Bishop and knight checkmate


The bishop and knight checkmate in chess is the checkmate of a lone king which can be by a bishop, knight, and king. With the stronger side to move and with perfect play, checkmate can be forced in at most thirty-three moves from any starting position where the defender cannot quickly win one of the pieces. The exceptions occur when the defending king may be forking the bishop and knight so that one of them is lost on the next move, or 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". These exceptions constitute about 0.5% of the positions. Checkmates are possible with the defending king on any square at the edge of the board, but can be forced only from positions with different or if the defending king is in a corner controlled by the bishop or on a square on the edge next to a corner, but mate adjacent to the corners not controlled by the bishop is only two moves deep, so is not generally encountered unless the defending side plays inaccurately. Although this is classified as one of the four basic or elementary checkmates , it occurs in practice approximately only once in every 6,000 games.

History and methods

A method for checkmate applicable when the lone king is in the corner of the opposite colour from the bishop, was given by Philidor in the 1777 update to his famous 1749 treatise, L'Analyse des Échecs. Another method, known as "Delétang's Method" or "Delétang's Triangles", applicable when the lone king is unable to reach the longest diagonal of the colour opposite to that of the bishop, involves confining the lone king in a series of three increasingly smaller triangles, ultimately forcing it into a corner of the same colour as the bishop. Some of the ideas of this method date back to 1780, but the complete system was first published in 1923 by Daniel Delétang. The method as propounded is not optimal, but it is relatively simple; so long as White has trapped the king behind the diagonal in a reasonable number of moves, it will lead to mate before the fifty-move rule takes effect. His "second triangle" or "middle triangle" occurs also in the analysis of play with the king in the corner of opposite colour to the bishop shown in, as well as in Philidor's analysis. Fine's analysis improves on Philidor's. Checkmate can be forced without using either method to complete the mate.

Importance

Opinions differ among chess authors 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 and that it occurs more often than the checkmate with two bishops. 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 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 only once or twice in his lifetime?

International Master Jonathan Hawkins has encountered the position only once in games. 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.
Although bishop, knight, and king versus king may never be encountered in the careers of many chess players, 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.

Method

Overview

It is assumed in this section that White has the bishop and knight.
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 "wrong" corner. Thus there are three phases in the checkmating process:
  1. Driving the opposing king to the edge of the board.
  2. Forcing the king out of the "wrong" corner to the "right" corner, if necessary.
  3. Delivering the checkmate.
Positions in this endgame fall into four categories:
  1. Positions which are stalemate, or in which White cannot prevent stalemate or the loss of one of their pieces, such as in the positions H and J. These form a significant minority, in almost all of which Black to play can immediately take a piece or irrecoverably fork the pieces or White to play has their pieces irrecoverably forked. Such positions are clearly drawn. All other positions can be won by White.
  2. Positions in which White can force mate on a square adjacent to the "wrong" corner, such as position K after 1...Kh8. These form a tiny minority.
  3. Positions in which White can prevent the black king reaching the longest diagonal of the colour opposite to that of the bishop, such as positions O–S. These form a substantial minority. In such positions White can force mate using Delétang's triangle method, described below.
  4. Positions in which the black king can reach the longest diagonal of the colour opposite to that of the bishop, such as positions L–N. These are the majority, in which the general method beginning with Phase 1 as described below can be applied.
Checkmate is usually quicker from the third type of position than the fourth type,, so White should usually aim for the former and Black the latter.

Phase 1

In the first phase, White uses their pieces to force the black king to the edge of the board. As noted above, White achieves mate more quickly by preventing the black king from reaching the longest diagonal of the colour opposite to that of the bishop.
Here is an example of how the first phase can be accomplished from the position [|L].
Keeping the black king out of the h8 corner. Now White can force the king to the a8 corner by one of the methods below, or by similar techniques.

