Suomeksi
Kemi Chess Club
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Chess Pieces



The Number of Simple Endgame Mates


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 even 984 mate positions with a king and two bishops against bare king, mostly helpmates. And 216 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. 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.

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 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 .
Copyright Ari Luiro and Aarre Tiilikainen 1995 - 2012.

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