The bishop and knight checkmate in chess is the checkmate of a lone king which can be forced by a bishop, knight, and king. With the stronger side to move and with perfect play, checkmate can be forced in at most thirtythree moves from any starting position where the defender cannot quickly win one of the pieces. The exceptions occur when (1) The defending king may be forking the bishop and knight so that one of them is lost on the next move, or (2) the knight may be trapped in a corner by the defending king and the knight is lost in one or two moves, and the position is not in the "stalemate trap" (see below). These exceptions constitute about 0.5% of the positions. Checkmate can be forced only with the defending king in a corner controlled by the bishop or on a square on the edge next to such a corner. Although this is classified as one of the four basic or elementary checkmates (Fine & Benko 2003:1) (the others being king and queen; king and rook; or king and two bishops against a lone king), it occurs in practice approximately only once in every 6000 games.
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A method for checkmate using the "W" method was given by Philidor in his famous 1749 treatise, Analyse du jeu des Échecs. Another method is known as Deletang's Method or Deletang's Triangles, involving confining the lone king in a series of three shrinking isosceles rightangled triangles, with the "right" corner at the 90degree angle of the triangle. Some of the ideas of this method date back to 1780 but the complete system was first published in 1923 by Daniel Deletang. This method takes five to ten more moves than Philidor's W method but there are fewer rules and it can still be accomplished before the fifty move rule takes effect. His "second triangle" or "middle triangle" comes up in the more standard methods (see below). Checkmate can be forced without strictly using either of the methods. Incidentally, checkmate can be delivered in 460 different ways (positions).
Opinions differ as to whether or not a player should learn this checkmate procedure. James Howell omits the checkmate with two bishops in his book because it rarely occurs but includes the bishop and knight checkmate. Howell says that he has had it three times (always on the defending side) and that it occurs more often than the checkmate with two bishops (Howell 1997:138). On the other hand, Jeremy Silman includes the checkmate with two bishops but not the bishop plus knight checkmate because he has encountered the latter only once and his friend John Watson has never encountered it (Silman 2007:33,188). Silman says:
[...] mastering it would take a significant chunk of time. Should the chess hopeful really spend many of his precious hours he's put aside for chess study learning an endgame he will achieve (at most) only once or twice in his lifetime?
International Master Jonathan Hawkins has encountered the position only once in games (Hawkins 2012:192). Grandmaster Andy Soltis says that he has never played this endgame and most players will never have it in their career. However, learning it teaches techniques that can be applied elsewhere (Soltis 2010:13).
Although king, bishop and knight versus king may never be encountered in the careers of many chessplayers, a notable example of it occurring in an important occasion was in Tal Shaked's victory over Alexander Morozevich in the penultimate round of the 1997 World Junior Chess Championship. Shaked knew the correct mating pattern; and his victory catapulted him to becoming World Junior Champion, whereas a draw would have prevented him from winning the title.
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Assume that White has the bishop and knight. In the first phase, White uses his pieces to force the black king to the edge of the board. There are four principles:
Here is an example of how the first phase can be accomplished from the diagrammed position.
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Keeping the black king out of the h8 corner. Now White can force the king to the a8 corner (a right corner for checkmate) by one of the methods below, or by similar techniques (Seirawan 2003:816).
Since checkmate can only be forced in the corner of the same colour as the squares on which the bishop moves, an opponent who is aware of this will try to stay first in the center of the board, and then in the wrongcolored corner. Thus there are three phases in the checkmating process (Müller & Lamprecht 2001:18):
The position on the right is one that typically arises after the first phase has been completed and the defender has headed to a corner of opposite colour to that of the bishop. The following method to push the king to the "right" corner is commonly given (Müller & Lamprecht 2001:18, Dvoretsky 2006:279):
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First White forces the king to leave the corner. The white bishop is positioned so that the next two moves, gaining control of g8, are possible.
A waiting move, forcing Black's king to move so White can play 3.Bh7, taking away g8 from the king.
The key to the standard winning method is the Nf7e5d7c5b7 movement of the knight, forming a "W" shape. Now there are two possible defenses:
Defense A: 4...Kf8 Black clings to the "safe" corner, but loses more quickly.
Defense B: 4...Kd8 Here, the defending king tries to break out from the edge. This holds out longer.
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Black's king is now restricted to the correctcolored corner. The perimeter is bounded by a6, b6, b5, c5, d5, d6, d7, e7, f7, f8. White's subsequent moves tighten this area further. Bb5 closes off c6; redeploying the knight to d5 (via f6) closes off d7 (and e8 by the bishop).
At this point two ways of continuing are possible.
Continue the W manoeuvre
One continuation from the position after Black's seventh move is to continue the "W" manoeuvre of the knight, by bringing it to c5 and b7. Müller & Lamprecht (2001:19) give 8.Be4 Kd8 9.Kd6 Ke8 10.Bg6+ Kd8 11.Bf7 Kc8 12.Nc5 Kd8 13.Nb7+ Kc8 14.Kc6 Kb8 15.Kb6 Kc8 16.Be6+ Kb8 17.Nc5 Ka8 18.Bd7 Kb8 19.Na6+ Ka8 20.Bc6# (the first checkmate diagram).
