Introduction
A chess endgame is the stage of a chess game when there are only a few pieces left on the board. There is, however, not a well-defined line distinguishing the endgame from its predecessor, the middle game. But all configurations just consisting of five pieces in total can safely be considered as endgames and this is what we will take a look at here. There are configurations—like a king and a rook vs. the lonely king—for which there exist easy strategies resulting almost always in a win—except for some obvious drawing positions. But there are configurations for which it is very hard to decide who will win with optimal play. The emergence of computers allowed to give a definite answer to this question for all configurations with up to five pieces— simply by brute force computation of the whole tree of possible moves. But even for this small number of pieces the computations are very elaborate and intelligent algorithms are necessary to make them possible. Richard Bellman, Thomas Ströhlein, and Ken Thompson are the fathers of computational endgames analysis (see also the Wikipedia article on endgame tablebases). Eugene Nalimov has created a famous database of chess endgames with up to five pieces, and it is his data that we will present here in a condensed form.
The computational endgame analysis revealed some surprises. For example, whereas the endgame king and queen vs. king and rook was always considered as resulting in a win for the king and queen side—again except for some obvious bad positions—it turned out that with optimal defense it is unbelievably hard to win. In 1978 there was a competition between Grandmaster Walter Browne and the chess computer BELLE using a table base for this endgame, and Browne gave up after 45 moves realizing that he would violate the 50 moves rule—although he could have won with perfect play after 31 moves.
More of these kinds of surprises can be found in the table below which shows for each configuration—always abbreviated in standard notation like KQ vs. K for king and queen vs lonely king—some theoretical statistics obtained by the computational results by Nalimov.
Hint: You can filter the table by entering any search string in the text field above the top right of the table.
Last updated: November 15, 2022 at 19:28 pm| Configuration | White configuration | Black configuration | Number of white pieces | Number of black pieces | Percentage of positions white to move wins | Percentage of positions white to move draws | Percentage of positions white to move loses | Longest mate for white to move | Percentage of positions black to move wins | Percentage of positions black to move draws | Percentage of