight  black knight  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  white king  black king  black king  black king 3
2  white queen  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Drawing fortress

In the two knights fortress, the knights are next to each other and their king should be between them and the attacking king. The defender must play accurately, though (Müller & Lamprecht 2001:339–41).

Lolli, 1763
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  white queen  black king  black king  black king  black king 5
4  black king  black knight  black knight  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  white king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Draw

There are several drawing positions with two knights against a queen. The best way is to have the knights adjacent to each other on a file or rank, with their king between them and the enemy king. This is not a true fortress since it is not static. The position of the knights may have to change depending on the opponent's moves. In this position (Lolli, 1763),

1. Qd1 Nd2+
2. Ke2 Nb3

and Black has an ideal defensive position.

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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black knight  black king  black king  black king 7
6  cross  cross  black king  black knight  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  white queen  black king  black king 4
3  black king  black king  black king  white king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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White to move. Draw with the queen anywhere except the marked squares.

If the knights cannot be adjacent to each other on a file or rank, the second best position is if they are next to each other diagonally (see diagram).

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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black knight  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black knight  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  white king  white queen  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Draw, but Black must keep his king in a good position.

The third type of defensive formation is with the knights protecting each other, but this method is more risky (Nunn 2002:300ff).

With pawns

Ree vs. Hort
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8  black king  black king  black king  black king  black king  black knight  white queen  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black rook 7
6  black king  black king  black king  black pawn  black king  black king  black pawn  black king 6
5  black king  black king  black bishop  white pawn  black king  black pawn  white pawn  black king 5
4  black king  black king  black king  black king  black king  white pawn  black king  white knight 4
3  black king  black king  black king  black king  black king  black king  white king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Black to play.
Ree vs. Hort
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8  black king  black king  black king  black king  black king  black knight  white queen  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black pawn  black king  black king  black pawn  black king 6
5  black king  black king  black king  white pawn  black king  black pawn  white pawn  black king 5
4  black king  black king  black king  black king  black king  white pawn  black king  black king 4
3  black king  black king  black king  black king  black king  white king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black bishop  black king  black king  black king  black king  black king  black king  black king 1
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Draw agreed. Black has established an impenetrable fortress.

Sometimes the two minor pieces can achieve a fortress against a queen even where there are pawns on the board. In Ree-Hort, Wijk aan Zee 1986 [3] (diagram at left), Black had the material disadvantage of rook and bishop against a queen. Dvoretsky writes that Black would probably lose after the natural 1...Bf2+? 2.Kxf2 Rxh4 because of 3.Kg3 Rh7 4.Kf3, followed by a king march to c6, or 3.Qg7!? Rxf4+ 4.Kg3 Rg4+ 5.Kf3, threatening 6.Qf6 or 6.Qc7 (Dvoretsky 2006:315). Instead, Hort forced a draw with 1...Rxh4!! 2.Kxh4 Bd4! (imprisoning White's queen) 3.Kg3 Ke7 4.Kf3 Ba1 (diagram at right), and the players agreed to a draw. White's queen has no moves, all of Black's pawns are protected, and his bishop will shuttle back and forth on the squares a1, b2, c3, and d4.


Knight versus a rook and pawn

Em. Lasker vs. Ed. Lasker
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  white knight  white king  black king  black king  black king  black king  black king  black king 4
3  black king  black pawn  black king  black king  black king  black king  black rook  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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A surprising draw.

At the great New York 1924 tournament, former world champion Emanuel Lasker was in trouble against his namesake Edward Lasker, but surprised everyone by discovering a new endgame fortress (Lasker 1951:314–15).[4] Despite having only a knight for a rook and pawn, White draws by moving his knight back and forth between b2 and a4. Black's only real winning try is to get his king to c2. However, to do so Black has to move his king so far from the pawn that White can play Ka3-b2 and Nc5xb3, when the rook versus knight ending is an easy draw. The game concluded:

93.Nb2 Ke4 94.Na4 Kd4 95.Nb2 Rf3 96.Na4 Re3 97.Nb2 Ke4 98.Na4 Kf3 99.Ka3! Ke4 If 99...Ke2, 100.Nc5 Kd2 101.Kb2! (101.Nxb3+?? Kc2 and Black wins) and 102.Nxb3 draws. 100.Kb4 Kd4 101.Nb2 Rh3 102.Na4 Kd3 103.Kxb3 Kd4+ ½–½