Deletang's second triangle
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Alternatively, from the position after Black's seventh move, Dvoretsky (2006:279) gives 8.Bb5 Kd8 9.Nf6 Kc7 10.Nd5+ Kd8, reaching this position. This bishop and knight configuration (right) is a very important position which can also be reached if the defender's king does not head for the "wrong" corner from the attacker's point of view (also known as Deletang's second triangle). Now 11.Kf7 Kc8 12.Ke7 Kb7 (12...Kb8 13.Ba6! Ka7 14.Bc8 Kb8 15.Kd7 as in the main variation) 13.Kd7 Kb8 (13...Ka7 14.Kc7 Ka8 15.Ne7 Ka7 16.Nc8+ Ka8 17.Bc6#; 13...Ka8 14.Kc8 Ka7 15.Kc7 is just a move slower) 14.Ba6! (or, Deletang's third triangle) 14...Ka7 15.Bc8 Kb8 16.Kd8 Ka8 (16...Ka7 17.Kc7 Ka8 18.Ne7 Ka7 19.Nc6+ Ka8 20.Bb7#) 17.Kc7 Ka7 18.Ne7 Ka8 19.Bb7+ Ka7 20.Nc6# (the second checkmate position)
Deletang's Triangle Method produces checkmate by confining the king in successively smaller areas. In the first set of three diagrams, the king is confined inside the marked area and a corner in which the checkmate can occur is in the area. The king cannot escape the area nor attack the bishop or knight. The second set of three diagrams shows the triangles and how the bishop controls the hypotenuse of the triangle (Pandolfini 2009:48ff).
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In the first net all three pieces are required to confine the king. In the second net only the bishop and knight are needed. In the third net, the king and bishop confine the king, allowing the knight to either checkmate or assist in the checkmate (de la Villa 2008:205). The winning procedure consists of making the king move so that the bishop can reach the hypotenuse of the next smaller triangle (Pandolfini 2009:48ff).
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Starting from the position of the first triangle, White wins:
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This game between Mika Karttunen and Vitezslav Rasik shows the knight's "W manoeuvre". The game continued:
84. Bc5 Kb7 85. Nd5 Kb8 86. Kc6 Ka8 87. Nc7+ Kb8 88. Bd4 Kc8 89. Ba7 Kd8 90. Nd5 Ke8 91. Kd6 Kf7 92. Ne7 Kf6 93. Be3 Kf7 94. Bd4 Ke8 95. Ke6 Kd8 96. Bb6+ Ke8 97. Nf5 Kf8 98. Bc7 Ke8 99. Ng7+ Kf8 100. Kf6 Kg8 101. Bd6 Kh7 102. Nf5 Kg8 103. Kg6 Kh8 104. Bc5 10 (Müller & Pajeken 2008:1067).
Checkmate follows after 104... Kg8, 105. Nh6+ Kh8 106. Bd4#.
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This position is from the blindfold game between Ljubomir Ljubojević and Judit Polgár, Monaco Amber 1994. Polgár did not use the standard method, but nevertheless coordinated the pieces effectively. Play continued: 84.Kd6 Kf6 85.Kc5 Ke5 86.Kc4 Bd5+ 87.Kd3 Nf4+ 88.Ke3 (White can resist about seven moves longer by 88. Kc3) Be4 89.Kd2 Kd4 90.Kc1 Kc3 91.Kd1 Bc2+ 92.Ke1 Kd3 93.Kf2 Ke4 94.Kg3 Bd1 95.Kf2 Nd3+ 96.Kg3 Ke3 97.Kh4 Kf4 98.Kh3 Ne1 99.Kh4 Ng2+ 100.Kh3 Kf3 101.Kh2 Kf2 102.Kh3 Be2 103.Kh2 Bg4 104.Kh1 Ne3 105.Kh2 Nf1+ 106.Kh1 Bf3# 01
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Delivering checkmate is difficult if the technique has not been studied and practiced. Even grandmasters, including GM Vladimir Epishin and Women's World Champion GM Anna Ushenina, have obtained the endgame but failed to win it.
In the Kempinski v. Epishin game, both players made suboptimal moves. The superior side had no idea how to win and ended up stalemating several moves after the inferior side could have claimed a draw under the fiftymove rule.