positions black to move loses | Longest mate for black to move | Percentage of positions white wins (no matter who moves first) | Percentage of positions white draws (no matter who moves first) | Percentage of positions white loses (no matter who moves first) | Number of mutual zugzwang positions | Number of mutual zugzwang positions in which white to move draws and black to move loses | Number of mutual zugzwang positions in which black to move draws and white to move loses | Number of mutual zugzwang positions in which whoever moves first loses | Occurrence of this endgame in games per million |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| KQK | KQ | K | 2 | 1 | 100.0 | 0.0 | 0.0 | 10 | 0.0 | 10.3 | 89.7 | - | 94.9 | 5.2 | 0.0 | 0 | 0 | 0 | 0 | 257 |
| KRK | KR | K | 2 | 1 | 100.0 | 0.0 | 0.0 | 16 | 0.0 | 9.9 | 90.1 | - | 95.1 | 5.0 | 0.0 | 0 | 0 | 0 | 0 | 542 |
| KBK | KB | K | 2 | 1 | 0.0 | 100.0 | 0.0 | - | 0.0 | 100.0 | 0.0 | - | 0.0 | 100.0 | 0.0 | 0 | 0 | 0 | 0 | 194 |
| KNK | KN | K | 2 | 1 | 0.0 | 100.0 | 0.0 | - | 0.0 | 100.0 | 0.0 | - | 0.0 | 100.0 | 0.0 | 0 | 0 | 0 | 0 | 224 |
| KPK | KP | K | 2 | 1 | 76.5 | 23.5 | 0.0 | 28 | 0.0 | 41.9 | 58.1 | - | 67.3 | 32.7 | 0.0 | 80 | 80 | 0 | 0 | 2352 |
| KQKQ | KQ | KQ | 2 | 2 | 41.7 | 57.8 | 0.5 | 13 | 41.7 | 57.8 | 0.5 | 13 | 21.1 | 57.8 | 21.1 | 0 | 0 | 0 | 0 | 222 |
| KQKR | KQ | KR | 2 | 2 | 99.0 | 0.8 | 0.2 | 35 | 28.7 | 5.8 | 65.5 | 19 | 82.3 | 3.3 | 14.5 | 0 | 0 | 0 | 0 | 400 |
| KQKB | KQ | KB | 2 | 2 | 99.7 | 0.3 | 0.0 | 17 | 0.0 | 23.1 | 76.9 | - | 88.3 | 11.7 | 0.0 | 0 | 0 | 0 | 0 | 25 |
| KQKN | KQ | KN | 2 | 2 | 99.3 | 0.7 | 0.0 | 21 | 0.0 | 19.5 | 80.5 | - | 89.9 | 10.1 | 0.0 | 0 | 0 | 0 | 0 | 74 |
| KQKP | KQ | KP | 2 | 2 | 99.4 | 0.6 | 0.0 | 28 | 7.7 | 12.1 | 80.2 | 29 | 89.8 | 6.4 | 3.9 | 0 | 0 | 0 | 0 | 937 |
| KRKR | KR | KR | 2 | 2 | 29.1 | 70.2 | 0.7 | 19 | 29.1 | 70.2 | 0.7 | 19 | 14.9 | 70.2 | 14.9 | 0 | 0 | 0 | 0 | 423 |
| KRKB | KR | KB | 2 | 2 | 35.2 | 64.8 | 0.0 | 29 | 0.0 | 96.8 | 3.2 | - | 19.2 | 80.8 | 0.0 | 5 | 5 | 0 | 0 | 322 |
| KRKN | KR | KN | 2 | 2 | 48.4 | 51.6 | 0.0 | 40 | 0.0 | 89.0 | 11.0 | 1 | 29.7 | 70.3 | 0.0 | 18 | 18 | 0 | 0 | 397 |
| KRKP | KR | KP | 2 | 2 | 91.4 | 8.4 | 0.2 | 26 | 16.4 | 17.5 | 66.1 | 43 | 78.8 | 13.0 | 8.3 | 12 | 12 | 0 | 0 | 2146 |
| KBKB | KB | KB | 2 | 2 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | 0 | 0 | 0 | 0 | 49 |
| KBKN | KB | KN | 2 | 2 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | 0 | 0 | 0 | 0 | 87 |
| KBKP | KB | KP | 2 | 2 | 0.0 | 94.8 | 5.2 | 1 | 23.6 | 76.4 | 0.0 | 29 | ? | ? | ? | 1 | 0 | 1 | 0 | 387 |
| KNKN | KN | KN | 2 | 2 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | 0 | 0 | 0 | 0 | 68 |