Bishop versus rook and bishop pawn on the sixth rank

from de la Villa, page 220
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  white rook  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black circle  white pawn  black king  black king 6
5  black king  black king  black king  black circle  white king  black king  black king  black king 5
4  black king  black king  black bishop  black king  black king  black king  black king  black king 4
3  black king  black circle  black king  black king  black king  black king  black king  black king 3
2  black circle  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Black draws

A bishop can make a fortress versus a rook and a bishop pawn on the sixth rank, if the bishop is on the color of the pawn's seventh rank square and the defending king is in front of the pawn. In this position, White would win if he gotten the king to the sixth rank ahead of the pawn. Black draws by keeping the bishop on the diagonal from a2 to e6, except when giving check. The bishop keeps the white king off e6 and checks him if he goes to g6, to drive him away. A possible continuation:

1... Ba2
2. Kf4 (2. f7 is an interesting attempt, but then Black moves 2... Kg7! and then 3... Bxf7, with a draw. 2... Kg7 prevents 3. Kf6, which would win.)
2... Bc4
3. Kg5 Bd5! (the only move to draw, since the bishop must be able to check the king if it goes to g6)
4. Rc7 Ba2!
5. Kg6 Bb1+!
6. Kh6 Ba2!
7. Ra7

if 7. f7 Bxf7!: the pawn can be safely captured when the white king is on h6

7... Bc4, draw because White cannot make progress (de la Villa 2008:219–21).

Defense perimeter (pawn fortress)

A defense perimeter is a drawing technique in which the side behind in material or otherwise at a disadvantage sets up a perimeter, largely or wholly composed of a pawn chain, that the opponent cannot penetrate. Unlike other forms of fortress, a defense perimeter can often be set up in the middlegame with several pieces remaining on the board.

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8  black king  black king  black king  white bishop  black king  black king  black king  black king 8
7  black king  black rook  black king  black king  black pawn  black king  black king  black king 7
6  black rook  black king  black king  black pawn  black king  black pawn  black king  black king 6
5  black bishop  black king  black pawn  black king  white pawn  black king  black pawn  black king 5
4  black king  black pawn  black king  white pawn  black king  white pawn  white pawn  black pawn 4
3  black pawn  black king  white pawn  black king  black king  black king  black king  white pawn 3
2  white pawn  white pawn  white bishop  black king  white king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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White, on move, forces a draw.
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black rook  black king  black king  black pawn  black king  black king  black king 7
6  black rook  black king  black king  black pawn  white pawn  black pawn  black king  black king 6
5  black bishop  black king  black pawn  white pawn  black king  white pawn  black pawn  black king 5
4  black king  black pawn  white pawn  black king  black king  black king  white pawn  black pawn 4
3  black pawn  white pawn  black king  black king  black king  black king  black king  white pawn 3
2  white pawn  black king  black king  black king  white king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Black, ahead two rooks and a bishop, cannot breach White's defense perimeter.

The position at left, a chess problem by W.E. Rudolph (La Strategie 1912), illustrates the defense perimeter. White already has a huge material disadvantage, but forces a draw by giving up his remaining pieces to establish an impenetrable defense perimeter with his pawns. White draws with 1.Ba4+! Kxa4 (1...Kc4 2.Bb3+! Kb5 3.Ba4+ repeats the position) 2.b3+ Kb5 3.c4+ Kc6 4.d5+ Kd7 5.e6+! Kxd8 6.f5! (diagram at right). Now Black is up two rooks and a bishop (normally an overwhelming material advantage) but has no hope of breaking through White's defense perimeter. The only winning attempts Black can make are to place his rooks on b5, c6, etc. and hope that White captures them. White draws by ignoring all such offers and simply shuffling his king about (Kmoch 1959:174–75).


Petrosian vs. Hazai
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black queen  black king  black pawn  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black pawn  black king  black king  black king  black king 6
5  black pawn  white pawn  black pawn  white pawn  black pawn  black king  black pawn  black pawn 5
4  white knight  black king  white pawn  black king  white pawn  black pawn  black king  black king 4
3  white pawn  black king  black king  black king  black king  white pawn  black bishop  white pawn 3
2  white king  white queen  black king  black king  black king  black king  white pawn  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Black to play.
Petrosian vs. Hazai
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black pawn  black king  black pawn  black king  black king  black king  black king 6
5  black pawn  white pawn  black pawn  white pawn  black pawn  black king  black king  black king 5
4  black king  black king  white pawn  black king  white pawn  black pawn  black king  black pawn 4
3  white pawn  black king  black king  black king  black king  white pawn  black bishop  white pawn 3
2  white king  black king  black king  white queen  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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White cannot penetrate Black's fortress.

