Basic chess endgames
This page shows the analysis of all pawnless four piece chess
endgames. Before looking at these details it's probably best to
first read the friendly introduction.
How to read the results table
Firstly, the entire table is from the perspective of White winning.
The Pieces column shows the pieces on the board: first
White's pieces, then an underscore character _, then
Black's pieces. The standard notation for pieces is used:
k for King, q for Queen, r for
Rook, b for Bishop and n for Knight. Next the
table is divided up into positions where it is White to
move and positions where it is Black to move. In
each division the max column shows the number of White
moves to achieve checkmate in the longest win, the
%win column shows the percentage of legal positions
that are winning for White, the #win column shows the
number of legal positions that are winning for White, and the
#legal column shows the number of legal positions.
Results table
Pieces | White to move | Black to move
---------+----------------------------------+----------------------------------
| max | %win | #win | #legal | max | %win | #win | #legal
|-----+------+----------+----------+-----+------+----------+----------
k_k | --- | --- | 0 | 3612 | --- | --- | 0 | 3612
k_kb | --- | --- | 0 | 223944 | --- | --- | 0 | 193284
k_kbb | --- | --- | 0 | 13660584 | --- | --- | 0 | 10164056
k_kbn | --- | --- | 0 | 13660584 | --- | --- | 0 | 10875504
k_kn | --- | --- | 0 | 223944 | --- | --- | 0 | 205496
k_knn | --- | --- | 0 | 13660584 | --- | --- | 0 | 11499304
k_kq | --- | --- | 0 | 223944 | --- | --- | 0 | 144508
k_kqb | --- | --- | 0 | 13660584 | --- | --- | 0 | 7698432
k_kqn | --- | --- | 0 | 13660584 | --- | --- | 0 | 8245296
k_kqq | --- | --- | 0 | 13660584 | --- | --- | 0 | 5657120
k_kqr | --- | --- | 0 | 13660584 | --- | --- | 0 | 6911296
k_kr | --- | --- | 0 | 223944 | --- | --- | 0 | 175168
k_krb | --- | --- | 0 | 13660584 | --- | --- | 0 | 9366840
k_krn | --- | --- | 0 | 13660584 | --- | --- | 0 | 9905048
k_krr | --- | --- | 0 | 13660584 | --- | --- | 0 | 8325184
kb_k | --- | --- | 0 | 193284 | --- | --- | 0 | 223944
kb_kb | 1 | 0% | 416 | 11832464 | 0 | 0% | 112 | 11832464
kb_kn | 1 | 0% | 16 | 11832464 | 0 | 0% | 8 | 12535256
kb_kq | --- | --- | 0 | 11832464 | --- | --- | 0 | 8952608
kb_kr | --- | --- | 0 | 11832464 | --- | --- | 0 | 10780728
kbb_k | 19 | 49% | 5007216 | 10164056 | 19 | 41% | 5628080 | 13660584
kbn_k | 33 | 100% | 10822184 | 10875504 | 33 | 82% | 11188168 | 13660584
kn_k | --- | --- | 0 | 205496 | --- | --- | 0 | 223944
kn_kb | 1 | 0% | 40 | 12535256 | 0 | 0% | 8 | 11832464
kn_kn | 1 | 0% | 40 | 12535256 | 0 | 0% | 8 | 12535256
kn_kq | --- | --- | 0 | 12535256 | --- | --- | 0 | 8952608
kn_kr | 1 | 0% | 32 | 12535256 | 0 | 0% | 8 | 10780728
knn_k | 1 | 0% | 1232 | 11499304 | 0 | 0% | 240 | 13660584
kq_k | 10 | ALL | 144508 | 144508 | 10 | 90% | 200896 | 223944
kq_kb | 17 | 100% | 8925252 | 8952608 | 17 | 77% | 9097332 | 11832464
kq_kn | 21 | 99% | 8894128 | 8952608 | 21 | 80% | 10088688 | 12535256
kq_kq | 13 | 42% | 3737092 | 8952608 | 12 | 0% | 40628 | 8952608
kq_kr | 35 | 99% | 8863768 | 8952608 | 35 | 66% | 7062680 | 10780728
kqb_k | 8 | ALL | 7698432 | 7698432 | 10 | 91% | 12379568 | 13660584
kqn_k | 9 | ALL | 8245296 | 8245296 | 10 | 90% | 12343856 | 13660584
kqq_k | 4 | ALL | 5657120 | 5657120 | 10 | 98% | 13378232 | 13660584
kqr_k | 6 | ALL | 6911296 | 6911296 | 16 | 99% | 13519192 | 13660584
kr_k | 16 | ALL | 175168 | 175168 | 16 | 90% | 201700 | 223944
kr_kb | 29 | 35% | 3787160 | 10780728 | 29 | 3% | 381888 | 11832464
kr_kn | 40 | 48% | 5210920 | 10780728 | 40 | 11% | 1364800 | 12535256
kr_kq | 19 | 29% | 3090088 | 10780728 | 18 | 0% | 17136 | 8952608
kr_kr | 19 | 29% | 3139232 | 10780728 | 19 | 1% | 72464 | 10780728
krb_k | 16 | ALL | 9366840 | 9366840 | 16 | 91% | 12458920 | 13660584
krn_k | 16 | ALL | 9905048 | 9905048 | 16 | 91% | 12406892 | 13660584
krr_k | 7 | ALL | 8325184 | 8325184 | 16 | 100% | 13621424 | 13660584
---------+-----+------+----------+----------+-----+------+----------+----------
Summary | 40 | 29% | 1.179E8 | 4.033E8 | 40 | 34% | 1.355E8 | 4.033E8
An endgame analysis
by Joe Leslie-Hurd on 8 February 2005.