| KNKP | KN | KP | 2 | 2 | 0.0 | 87.1 | 12.9 | 7 | 32.6 | 67.4 | 0.0 | 29 | 0.0 | 77.3 | 22.8 | 29 | 22 | 7 | 0 | 497 |
| KPKP | KP | KP | 2 | 2 | 43.4 | 33.3 | 23.2 | 33 | 43.4 | 33.3 | 23.2 | 33 | 33.3 | 33.3 | 33.3 | 121 | 106 | 106 | 15 | 2810 |
| KQQK | KQQ | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 4 | 0.0 | 2.1 | 97.9 | - | 99.0 | 1.1 | 0.0 | 0 | 0 | 0 | 0 | 13 |
| KQRK | KQR | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 6 | 0.0 | 1.1 | 98.9 | - | 99.5 | 0.6 | 0.0 | 0 | 0 | 0 | 0 | 18 |
| KQBK | KQB | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 8 | 0.0 | 9.4 | 90.6 | - | 95.3 | 4.7 | 0.0 | 0 | 0 | 0 | 0 | 36 |
| KQNK | KQN | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 9 | 0.0 | 9.7 | 90.3 | - | 95.2 | 4.9 | 0.0 | 0 | 0 | 0 | 0 | 41 |
| KQPK | KQP | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 10 | 0.0 | 2.8 | 97.2 | - | 98.6 | 1.4 | 0.0 | 0 | 0 | 0 | 0 | 156 |
| KRRK | KRR | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 7 | 0.0 | 0.3 | 99.7 | - | 99.9 | 0.2 | 0.0 | 0 | 0 | 0 | 0 | 8 |
| KRBK | KRB | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 16 | 0.0 | 8.8 | 91.2 | - | 95.6 | 4.4 | 0.0 | 0 | 0 | 0 | 0 | 46 |
| KRNK | KRN | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 16 | 0.0 | 9.2 | 90.8 | - | 95.4 | 4.6 | 0.0 | 0 | 0 | 0 | 0 | 15 |
| KRPK | KRP | K | 3 | 1 | 100.0 | 0.0 | 0.0 | 16 | 0.0 | 2.5 | 97.5 | - | 98.8 | 1.3 | 0.0 | 0 | 0 | 0 | 0 | 333 |
| KBBK | KBB | K | 3 | 1 | 49.3 | 50.7 | 0.0 | 19 | 0.0 | 58.8 | 41.2 | - | 45.3 | 54.8 | 0.0 | 0 | 0 | 0 | 0 | 31 |
| KBNK | KBN | K | 3 | 1 | 99.5 | 0.5 | 0.0 | 33 | 0.0 | 18.1 | 81.9 | - | 90.7 | 9.3 | 0.0 | 0 | 0 | 0 | 0 | 206 |
| KBPK | KBP | K | 3 | 1 | 96.0 | 4.0 | 0.0 | 31 | 0.0 | 16.8 | 83.2 | - | 89.6 | 10.4 | 0.0 | 0 | 0 | 0 | 0 | 453 |
| KNNK | KNN | K | 3 | 1 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | - | 0.0 | 100.0 | 0.0 | 0 | 0 | 0 | 0 | 20 |
| KNPK | KNP | K | 3 | 1 | 96.3 | 3.7 | 0.0 | 27 | 0.0 | 18.5 | 81.5 | - | 88.9 | 11.1 | 0.0 | 75 | 75 | 0 | 0 | 426 |
| KPPK | KPP | K | 3 | 1 | 98.4 | 1.6 | 0.0 | 32 | 0.0 | 7.9 | 92.1 | - | 95.3 | 4.8 | 0.0 | 43 | 43 | 0 | 0 | 563 |
| KQQKQ | KQQ | KQ | 3 | 2 | 99.1 | 0.8 | 0.1 | 30 | 0.6 | 32.8 | 66.6 | 13 | 82.9 | 16.8 | 0.4 | 0 | 0 | 0 | 0 | 51 |
| KQQKR | KQQ | KR | 3 | 2 | 100.0 | 0.0 | 0.0 | 35 | 0.1 | 0.2 | 99.7 | 19 | 99.9 | 0.1 | 0.1 | 0 | 0 | 0 | 0 | 0 |
| KQQKB | KQQ | KB | 3 | 2 | 100.0 | 0.0 | 0.0 | 15 | 0.0 | 0.1 | 99.9 | - | 100.0 | 0.1 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KQQKN | KQQ | KN | 3 | 2 | 100.0 | 0.0 | 0.0 | 19 | 0.0 | 0.1 | 99.9 | - | 100.0 | 0.1 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KQQKP | KQQ | KP | 3 | 2 | 100.0 | 0.0 | 0.0 | 22 | 0.0 | 0.7 | 99.3 | 13 | 99.7 | 0.4 | 0.0 | 0 | 0 | 0 | 0 | 7 |