The above example may seem fanciful, but Black achieved a similar defense perimeter in Arshak Petrosian-Hazai, Schilde 1970[5] (diagram at left). In the position at left, Black has a difficult endgame, since White can attack and win his a-pawn by force, and he has no counterplay. Black tried the extraordinary 45...Qb6!?, to which White replied with the obvious 46.Nxb6+? This is actually a critical mistake, enabling Black to establish an impenetrable fortress. White should have carried out his plan of winning Black's a-pawn, for example with 46.Qc1 (threatening 47.Nxb6+ cxb6 48.h4! gxh4 49.Qh1 and Qh3, winning) Qa7 47.Qd2 followed by Kb3, Nc3, Ka4, and Na2-c1-b3. 46...cxb6 Now Black threatens 47...h4, locking down the entire board with his pawns, so White tries to break the position open. 47.h4 gxh4 48.Qd2 h3! 49.gxh3 Otherwise 49...h2 draws. 49...h4! (diagram at right) Black has established his fortress, and now can draw by simply moving his king around. The only way White could attempt to breach the fortress would be a queen sacrifice at some point (for example Qxa5 or Qxe5), but none of these give White winning chances as long as Black keeps his king near the center. The players shuffled their kings, and White's queen, around for six more moves before agreeing to a draw (Müller & Pajeken 2008:200) (Soltis 1975:249–50).


Smirin vs. HIARCS
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black pawn  black king  black king  black bishop  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black pawn  black king  black king 5
4  black pawn  black king  black king  white knight  black bishop  white pawn  black pawn  black pawn 4
3  white pawn  black king  white bishop  black king  white pawn  black king  black king  black king 3
2  black king  white pawn  black king  black king  black king  black king  black king  white pawn 2
1  black king  black king  black king  black king  black king  black king  white king  black king 1
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Position after 46.Kg1: White envisions a pawn fortress.
Smirin vs. HIARCS
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black pawn  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black pawn  black king  black king 5
4  black pawn  white pawn  black king  black king  black bishop  white pawn  black king  black pawn 4
3  black king  black king  white knight  black king  white pawn  black king  black pawn  white pawn 3
2  black king  white pawn  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  white king  black king 1
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Position after 51.Nc3: the fortress is complete.

In Smirin-HIARCS, Smirin-Computers match 2002,[6] the super-grandmaster looked to be in trouble against the computer, which has the bishop pair, can tie White's king down with ...g3, and threatens to invade with its king on the light squares. Smirin, however, saw that he could set up a fortress with his pawns. The game continued 46...g3 47.h3! A surprising move, giving Black a formidable protected passed pawn on the sixth rank, but it begins to build White's fortress, keeping Black's king out of g4. 47...Bc5 48.Bb4! Now Smirin gives HIARCS the choice between an opposite-colored bishops endgame (in which, moreover, White will play Be7 and win the h-pawn if Black's king comes to the center) and a bishop versus knight ending in which Smirin envisions a fortress. 48...Bxb4 49.axb4 Kf7 Black could try to prevent White's coming maneuver with 49...Bd3, but then White could play 50.Nf3 Kh5 (forced) 51.Nd4. 50.Nb5! Ke6 51.Nc3! Completing the fortress. Now Black's king has no way in, and his bishop can do nothing, since White's King can prevent ...Bf1, attacking White's only pawn on a light square. The game concluded: 51...Bc2 52.Kg2 Kd6 53.Kg1 Kc6 54.Kg2 b5 55.Kg1 Bd3 56.Kg2 Be4+ 57.Kg1 Bc2 58.Kg2 Bd3 59.Kg1 Be4 60.Kf1 ½–½

Other examples

Here are some other drawing fortresses (Müller & Pajeken 2008:183–215).

