Bishop and knight checkmate

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 thirty-three 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. 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 material 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 (with the same material), so is not generally encountered unless the defending side plays inaccurately. 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.

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'Analyze des Echecs. A method, known as Delétang's Method or Delétang's Triangles, applicable when the lone king is trapped behind one of the seven square diagonals of the same colour as the bishop, involves confining the lone king in a series of three shrinking isosceles right-angled triangles, with the "right" corner at the 90-degree 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 Delétang. The method as propounded is not optimal but is relatively simple and, so long as White has trapped the king behind the diagonal in a reasonable number of moves, will still 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 Fine (1941:4), as well as in Philidor's analysis (see below). Fine's analysis improves on Philidor's. Checkmate can be forced without using either method to complete the mate.

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