| KQRKQ | KQR | KQ | 3 | 2 | 97.0 | 2.8 | 0.2 | 67 | 24.4 | 21.2 | 54.4 | 38 | 75.7 | 12.0 | 12.3 | 1 | 1 | 0 | 0 | 36 |
| KQRKR | KQR | KR | 3 | 2 | 99.8 | 0.1 | 0.0 | 34 | 0.3 | 17.1 | 82.1 | 20 | 91.0 | 8.6 | 0.2 | 0 | 0 | 0 | 0 | 132 |
| KQRKB | KQR | KB | 3 | 2 | 100.0 | 0.0 | 0.0 | 29 | 0.0 | 11.6 | 88.4 | - | 94.2 | 5.8 | 0.0 | 0 | 0 | 0 | 0 | 12 |
| KQRKN | KQR | KN | 3 | 2 | 99.9 | 0.1 | 0.0 | 1 | 0.0 | 7.7 | 92.3 | 1 | 96.1 | 3.9 | 0.0 | 0 | 0 | 0 | 0 | 2 |
| KQRKP | KQR | KP | 3 | 2 | 100.0 | 0.0 | 0.0 | 40 | 0.3 | 1.4 | 98.3 | 43 | 99.2 | 0.7 | 0.2 | 0 | 0 | 0 | 0 | 25 |
| KQBKQ | KQB | KQ | 3 | 2 | 55.7 | 44.0 | 0.3 | 33 | 30.5 | 62.3 | 7.2 | 24 | 31.5 | 53.2 | 15.4 | 25 | 25 | 0 | 0 | 28 |
| KQBKR | KQB | KR | 3 | 2 | 99.3 | 0.6 | 0.0 | 40 | 0.7 | 27.5 | 71.8 | 30 | 85.6 | 14.1 | 0.4 | 0 | 0 | 0 | 0 | 21 |
| KQBKB | KQB | KB | 3 | 2 | 99.7 | 0.3 | 0.0 | 17 | 0.0 | 19.8 | 80.2 | 2 | 90.0 | 10.1 | 0.0 | 0 | 0 | 0 | 0 | 2 |
| KQBKN | KQB | KN | 3 | 2 | 99.5 | 0.5 | 0.0 | 21 | 0.0 | 16.7 | 83.3 | 1 | 91.4 | 8.6 | 0.0 | 0 | 0 | 0 | 0 | 2 |
| KQBKP | KQB | KP | 3 | 2 | 100.0 | 0.0 | 0.0 | 32 | 1.0 | 14.1 | 84.9 | 24 | 92.5 | 7.1 | 0.5 | 0 | 0 | 0 | 0 | 25 |
| KQNKQ | KQN | KQ | 3 | 2 | 50.1 | 49.6 | 0.3 | 41 | 33.5 | 62.2 | 4.3 | 24 | 27.2 | 55.9 | 16.9 | 38 | 38 | 0 | 0 | 74 |
| KQNKR | KQN | KR | 3 | 2 | 99.2 | 0.7 | 0.0 | 38 | 3.0 | 27.2 | 69.8 | 41 | 84.5 | 14.0 | 1.5 | 0 | 0 | 0 | 0 | 12 |
| KQNKB | KQN | KB | 3 | 2 | 99.8 | 0.2 | 0.0 | 17 | 0.0 | 20.9 | 79.1 | 1 | 89.5 | 10.6 | 0.0 | 0 | 0 | 0 | 0 | 7 |
| KQNKN | KQN | KN | 3 | 2 | 99.4 | 0.6 | 0.0 | 21 | 0.0 | 17.8 | 82.2 | 1 | 90.8 | 9.2 | 0.0 | 0 | 0 | 0 | 0 | 13 |
| KQNKP | KQN | KP | 3 | 2 | 99.9 | 0.1 | 0.0 | 30 | 1.9 | 15.0 | 83.1 | 29 | 91.5 | 7.6 | 1.0 | 0 | 0 | 0 | 0 | 46 |
| KQPKQ | KQP | KQ | 3 | 2 | 68.4 | 31.2 | 0.4 | 124 | 35.2 | 51.2 | 13.6 | 29 | 41.0 | 41.2 | 17.8 | 640 | 640 | 0 | 0 | 1179 |
| KQPKR | KQP | KR | 3 | 2 | 99.6 | 0.3 | 0.1 | 38 | 19.7 | 6.1 | 74.1 | 33 | 86.9 | 3.2 | 9.9 | 1 | 1 | 0 | 0 | 216 |
| KQPKB | KQP | KB | 3 | 2 | 99.9 | 0.1 | 0.0 | 28 | 0.0 | 16.7 | 83.3 | 2 | 91.6 | 8.4 | 0.0 | 0 | 0 | 0 | 0 | 16 |
| KQPKN | KQP | KN | 3 | 2 | 99.7 | 0.3 | 0.0 | 30 | 0.0 | 12.5 | 87.5 | 8 | 93.6 | 6.4 | 0.0 | 0 | 0 | 0 | 0 | 41 |
| KQPKP | KQP | KP | 3 | 2 | 100.0 | 0.0 | 0.0 | 105 | 3.3 | 7.3 | 89.4 | 34 | 94.7 | 3.7 | 1.7 | 0 | 0 | 0 | 0 | 622 |
| KRRKQ | KRR | KQ | 3 | 2 | 58.2 | 36.8 | 5.1 | 29 | 52.0 | 37.0 | 11.0 | 49 | 34.6 | 36.9 | 28.6 | 10 | 10 | 0 | 0 | 8 |
| KRRKR | KRR | KR | 3 | 2 | 99.2 | 0.7 | 0.0 | 31 | 0.4 | 33.4 | 66.2 | 20 | 82.7 | 17.1 | 0.2 | 0 | 0 | 0 | 0 | 38 |