Phases 2 and 3

W manoeuvre

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 :
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 Nf7–e5–d7–c5–b7 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 leave the edge of the board. This holds out longer.
Black's king is now restricted to the correct-coloured corner. The perimeter is bounded by a6, b6, b5, c5, d5, d6, d7, e7, f7, f8. White's subsequent moves tighten this area further.
At this point several ways of continuing are possible, including the following two:
Continue the W manoeuvre
One continuation from the position after Black's eighth move is to continue the W manoeuvre of the knight, by bringing it to c5 and b7. give 9. Bf3 Kd8 10. Kd6 Ke8 11. Bh5+ Kd8 12. Bf7 Kc8 13. Nc5 Kd8 14. Nb7+ Kc8 15. Kc6 Kb8 16. Kb6 Kc8 17. Be6+ Kb8 18. Nc5 Ka8 19. Bd7 Kb8 20. Na6+ Ka8 21. Bc6#. However, the following improvement is possible 12. Nc5 after which continues 12... Kc8 13. Be2 Kd8 14. Bb5 Kc8 15. Bd7+ Kb8 16. Kc6 Ka7 17. Kc7 Ka8 18. Kb6 Kb8 19. Na6+ Ka8 20. Bc6#.
Delétang's second triangle
Alternatively, from the position after Black's eighth move, shows that Delétang's triangle method is equally quick: 9. Bb5 Kd8 10. Nb6 Kc7 11. Nd5+ Kd8 12. Kd6 Kc8 13. Ke7 Kb7 14. Kd7 Kb8 15. Ba6 15... Ka7 16. Bc8 Kb8 17. Ne7 Ka7 18. Kc7 Ka8 19. Bb7+ Ka7 20. Nc6#.

Delétang's triangle method

Delétang'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.
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. The winning procedure consists of making the king move so that the bishop can reach the hypotenuse of the next smaller triangle.
Starting from the position of the first triangle, White wins:

Examples from games

The comments in this section are mostly editorial additions and not included in the cited references.

Karttunen vs. Rasik

The ending of the game between Mika Karttunen and Vitezslav Rasik at the 2003 European Chess Club Cup shows the knight's W manoeuvre. From position Y1, it 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 .
Checkmate follows after 104...Kg8, 105.Nh6+ Kh8 106.Bd4#. Black could have held out a move longer with 92...Kg7 and 98.Bd8 would have been faster for White.

Ljubojević vs. J. Polgár

Position Y2 is from a blindfold game between Ljubomir Ljubojević and Judit Polgár at the 1994 Amber chess tournament. Play continued:
84. Kd6 Kf6? 85. Kc5 Ke5 86. Kc4 Bd5+ 87. Kd3 Nf4+ 88. Ke3? 88... 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#

Grandmasters failing to mate

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 vs. Epishin game, both players made suboptimal moves. The superior side was unable to win and ended up stalemating several moves after the inferior side could have claimed a draw under the fifty-move rule.
Robert Kempinski – Vladimir Epishin
Bundesliga 0001 Germany, 07.01.2001
From position Z: 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 Bd8? 140...Nc4 141.Ka7 Nb6 142.Ka6 Bb8 is optimal. 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. 150. Ka5 Kc5? 150...Nd5 is optimal. 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 156.Ka4. 156... Bc7? Missing 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
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 a winning line, up to a point, but then failed to find 156...Nb4+ and instead tried again to mate in the wrong corner.
In the Anna Ushenina vs. Olga Girya game, played in the Geneva tournament of the FIDE Women's Grand Prix 2013–2014, White started [|Phase 2] correctly but missed two chances to finish it.
From position AB: 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 82. Bd5? 82... Kg3 83. Ke3 83... Kg4 84. Be4 84... 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 93... Kg8 94. Nh3 Kh8 95. Ng5 Kg8 96. Nf7 Kf8 97. Bh7 Ke8 98. Bf5 98... Kf8 99. Nh6 Ke8 100. Nf7 100... Kf8 101. Ne5 Kg8 102. Ng6 102... Kh7 103. Be6 103... Kh6 104. Bg8 Kh5 105. Ne5 Kh4 106. Kf5 Kg3 107. Bc4? 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+ 122... Kc1 123. Nc4 Kd1 124. Kd3 Kc1 125. Kc3 Kd1 126. Bd3 ½–½

A stalemate trap

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 in diagram AD, after 1...Nb6+ 2.Kb7?? Nd5, Black would be well on their way to setting up Delétang's second triangle. However, White draws instantly with 2.Kd8!, when the only way for Black to save their bishop is to move it, resulting in stalemate. The position 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 diagram AE occurs at the climax of a study by A. H. Branton, second prize, New Statesman, 1966 , though it may have been known even earlier.
From the position in diagram AD, instead of 1...Nb6+??, Black would win quickly by threatening mate on d8 by 1...Na5, e.g. 2.Kd8 Ba4 3.Kc8 Bd7+ and the white king is forced to b8 with mate in 6.

Quotations