We present here the number of checkmate positions with a queen, rook, two bishops, bishop and knight and two knights against bare king. The adversary has only his king on the board. All the possible mates have been taken into the calculations, including helpmates.

In some cases all the mates are actually helpmates (two knights vs king), in other cases most (two bishops vs king, see the figure below) or none (all the heavy pieces vs king).

There are 364 mate positions with a king and queen against bare king. And 216 mate positions with a king and rook against bare king.

There are 1008 mate positions with a king and two bishops against bare king, mostly helpmates. And 360 mate positions with a king, bishop and knight against bare king, mostly helpmates. All the 120 mate positions with a king and two knights are helpmates.

This study of endgame mates has been made by Ari Luiro and Aarre Tiilikainen on September 9, 1995 and partly by Seppo Savikko on January 27, 2018. Originally this article has been published in Finnish on the web sites of Järvenpää Chess Club and Kemi Chess Club and the Finnish web chess magazine Avoin Linja in 1999.

Ari Luiro

Aarre Tiilikainen

We use the next concepts depending on the square of the opponent's king: corner square (opponent's king on the squares a1, h1, a8 and h8), knight's square (squares b8, g8, a7, h7, a2, h2, b1 and g1), bishop's square (squares c1, f1, a3, h3, a6, h6, c8 and f8) and center square (squares d1, e1, a4, a5, h4, h5, d8 and e8).

Please note that the concept of the center square is different to the usual chess game, because the mates are possible on the edge of the board only.
In the case of the mate with one bishop and one knight we have additionally used the concepts the corner of the bishop's color = C and the corner of the bishop's opposite color = E.

Pieces	opp. king on corner sq.	opp. king on knight's sq.	opp. king on bishop's sq.	opp. king on d8/e8/a4/a5 etc sq.	Mates
K + Q vs K	17 × 8	9 × 8	10 × 8	10 × 8	364
K + R vs K	12 × 8	5 × 8	5 × 8	5 × 8	216
K + 2 B vs K	336 + 392	40 + 48	32 + 56 + 8	24 + 56 + 16	1008
K + 2 N vs K	7 × 8	0	4 × 8	4 × 8	120
K + B + N vs K	240	48	36	36	360

Calculations

K + Q vs K: one corner mate can be made on either side, eg Kf6 + Qg7 vs Kh8, this need to be diminished
(17 + 9 + 10 + 10 =) 46 × 8 - 4 = 364 mates

K + R vs K: 12 × 8 + 5 × 24 = 96 + 120 = 216 mates

K + 2 Ns vs K: 7 × 8 + 0 × 8 + 4 × 8 + 4 × 8 = 56 + 0 + 32 + 32 = 120 mates
 

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Copyright Ari Luiro and Aarre Tiilikainen 1995, 2018.