| KRRKB | KRR | KB | 3 | 2 | 99.3 | 0.7 | 0.0 | 29 | 0.0 | 22.4 | 77.6 | - | 88.5 | 11.6 | 0.0 | 1 | 1 | 0 | 0 | 8 |
| KRRKN | KRR | KN | 3 | 2 | 99.7 | 0.3 | 0.0 | 40 | 0.0 | 15.0 | 85.0 | 1 | 92.4 | 7.7 | 0.0 | 0 | 0 | 0 | 0 | 8 |
| KRRKP | KRR | KP | 3 | 2 | 100.0 | 0.0 | 0.0 | 33 | 1.0 | 5.7 | 93.3 | 50 | 96.7 | 2.9 | 0.5 | 0 | 0 | 0 | 0 | 3 |
| KRBKQ | KRB | KQ | 3 | 2 | 38.7 | 48.0 | 13.4 | 21 | 71.2 | 25.6 | 3.2 | 70 | 21.0 | 36.8 | 42.3 | 372 | 372 | 0 | 0 | 23 |
| KRBKR | KRB | KR | 3 | 2 | 41.3 | 58.7 | 0.0 | 65 | 0.8 | 94.1 | 5.1 | 30 | 23.2 | 76.4 | 0.4 | 17 | 17 | 0 | 0 | 649 |
| KRBKB | KRB | KB | 3 | 2 | 98.2 | 1.8 | 0.0 | 30 | 0.0 | 31.1 | 68.9 | 2 | 83.6 | 16.5 | 0.0 | 0 | 0 | 0 | 0 | 20 |
| KRBKN | KRB | KN | 3 | 2 | 98.9 | 1.1 | 0.0 | 40 | 0.0 | 24.0 | 76.0 | 1 | 87.5 | 12.6 | 0.0 | 0 | 0 | 0 | 0 | 5 |
| KRBKP | KRB | KP | 3 | 2 | 99.1 | 0.9 | 0.0 | 28 | 2.4 | 17.1 | 80.5 | 70 | 89.8 | 9.0 | 1.2 | 1 | 1 | 0 | 0 | 33 |
| KRNKQ | KRN | KQ | 3 | 2 | 35.4 | 41.1 | 23.4 | 20 | 78.2 | 19.7 | 2.1 | 69 | 18.8 | 30.4 | 50.8 | 455 | 0 | 455 | 0 | 15 |
| KRNKR | KRN | KR | 3 | 2 | 36.7 | 63.3 | 0.1 | 37 | 3.2 | 93.6 | 3.2 | 41 | 20.0 | 78.5 | 1.7 | 10 | 10 | 0 | 0 | 430 |
| KRNKB | KRN | KB | 3 | 2 | 97.7 | 2.3 | 0.0 | 31 | 0.0 | 32.4 | 67.6 | 1 | 82.7 | 17.4 | 0.0 | 0 | 0 | 0 | 0 | 7 |
| KRNKN | KRN | KN | 3 | 2 | 99.0 | 1.0 | 0.0 | 37 | 0.0 | 24.6 | 75.4 | 1 | 87.2 | 12.8 | 0.0 | 3 | 3 | 0 | 0 | 12 |
| KRNKP | KRN | KP | 3 | 2 | 98.5 | 1.5 | 0.0 | 29 | 4.5 | 17.1 | 78.4 | 68 | 88.5 | 9.3 | 2.3 | 0 | 0 | 0 | 0 | 32 |
| KRPKQ | KRP | KQ | 3 | 2 | 37.7 | 11.8 | 50.5 | 68 | 91.0 | 7.1 | 1.8 | 104 | 19.8 | 9.5 | 70.8 | 243 | 2 | 241 | 0 | 367 |
| KRPKR | KRP | KR | 3 | 2 | 66.6 | 33.0 | 0.4 | 74 | 20.1 | 54.4 | 25.5 | 33 | 46.1 | 43.7 | 10.3 | 209 | 209 | 0 | 0 | 9184 |
| KRPKB | KRP | KB | 3 | 2 | 96.4 | 3.6 | 0.0 | 73 | 0.0 | 32.6 | 67.4 | 2 | 81.9 | 18.1 | 0.0 | 225 | 225 | 0 | 0 | 626 |
| KRPKN | KRP | KN | 3 | 2 | 97.5 | 2.5 | 0.0 | 54 | 0.0 | 24.7 | 75.3 | 8 | 86.4 | 13.6 | 0.0 | 413 | 413 | 0 | 0 | 397 |
| KRPKP | KRP | KP | 3 | 2 | 99.4 | 0.4 | 0.3 | 56 | 10.0 | 6.6 | 83.5 | 103 | 91.5 | 3.5 | 5.2 | 3 | 0 | 2 | 1 | 1092 |
| KBBKQ | KBB | KQ | 3 | 2 | 15.3 | 20.2 | 64.5 | 21 | 96.5 | 2.9 | 0.6 | 81 | 8.0 | 11.6 | 80.5 | 1 | 0 | 1 | 0 | 3 |
| KBBKR | KBB | KR | 3 | 2 | 16.5 | 83.4 | 0.1 | 23 | 1.3 | 97.2 | 1.5 | 31 | 9.0 | 90.3 | 0.7 | 3 | 3 | 0 | 0 | 13 |
| KBBKB | KBB | KB | 3 | 2 | 15.6 | 84.3 | 0.0 | 22 | 0.0 | 98.6 | 1.4 | 2 | 8.5 | 91.5 | 0.0 | 0 | 0 | 0 | 0 | 35 |
| KBBKN | KBB | KN | 3 | 2 | 48.2 | 51.8 | 0.0 | 78 | 0.0 | 66.1 | 33.9 | 1 | 41.1 | 59.0 | 0.0 | 1 | 1 | 0 | 0 | 28 |
