In chess, the bishop and knight checkmate is the checkmate of a lone king by an opposing king, bishop, and knight. With the stronger side to move, checkmate can be forced in at most thirty-three moves from almost any starting position.[1][2] Although this is classified as one of the four basic checkmates[3][a], it occurs in practice only approximately once in every 6,000 games.[4]
A method for checkmate applicable when the lone king is in the corner of the opposite color from the bishop (the "wrong" corner, where checkmate cannot be forced), was given by François-André Danican Philidor in the 1777 update[5] to his famous 1749 treatise, L'Analyse des Échecs.[6] He called attention to the route of the knight now known as the "W Manoeuvre".
Another method, known as "Delétang's Method" or "Delétang's Triangles",[7] applicable when the lone king is unable to reach the longest diagonal of the color 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 color as the bishop (the "right" corner). Some of the ideas of this method date back to 1780, but the complete system was first published in 1923 by Daniel Delétang.[8] 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.
In 1983, Julius Telesin showed that a king, bishop, and knight can force checkmate on the lone enemy king on an arbitrarily large board, as long as it contains a corner of the color that the bishop travels on. While the other basic checkmates[a] can be forced in O(n) moves on an n × n board, Telesin's method gives an O(n2) bound for a bishop and knight checkmate, though it has not been proven whether this is optimal.[9][10]
Opinions differ among chess authors as to whether or not a player should learn this checkmate procedure.
Jeremy Silman omitted the bishop-and-knight checkmate from his Complete Endgame Course, claiming he had encountered it only once, and that his friend John Watson had never encountered it.[11] Silman said, "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?" Similarly, International Master Jonathan Hawkins reported only ever encountering the position in a game once.[12]
On the other hand, while Grandmaster Andy Soltis concedes that he has never played this endgame and most players will never have it in their career, he argues that learning the checkmate teaches techniques that can be applied elsewhere.[13] James Howell includes the bishop-and-knight checkmate in his book, saying that he has defended against it three times and that it occurs more often than the checkmate with two bishops; he omits the latter from his book.[14] Finally, the checkmate occurred in at least one very notable case: Tal Shaked's victory over Alexander Morozevich in the penultimate round of the 1997 World Junior Chess Championship.[15] 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.[16]
This section is adapted from Yasser Seirawan's Winning Chess Endings. It is assumed that White has the bishop and knight.
Since checkmate can only be forced in the corner of the same color as the squares on which the bishop moves (the "right" corner), an opponent who is aware of this will first try to stay in the center of the board, and then move into in the "wrong" corner. Thus, there are three phases in the checkmating process:
To start, White uses their pieces to force the black king to the edge of the board.
From the diagrammed starting position:
Now that the black king is trapped on the edge of the board and cannot use the h8 square, white's pieces will coordinate to force the black king to the a8 corner of the board.
In this position, white has completed Phase 1, driving the black king to the h8 corner. Since white has a light-squared bishop, the knight must be used to control the dark squares on the 8th rank, forcing the black king to the h1 square. One of the most efficient ways for white to do this is by moving the knight along the board in a pattern known as the "W maneuver".[20][21]
With Delétang's triangle method, white confines the black king into a series of shrinking triangles; the bishop controls the hypotenuse of each triangle, while the knight and king control other squares that prevent the black king's escape. The winning procedure consists of forcing the king to move towards the corner so that the bishop can reach the hypotenuse of the next smaller triangle.[23]
From this position, in which black already cannot cross the b1-h7 diagonal, white wins the following way:
A drawing trap 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.[b] In the first position here, White would err with 1. Nb6+??. Rather than the expected 1... Kb7??, when White could continue on to hem in the opposing king, Black instead forces a draw with 1... Kd8!. Black is attacking the white bishop, and any bishop move results in stalemate, while a move by any other piece allows 2... Kxe8, an instant draw since the king and knight cannot checkmate by themselves.
The comments in this section are mostly editorial additions and not included in the cited references.
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The ending of the game between Mika Karttunen and Vitezslav Rasik[26] at the 2003 European Chess Club Cup shows the knight's W manoeuvre. From position U, it continued:
Checkmate follows after 104...Kg8, 105.Nh6+ Kh8 106.Bd4#. Black could have held out a move longer with 92...Kg7,[28] and 98.Bd8 would have been faster for White.
PGN text ➤
Position V is from a blindfold game between Ljubomir Ljubojević and Judit Polgár at the 1994 Amber chess tournament.[29] Play continued:84. Kd6 Kf6? (Better is 84...Nf4. The white king should run towards a1 so the black king should follow it towards that corner.) 85. Kc5 Ke5 86. Kc4 Bd5+ 87. Kd3 Nf4+ 88. Ke3? (White can resist about seven moves longer by 88.Kc3) 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# 0–1
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.[30]
Robert Kempinski (2498) – Vladimir Epishin (2567) [E60]Bundesliga 0001 Germany (5.3), 07.01.2001
From position W.
140...Nc4 141.Ka7 Nb6 142.Ka6 Bb8 is optimal.
Reaching the same position Black could have forced earlier (see previous note).
150...Nd5 is optimal.
Reaching the same position as after Black's 149th move.
Belatedly finding the winning move he missed five moves ago.
Objectively best was 156.Ka4.
Missing 156...Nb4+.
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 Y:
White should have played this move in place of the previous move or should now continue the W manoeuvre 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. Playing Bd5 at this stage is six moves slower than continuing the W manoeuvre, but White can still continue to mate in the h1 corner by e.g. Ne6, Bc4 sealing the black king behind the b1–h7 diagonal and leading to Delétang's first net.
After this move, White cannot prevent the black king escaping the b1–h7 diagonal. The black king can play up the g-file to g6 and the white king has no option but to follow with opposition on the e-file to at least e5, otherwise the black king can escape to the third perimeter at f5 or f6.
The black king can now escape to f6.
This wastes two moves because the knight needs three moves to reach e7 instead of one to reach g6. White should have immediately started the W manoeuvre along the h8–h1 edge, e.g. 94.Bf7 reproducing the position after White's move 77.
Quickest is to continue the W manoeuvre with Ne5, but White plans to control g8 with knight instead of bishop, which is three moves slower.
Now 100.Be6 would seal the king behind the a2–g8 diagonal. White has time to relocate the knight to d3 reaching Delétang's first net.
White instead abandons the idea.
On both preceding moves, playing the W manoeuvre along the h8–a8 edge would have been best.
White could have reached this position in two moves after move 92.
Missing a second chance to continue the W manoeuvre with 107.Ng4!. After White missed this opportunity, Black can now with best play stave off checkmate long enough for the 50-move draw to come into effect.
As both players now have made fifty consecutive moves without a capture or pawn move, Black could claim the draw now by the 50-move rule. Girya played on for another four moves before actually taking the draw.
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