Fortresses against a bishop

Müller & Pajeken
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  white bishop  black king  black king  black king  black king  black king  black king  black king 7
6  black king  white pawn  black king  black king  black king  black king  black king  black king 6
5  black king  black king  white king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Draw
Müller & Pajeken
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black pawn  black king  black king  black king  black king  black king  black king  black king 7
6  white pawn  black king  black king  black king  black king  black king  black king  black king 6
5  black king  white king  black king  white bishop  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Draw

Fortresses against a knight

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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black knight  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  white king  black king  black king  black pawn 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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White draws with 1.Kf2!
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black pawn  black king  black king  black king  black king  black king  black king  black king 7
6  white pawn  black king  white king  black king  black king  black king  black king  black king 6
5  black king  white knight  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Draw with either side to move

Fortresses against a rook

Solid white.svg a b c d e f g h Solid white.svg
8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  white rook  black king  black king  black king  black king  black king  black pawn 7
6  black king  black king  black king  black king  white king  black king  black pawn  black king 6
5  black king  black king  black king  black king  black king  black king  black king  black king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black bishop  black king  black king  black king  white pawn  black king 3
2  black king  black king  black king  black king  black king  black king  black king  white pawn 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Black to move draws with 1... h5!

A semi-fortress

Kling & Horowitz, 1851
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8 a8 black king b8 black king c8 black king d8 black king e8 black king f8 white bishop g8 black king h8 black king 8
7 a7 black king b7 black knight c7 black king d7 black king e7 black king f7 black king g7 black king h7 black king 7
6 a6 black king b6 black king c6 black king d6 black king e6 black king f6 black king g6 black king h6 black king 6
5 a5 black king b5 black king c5 black king d5 white king e5 black king f5 black king g5 black king h5 black king 5
4 a4 white bishop b4 black king c4 black king d4 black king e4 black king f4 black king g4 black king h4 black king 4
3 a3 black king b3 black king c3 black king d3 black king e3 black king f3 black king g3 black king h3 black king 3
2 a2 black king b2 black king c2 black king d2 black king e2 black king f2 black king g2 black king h2 black king 2
1 a1 black king b1 black king c1 black king d1 black king e1 black king f1 black king g1 black king h1 black king 1
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This is a semi-fortress, but White wins in 45 moves.

The endgame of two bishops versus a knight was thought to be a draw for more than one hundred years. It was known that the temporary defensive fortress in this position could be broken down after a number of moves, but it was assumed that the fortress could be reformed in another corner. Computer endgame tablebases show that the bishops generally win, but it takes up to 66 moves. It takes several moves to force Black out of the temporary fortress in the corner; then precise play with the bishops prevents Black from forming the fortress in another corner. The position in the diagram was thought to be a draw by Kling and Horwitz but computer analysis shows that White wins in 45 moves (either by checkmate or by winning the knight). All of the long wins in this endgame go through this type of semi-fortress position (Nunn 1995:265ff).

Pintér vs. Bronstein
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8 a8 black king b8 black king c8 black king d8 black king e8 black king f8 black king g8 black king h8 black king 8
7 a7 black king b7 black king c7 black bishop d7 black king e7 black king f7 black king g7 black king h7 black king 7
6 a6 black king b6 black king c6 black king d6 black king e6 black king f6 black king g6 black king h6 black king 6
5 a5 black king b5 black king c5 black king d5 black bishop e5 black king f5 black king g5 black king h5 black king 5
4 a4 black king b4 black king c4 black king d4 black king e4 black king f4 black king g4 black king h4 black king 4
3 a3 black king b3 black king c3 black king d3 black king e3 white king f3 black king g3 black king h3 black king 3
2 a2 black king b2 black king c2 black king d2 black king e2 black king f2 black king g2 black king h2 black king 2
1 a1 black king b1 black king c1 black king d1 black king e1 white knight f1 black king g1 black king h1 black king 1
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Position before 68. Kd4

This game[7] between József Pintér and David Bronstein demonstrates the human play of the endgame. The defender has two ideas: (1) keep the king off the edge of the board and (2) keep the knight close to the king. White reaches the semi-fortress after 71. Nb2!, which falls after 75... Kb5!. White gets to a semi-fortress again in another corner after 90. Ng2+. After 100. Ke3 White cannot hold that semi-fortress any longer, but forms one in another corner after 112. Nb7!. On move 117 White claimed a draw by the fifty move rule (Benko 2007:132–34,137).

Positional draw

Portisch vs. Kavalek, 1978
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black knight  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  white bishop  black king  black king  black king  black king  black king  black king  black king 5
4  white pawn  black king  black king  white king  black king  black king  black king  black king 4
3  black king  white pawn  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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White to move

A "positional draw" is a concept most commonly used in endgame studies and describes an impasse other than stalemate. It usually involves the repetition of moves in which neither side can make progress or safely deviate. Typically a material advantage is balanced by a positional advantage. Fortresses and perpetual check are examples of positional draws (Hooper & Whyld 1992). Sometimes they salvage a draw from a position that seems hopeless because of a material deficit (Giddens 2007:111). Grandmaster John Nunn describes a positional draw as a position in which one side has enough material to normally win and he is not under direct attack, but some special feature of the position (often a blockade) prevents him from winning (Nunn 1981:68).

