We present here the number of endgame mates with a queen, a rook,
two bishops, a bishop and a knight and finally with two knights.
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).
This study of endgame mates has been made by Ari Luiro and Aarre Tiilikainen on September 9, 1995. Originally this article has been published in Finnish on the web site of the Järvenpää Chess Club and the Finnish web chess magazine Avoin Linja in 1999.
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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 corner sq. |
opp. king knight's sq. |
opp. king bishop's sq. |
opp. king center sq. |
Mates | |
| K + Q vs K | 17 | 9 | 10 | 10 | 364 | |
| K + R vs K | 12 | 5 | 5 | 5 | 216 | |
| K + 2 B vs K | 89 | 11 | 11 | 12 | 984 | |
| K + 2 N vs K | 7 | 0 | 4 | 4 | 120 | |
| K + B + N vs K | C 20
E 8 |
C 3
E 5 |
C 5
E 4 |
C 4
E 5 |
216 | bishop's
color |
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 x 8 - 4 = 364 mates
K + R vs K: 12 x 8 + 5 x 24 = 96 + 120 = 216 mates
K + 2 B vs K: on the corner there are (42 Ka8 Ka6 + 47 Ka8 Kb6
=) 89 mates
89 x 8 + 11 x 8 + 11 x 8 + 12 x 8 = 712 + 88 + 88 + 96 = 984 mates
K + 2 N vs K: 7 x 8 + 0 x 8 + 4 x 8 + 4 x 8 = 56 + 0 + 32 + 32 = 120 mates
K + B + N vs K: 20 x 4 + 8 x 4 + 3 x 4 + 5 x 4 + 5 x 4 + 4 x
4 + 4 x 4 + 5 x 4 = 80 + 32 + 12 + 20 + 20 + 16 + 16 + 20 = 216 mates
Your feedback is welcome to my e-mail
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Copyright Ari Luiro and Aarre Tiilikainen 1995 - 2006.