| KBBKP | KBB | KP | 3 | 2 | 48.0 | 50.2 | 1.8 | 74 | 11.4 | 54.1 | 34.5 | 83 | 41.3 | 52.2 | 6.6 | 1 | 1 | 0 | 0 | 23 |
| KBNKQ | KBN | KQ | 3 | 2 | 25.0 | 6.4 | 68.6 | 36 | 97.6 | 1.7 | 0.7 | 53 | 12.9 | 4.1 | 83.1 | 1 | 0 | 1 | 0 | 13 |
| KBNKR | KBN | KR | 3 | 2 | 26.0 | 73.8 | 0.2 | 36 | 3.8 | 94.6 | 1.6 | 41 | 13.8 | 84.2 | 2.0 | 8 | 6 | 2 | 0 | 64 |
| KBNKB | KBN | KB | 3 | 2 | 25.5 | 74.5 | 0.0 | 39 | 0.0 | 98.8 | 1.2 | 2 | 13.4 | 86.7 | 0.0 | 45 | 45 | 0 | 0 | 54 |
| KBNKN | KBN | KN | 3 | 2 | 32.2 | 67.8 | 0.0 | 107 | 0.0 | 96.1 | 3.9 | 1 | 18.1 | 82.0 | 0.0 | 922 | 922 | 0 | 0 | 36 |
| KBNKP | KBN | KP | 3 | 2 | 91.4 | 5.5 | 3.2 | 104 | 14.7 | 23.0 | 62.4 | 55 | 76.9 | 14.3 | 9.0 | 62 | 61 | 1 | 0 | 165 |
| KBPKQ | KBP | KQ | 3 | 2 | 21.3 | 11.5 | 67.2 | 35 | 96.8 | 2.8 | 0.4 | 50 | 10.9 | 7.2 | 82.0 | 16 | 0 | 16 | 0 | 117 |
| KBPKR | KBP | KR | 3 | 2 | 30.9 | 67.3 | 1.8 | 45 | 23.4 | 73.1 | 3.5 | 39 | 17.2 | 70.2 | 12.6 | 306 | 4 | 302 | 451 | |
| KBPKB | KBP | KB | 3 | 2 | 41.3 | 58.7 | 0.0 | 51 | 0.0 | 86.9 | 13.1 | 3 | 27.2 | 72.8 | 0.0 | 160 | 160 | 0 | 0 | 570 |
| KBPKN | KBP | KN | 3 | 2 | 53.7 | 46.3 | 0.0 | 105 | 0.0 | 76.4 | 23.6 | 8 | 38.7 | 61.4 | 0.0 | 2125 | 2112 | 13 | 0 | 497 |
| KBPKP | KBP | KP | 3 | 2 | 86.4 | 9.5 | 4.1 | 67 | 16.7 | 24.1 | 59.2 | 51 | 72.8 | 16.8 | 10.4 | 406 | 403 | 2 | 1 | 1443 |
| KNNKQ | KNN | KQ | 3 | 2 | 0.0 | 42.8 | 57.1 | 1 | 94.0 | 6.0 | 0.0 | 72 | 0.0 | 24.4 | 75.6 | 229 | 0 | 229 | 0 | 5 |
| KNNKR | KNN | KR | 3 | 2 | 0.0 | 99.6 | 0.4 | 3 | 6.3 | 93.7 | 0.0 | 41 | 0.0 | 96.7 | 3.4 | 25 | 0 | 25 | 0 | 15 |
| KNNKB | KNN | KB | 3 | 2 | 0.0 | 100.0 | 0.0 | 4 | 0.0 | 100.0 | 0.0 | 1 | 0.0 | 100.0 | 0.0 | 0 | 0 | 0 | 0 | 2 |
| KNNKN | KNN | KN | 3 | 2 | 0.1 | 99.9 | 0.0 | 7 | 0.0 | 100.0 | 0.0 | 1 | 0.1 | 100.0 | 0.0 | 362 | 362 | 0 | 0 | 8 |
| KNNKP | KNN | KP | 3 | 2 | 31.3 | 66.4 | 2.3 | 115 | 12.8 | 73.6 | 13.6 | 74 | 22.5 | 70.0 | 7.6 | 3143 | 3124 | 19 | 0 | 71 |
| KNPKQ | KNP | KQ | 3 | 2 | 17.9 | 11.9 | 70.2 | 41 | 97.2 | 2.3 | 0.5 | 55 | 9.2 | 7.1 | 83.7 | 52 | 0 | 52 | 0 | 130 |
| KNPKR | KNP | KR | 3 | 2 | 26.7 | 69.3 | 4.0 | 44 | 29.3 | 68.5 | 2.2 | 67 | 14.5 | 68.9 | 16.7 | 1181 | 23 | 1158 | 0 | 433 |
| KNPKB | KNP | KB | 3 | 2 | 38.8 | 61.2 | 0.0 | 42 | 0.0 | 88.1 | 11.9 | 9 | 25.4 | 74.7 | 0.0 | 642 | 640 | 2 | 0 | 728 |
| KNPKN | KNP | KN | 3 | 2 | 49.2 | 50.8 | 0.0 | 97 | 0.0 | 77.2 | 22.8 | 7 | 36.0 | 64.0 | 0.0 | 4191 | 4128 | 63 | 0 | 781 |
| KNPKP | KNP | KP | 3 | 2 | 78.3 | 13.6 | 8.1 | 57 | 21.8 | 27.6 | 50.6 | 58 | 64.5 | 20.6 | 15.0 | 2303 | 2281 | 14 | 8 | 1410 |
| KPPKQ | KPP | KQ | 3 | 2 | 16.0 | 12.6 | 71.4 | 124 | 98.4 | 1.5 | 0.1 | 41 | 8.1 | 7.1 | 84.9 | 2 | 0 | 2 | 0 | 726 |