A simple example is shown in the game between Lajos Portisch and Lubomir Kavalek. White could have won easily with 1. Be1 Kc6 2. b4. However, play continued 1. b4? Nb8 2. b5 Nc6+! The only way to avoid the threatened 3...Nxa5 is 3. bxc6 Kxc6, but the resultant position is a draw because the bishop is on the wrong color to be able to force the rook pawn's promotion (see above, wrong bishop, and wrong rook pawn) (Beliavsky & Mikhalchishin 2003:130–35).

Pachman
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black king  black king  black king  black king 7
6  black king  black king  black king  black king  black king  black king  black king  black king 6
5  black king  black king  black king  black king  black rook  black king  black king  black king 5
4  black king  black king  black king  black king  black rook  black king  black king  black king 4
3  black king  black king  black king  white bishop  black king  white king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
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Either side to move

Luděk Pachman cites the endgame position at right as a simple example of a positional draw. White on move simply plays waiting moves with the bishop (Bb1-c2-d3). As for Black, "If he is unwilling to allow the transition to the drawn ending of Rook versus Bishop, nothing else remains for him but to move his Rook at [e5] continuously up and down the [e-file]." Pachman explains, "The indecisive result here contradicts the principles concerning the value of the pieces and is caused by the bad position of the black pieces (pinned rook at [e4])." (Pachman 1973:186–87).

Botvinnik vs. Keres, 1951
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8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  white rook  white pawn  black king  black king  black knight  black king  black pawn 7
6  black king  black king  black king  black pawn  black king  black pawn  black pawn  black king 6
5  black king  black king  black king  white pawn  black pawn  black king  black king  black king 5
4  black king  black king  black king  black king  white pawn  white pawn  black king  black king 4
3  black king  black king  black rook  black king  black king  black king  white pawn  black king 3
2  black king  black king  black king  black king  black king  black king  black king  white pawn 2
1  black king  black king  black king  black king  black king  black king  black king  white king 1
Solid white.svg a b c d e f g h Solid white.svg
Position after 52. c7, draw

This position from a game between Mikhail Botvinnik and Paul Keres in the 1951 USSR Championship is drawn because the black king cannot get free and the rook must stay on the c-file. The players agreed to a draw four moves later (Giddens 2007:111).

Tal vs. Fischer, 1962
Solid white.svg a b c d e f g h Solid white.svg
8 a8 black king b8 black king c8 black rook d8 black king e8 black king f8 black king g8 black king h8 black king 8
7 a7 black king b7 black king c7 black king d7 black king e7 black king f7 black pawn g7 black king h7 black king 7
6 a6 black king b6 black king c6 black king d6 black pawn e6 black king f6 white pawn g6 black king h6 black king 6
5 a5 black king b5 black king c5 black king d5 white pawn e5 black king f5 black king g5 black pawn h5 black king 5
4 a4 black king b4 black king c4 black king d4 black king e4 white rook f4 black bishop g4 black king h4 black pawn 4
3 a3 black king b3 black king c3 black king d3 black king e3 black king f3 black king g3 black king h3 white pawn 3
2 a2 black king b2 black king c2 black king d2 black king e2 black king f2 black king g2 white pawn h2 black king 2
1 a1 black king b1 white rook c1 black king d1 black king e1 black king f1 black king g1 white king h1 black king 1
Solid white.svg a b c d e f g h Solid white.svg
Position after 47...Kh7
Tal vs. Fischer
Solid white.svg a b c d e f g h Solid white.svg
8 a8 black king b8 black king c8 black king d8 black king e8 black king f8 black king g8 black king h8 black king 8
7 a7 black king b7 black king c7 black king d7 black king e7 black king f7 black king g7 black king h7 black king 7
6 a6 black king b6 black king c6 black king d6 black pawn e6 black king f6 black king g6 black king h6 black king 6
5 a5 black king b5 black king c5 black king d5 white rook e5 black king f5 black king g5 black pawn h5 black king 5
4 a4 black king b4 black king c4 black king d4 black king e4 white rook f4 black bishop g4 black king h4 black pawn 4
3 a3 black king b3 black rook c3 black king d3 black king e3 black king f3 black king g3 black king h3 white pawn 3
2 a2 black king b2 black king c2 black king d2 black king e2 black king f2 white king g2 white pawn h2 black king 2
1 a1 black king b1 black king c1 black king d1 black king e1 black king f1 black king g1 black king h1 black king 1
Solid white.svg a b c d e f g h Solid white.svg
Final position, after 58. Re4

The first diagram shows a position from a game between former World Champion Mikhail Tal and future World Champion Bobby Fischer from the 1962 Candidates Tournament in Curaçao.[8] After 41 moves Tal had the advantage but Fischer sacrificed the exchange (a rook for a knight). The game was drawn on the 58th move (Timman 2005:55–56).