Robert Kempinski (2498)  Vladimir Epishin (2567) [E60] Bundesliga 0001 Germany (5.3), 07.01.2001
(Diagrammed position on left) 127.Kf3 Bc5 128.Ke4 Kc4 129.Kf5 Kd5 130.Kf6 Bd6 131.Kf7 Ne5+ 132.Ke8 Ke6 133.Kd8 Nf7+ 134.Kc8 Kd5 135.Kb7 Kc5 136.Ka6 Bc7 137.Kb7 Kd6 138.Ka6 Kc6 139.Ka7 Nd6 140.Ka8 (see diagram at right) Bd8? 140...Nc4 141.Ka7 Nb6 142.Ka6 Bb8 is the standard win. 141.Ka7 Kb5 142.Kb8 Kb6 143.Ka8 Nb7 144.Kb8 Bc7+ 145.Ka8 Kc6 146.Ka7 Nc5 147.Ka8 Nd7 148.Ka7 Nb6 149.Ka6 Bb8! Reaching the same position Black could have forced earlier (see previous note). 150.Ka5 Kc5? 150...Nd5 is the standard win. 151.Ka6 Bd6? 152.Kb7 Kb5 153.Ka7 Kc6 154.Ka6 Bb8! Reaching the same position as after Black's 149th move. 155.Ka5 Nd5! Belatedly finding the winning move he missed five moves ago. 156.Ka6 Objectively best was 151.Ka4. Bc7? Missing the standard 156...Nb4+. 157.Ka7 Bb6+ 158.Kb8 Bc5 159.Ka8 Nc7+ 160.Kb8 Nb5 161.Ka8 Kb6 162.Kb8 Na7 163.Ka8 Ka6 164.Kb8 Bb6 165.Ka8 Nb5 166.Kb8 Nd6 167.Ka8 Kb5 168.Kb8 Kc6 169.Ka8 Bc7 170.Ka7 Nb7 171.Ka8 Nc5 172.Ka7 Bb6+ 173.Ka8 Bc7 174.Ka7 Nd7 175.Ka8 Bd6 176.Ka7 Nb6 177.Ka6 Bb8 178.Ka5 Bc7 179.Ka6 Nc8 stalemate ½½
After the basic king, bishop, and knight versus king position arrived, White was kind enough to allow his king to retreat to the last rank in only six moves. But Black seemed to try to mate White in the wrong corner. Black eventually found the standard winning line, up to a point, but then failed to find 156... Nb4+ and instead tried again to mate in the wrong corner.
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In the Anna Ushenina v. Olga Girya game, played in the Geneva tournament of the FIDE Women's Grand Prix 20132014, White started the Wmaneuver correctly but missed two chances to finish it.
72... Ka1 73. Nd1 Ka2 74. Bc2 Ka1 75. Kc3 Ka2 76. Bb3+ Ka1 77. Ne3 Kb1 78. Nc2 Kc1 79. Ba2 Kd1 80. Nd4 Ke1 81. Kd3 Kf2 (see diagram) 82. Bd5? (White should continue with the Wmaneuver here with 82. Ne2! It looks at first as if the black king might run away with 82...Kf3 or 82...Kg2, but in either case 83.Be6 reins it in again.) 82...Kg3 83. Ke3 Kg4 84. Be4 Kg5 85. Kf3 Kf6 86. Kf4 Kg7 87. Kg5 Kf7 88. Kf5 Kg7 89. Bd5 Kh6 90. Ne6 Kh7 91. Kf6 Kg8 92. Nf4+ Kh8 93. Be4 Kg8 94. Nh3 Kh8 95. Ng5 Kg8 96. Nf7 Kf8 97. Bh7 Ke8 98. Bf5 Kf8 99. Nh6 Ke8 100. Nf7 Kf8 101. Ne5 Kg8 102. Ng6 Kh7 103. Be6 Kh6 104. Bg8 Kh5 105. Ne5 Kh4 106. Kf5 Kg3 107. Bc4? (Missing a second chance to complete the Wmaneuver with 107.Ng4!. After White missed this opportunity, Black can now with best play stave off checkmate long enough for the 50move draw to come into effect.) 107...Kf2 108. Kf4 Ke1 109. Ke3 Kd1 110. Bd3 Kc1 111. Nc4 Kd1 112. Nb6 Kc1 113. Na4 Kd1 114. Be4 Kc1 115. Bd3 Kd1 116. Nb2+ Kc1 117. Nc4 Kd1 118. Bg6 Kc1 119. Bf5 Kd1 120. Nb6 Kc1 121. Na4 Kd1 122. Nb2+ (As both players now have made fifty consecutive moves without a capture or pawn move, Black could claim the draw now by the 50move rule. Girya played on for another four moves, before actually taking the draw.) 122... Kc1 123. Nc4 Kd1 124. Kd3 Kc1 125. Kc3 Kd1 126. Bd3 ½½
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A surprising stalemate trap, not mentioned in endgame treatises, was noted by the American master Frederick Rhine in 2000 and published in Larry Evans' "What's the Best Move?" column in Chess Life magazine. In the position at left, after 1...Nb6+?? 2.Kb7?? Nd5, Black would be well on his way to setting up Deletang's second triangle. However, White draws instantly with 2.Kd8! (position at right), when the only way for Black to save his bishop is to move it, resulting in stalemate. The position at right would also be drawn if the knight were at a7 or e7 instead. Rhine later used this discovery as the basis for a "White to play and draw" composition. A stalemate idea essentially identical to that shown in the diagram at right occurs at the climax of a study by A. H. Branton, second prize, New Statesman, 1966 (Roycroft 1972:246) (White: king on c1; Black: king on c3, knight on a3, bishop on d1), though it may have been known even earlier.
From the diagram position at left, instead of 1...Nb6+??, Black would win quickly by setting up Deletang's second triangle via the alternate route 1...Ne3, e.g. 2.Kd8 Bb5 3.Kc8 Nd5.