| KPPKR | KPP | KR | 3 | 2 | 35.4 | 20.1 | 44.5 | 54 | 75.2 | 18.2 | 6.6 | 40 | 21.0 | 19.2 | 59.9 | 119 | 18 | 99 | 2 | 1652 |
| KPPKB | KPP | KB | 3 | 2 | 54.4 | 45.6 | 0.0 | 43 | 0.0 | 75.4 | 24.6 | 4 | 39.5 | 60.5 | 0.0 | 212 | 211 | 1 | 0 | 519 |
| KPPKN | KPP | KN | 3 | 2 | 64.7 | 35.3 | 0.0 | 50 | 0.0 | 62.4 | 37.6 | 17 | 51.2 | 48.9 | 0.0 | 1077 | 920 | 157 | 0 | 705 |
| KPPKP | KPP | KP | 3 | 2 | 77.1 | 10.3 | 12.6 | 127 | 27.7 | 19.1 | 53.2 | 43 | 65.2 | 14.7 | 20.2 | 4237 | 4179 | 52 | 6 | 5080 |
| KQQQK | KQQQ | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 3 | 0.0 | 4.0 | 96.0 | - | 98.0 | 2.0 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KQQRK | KQQR | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 4 | 0.0 | 3.1 | 96.9 | - | 98.5 | 1.6 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KQQBK | KQQB | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 4 | 0.0 | 2.7 | 97.3 | - | 98.7 | 1.4 | 0.0 | 0 | 0 | 0 | 0 | 3 |
| KQQNK | KQQN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 4 | 0.0 | 2.4 | 97.6 | - | 98.8 | 1.2 | 0.0 | 0 | 0 | 0 | 0 | 2 |
| KQQPK | KQQP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 4 | 0.0 | 2.1 | 97.9 | - | 99.0 | 1.1 | 0.0 | 0 | 0 | 0 | 0 | 12 |
| KQRRK | KQRR | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 4 | 0.0 | 2.0 | 98.0 | - | 99.0 | 1.0 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KQRBK | KQRB | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 5 | 0.0 | 1.7 | 98.3 | - | 99.2 | 0.9 | 0.0 | 0 | 0 | 0 | 0 | 3 |
| KQRNK | KQRN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 5 | 0.0 | 1.4 | 98.6 | - | 99.3 | 0.7 | 0.0 | 0 | 0 | 0 | 0 | 3 |
| KQRPK | KQRP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 7 | 0.0 | 1.1 | 98.9 | - | 99.5 | 0.6 | 0.0 | 0 | 0 | 0 | 0 | 26 |
| KQBBK | KQBB | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 6 | 0.0 | 5.0 | 95.0 | - | 97.5 | 2.5 | 0.0 | 0 | 0 | 0 | 0 | 3 |
| KQBNK | KQBN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 7 | 0.0 | 1.1 | 98.9 | - | 99.5 | 0.6 | 0.0 | 0 | 0 | 0 | 0 | 5 |
| KQBPK | KQBP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 9 | 0.0 | 1.2 | 98.8 | - | 99.4 | 0.6 | 0.0 | 0 | 0 | 0 | 0 | 31 |
| KQNNK | KQNN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 8 | 0.0 | 9.1 | 90.9 | - | 95.5 | 4.6 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KQNPK | KQNP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 9 | 0.0 | 1.0 | 99.0 | - | 99.5 | 0.5 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KQPPK | KQPP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 9 | 0.0 | 0.7 | 99.3 | - | 99.7 | 0.4 | 0.0 | ? | ? | ? | ? | 64 |