Benko vs. Bonin
Solid white.svg a b c d e f g h Solid white.svg
8  black king  black king  black king  black king  black king  black king  black king  black king 8
7  black king  black king  black king  black king  black knight  black king  black king  black king 7
6  black king  black king  black rook  black king  black king  black king  black king  black king 6
5  black king  black king  white pawn  black king  white queen  black king  white pawn  white king 5
4  black king  black king  black king  black king  black king  black king  black king  black king 4
3  black king  black king  black king  black king  black king  black king  black king  black king 3
2  black king  black king  black king  black king  black king  black king  black king  black king 2
1  black king  black king  black king  black king  black king  black king  black king  black king 1
Solid white.svg a b c d e f g h Solid white.svg
Draw

In this position from a game between Pal Benko and International Master Jay Bonin, White realized that the blockade cannot be broken and the game is a draw despite the extra material (Benko 2009:40).

See also

Notes

References

  • Beliavsky, Alexander; Mikhalchishin, Adrian (2003), Modern Endgame Practice, Batsford, ISBN 0-7134-8740-2 
  • Benko, Pal (2007), Pal Benko's Endgame Laboratory, Ishi Press, ISBN 0-923891-88-9 
  • Benko, Pal (August 2009), "Endgame Lab", Chess Life (8): 40–41 
  • de la Villa, Jesús (2008), 100 Endgames You Must Know, New in Chess, ISBN 978-90-5691-244-4 
  • Dvoretsky, Mark (2006), Dvoretsky's Endgame Manual (2nd ed.), Russell Enterprises, ISBN 1-888690-28-3 
  • Giddins, Steve (2007), 101 Chess Endgame Tips: Golden nuggets of endgame wisdom, Gambit Publications, ISBN 978-1-904600-66-4 
  • Grivas, Efstratios (2008), Practical Endgame Play - mastering the basics, Everyman Chess, ISBN 978-1-85744-556-5 
  • Hooper, David; Whyld, Kenneth (1992), The Oxford Companion to Chess (2nd ed.), Oxford University Press, ISBN 0-19-280049-3 
  • Kmoch, Hans (1959), Pawn Power in Chess, David McKay 
  • Lasker, Edward (1951), Chess Secrets, David McKay 
  • Mednis, Edmar (1982), Practical Rook Endings, Chess Enterprises, Inc., ISBN 0-931462-16-9 
  • Müller, Karsten; Lamprecht, Frank (2001), Fundamental Chess Endings, Gambit Publications, ISBN 1-901983-53-6 
  • Müller, Karsten; Pajeken, Wolfgang (2008), How to Play Chess Endings, Gambit Publications, ISBN 978-1-904600-86-2  (chapter 11 gives extensive coverage to fortresses)
  • Nunn, John (1981), Tactical Chess Endings, Batsford, ISBN 0-7134-5937-9 
  • Nunn, John (1995), Secrets of Minor-Piece Endings, Batsford, ISBN 0-8050-4228-8 
  • Nunn, John (2002), Secrets of Pawnless Endings, Gambit Publications, ISBN 1-901983-65-X 
  • Pachman, Luděk (1973), Attack and Defense in Modern Chess Tactics, David McKay 
  • Seirawan, Yasser (2003), Winning Chess Endings, Everyman Chess, ISBN 1-85744-348-9 
  • Snape, Ian (2003), Chess Endings Made Simple: How to Approach the Endgame with Confidence, Gambit Publications, ISBN 1-901983-97-8 
  • Soltis, Andrew (1975), The Art of Defense in Chess, David McKay, ISBN 0-679-13043-8 
  • Timman, Jan (2005), Curaçao 1962: The Battle of Minds that Shook the Chess World, New in Chess, ISBN 978-90-5691-139-2 
  • Whitaker, Norman T.; Hartleb, Glenn E. (1960), 365 Selected Endings: One For Each Day of the Year, ISBN 0-923891-84-6 

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