| KRRRK | KRRR | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 5 | 0.0 | 0.9 | 99.1 | - | 99.6 | 0.5 | 0.0 | 0 | 0 | 0 | 0 | 2 |
| KRRBK | KRRB | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 10 | 0.0 | 0.8 | 99.2 | - | 99.6 | 0.4 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KRRNK | KRRN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 10 | 0.0 | 0.6 | 99.4 | - | 99.7 | 0.3 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KRRPK | KRRP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 14 | 0.0 | 0.3 | 99.7 | - | 99.9 | 0.2 | 0.0 | 0 | 0 | 0 | 0 | 7 |
| KRBBK | KRBB | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 12 | 0.0 | 4.3 | 95.7 | - | 97.9 | 2.2 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KRBNK | KRBN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 29 | 0.0 | 0.5 | 99.5 | - | 99.8 | 0.3 | 0.0 | 0 | 0 | 0 | 0 | 3 |
| KRBPK | KRBP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 16 | 0.0 | 0.6 | 99.4 | - | 99.7 | 0.3 | 0.0 | 0 | 0 | 0 | 0 | 23 |
| KRNNK | KRNN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 15 | 0.0 | 8.5 | 91.5 | - | 95.8 | 4.3 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KRNPK | KRNP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 17 | 0.0 | 0.5 | 99.5 | - | 99.8 | 0.3 | 0.0 | 0 | 0 | 0 | 0 | 15 |
| KRPPK | KRPP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 15 | 0.0 | 0.2 | 98.8 | - | 99.4 | 0.1 | 0.0 | 0 | 0 | 0 | 0 | 119 |
| KBBBK | KBBB | K | 4 | 1 | 74.0 | 26.0 | 0.0 | 16 | 0.0 | 31.6 | 68.4 | - | 71.2 | 28.8 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KBBNK | KBBN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 33 | 0.0 | 4.1 | 95.9 | - | 98.0 | 2.1 | 0.0 | 0 | 0 | 0 | 0 | 3 |
| KBBPK | KBBP | K | 4 | 1 | 98.3 | 1.7 | 0.0 | 30 | 0.0 | 6.8 | 93.2 | - | 95.8 | 4.3 | 0.0 | 0 | 0 | 0 | 0 | 5 |
| KBNNK | KBNN | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 34 | 0.0 | 8.4 | 91.6 | - | 95.8 | 4.2 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KBNPK | KBNP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 33 | 0.0 | 0.8 | 99.2 | - | 99.6 | 0.4 | 0.0 | 0 | 0 | 0 | 0 | 26 |
| KBPPK | KBPP | K | 4 | 1 | 99.8 | 0.2 | 0.0 | 25 | 0.0 | 1.3 | 98.7 | - | 99.3 | 0.8 | 0.0 | 6 | 6 | 0 | 0 | 100 |
| KNNNK | KNNN | K | 4 | 1 | 98.7 | 1.3 | 0.0 | 21 | 0.0 | 25.0 | 75.0 | - | 86.9 | 13.2 | 0.0 | 0 | 0 | 0 | 0 | 0 |
| KNNPK | KNNP | K | 4 | 1 | 98.4 | 1.6 | 0.0 | 28 | 0.0 | 12.8 | 88.0 | - | 93.2 | 7.2 | 0.0 | 0 | 0 | 0 | 0 | 7 |
| KNPPK | KNPP | K | 4 | 1 | 100.0 | 0.0 | 0.0 | 32 | 0.0 | 1.0 | 99.0 | - | 99.5 | 0.5 | 0.0 | 93 | 93 | 0 | 0 | 96 |
| KPPPK | KPPP | K | 4 | 1 | 99.9 | 0.1 | 0.0 | 33 | 0.0 | 0.6 | 99.4 | - | 99.7 | 0.4 | 0.0 | 11 | 11 | 0 | 0 | 97 |