Theoretical statistics for chess endgames with up to five pieces

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. The legend for the table is below.

Table

ABCDEFGHIJKLMNOPQRST
KQK21100.00.00.0100.010.389.7-94.95.20.00000257
KRK21100.00.00.0160.09.990.1-95.15.00.00000542
KBK210.0100.00.0-0.0100.00.0-0.0100.00.00000194
KNK210.0100.00.0-0.0100.00.0-0.0100.00.00000224
KPK2176.523.50.0280.041.958.1-67.332.70.08080002352
KQKQ2241.757.80.51341.757.80.51321.157.821.10000222
KQKR2299.00.80.23528.75.865.51982.33.314.50000400
KQKB2299.70.30.0170.023.176.9-88.311.70.0000025
KQKN2299.30.70.0210.019.580.5-89.910.10.0000074
KQKP2299.40.60.0287.712.180.22989.86.43.90000937
KRKR2229.170.20.71929.170.20.71914.970.214.90000423
KRKB2235.264.80.0290.096.83.2-19.280.80.05500322
KRKN2248.451.60.0400.089.011.0129.770.30.0181800397
KRKP2291.48.40.22616.417.566.14378.813.08.31212002146
KBKB220.0100.00.010.0100.00.010.0100.00.0000049
KBKN220.0100.00.010.0100.00.010.0100.00.0000087
KBKP2243.433.323.23343.433.323.23333.333.333.31010387
KNKN220.0100.00.010.0100.00.010.0100.00.0000068
KNKP220.087.112.9732.667.40.0290.077.322.8292270497
KPKP2243.433.323.23343.433.323.23333.333.333.3121106106152810
KQQK31100.00.00.0400.02.197.9-99.01.10.0000013
KQRK31100.00.00.060.01.198.9-99.50.60.0000018
KQBK31100.00.00.080.09.490.6-95.34.70.0000036
KQNK31100.00.00.090.09.790.3-95.24.90.0000041
KQPK31100.00.00.0100.02.897.2-98.61.40.00000156
KRRK31100.00.00.070.00.399.7-99.90.20.000008
KRBK31100.00.00.0160.08.891.2-95.64.40.0000046
KRNK31100.00.00.0160.09.290.8-95.44.60.0000015
KRPK31100.00.00.0160.02.597.5-98.81.30.00000333
KBBK3149.350.70.0190.058.841.2-45.354.80.0000031
KBNK3199.50.50.0330.018.181.9-90.79.30.00000206
KBPK3196.04.00.0310.016.883.2-89.610.40.00000453
KNNK310.0100.00.010.0100.00.0-0.0100.00.0000020
KNPK3196.33.70.0270.018.581.5-88.911.10.0757500426
KPPK3198.41.60.0320.07.992.1-95.34.80.0434300563
KQQKQ3299.10.80.1300.632.866.61382.916.80.4000051
KQQKR32100.00.00.0350.10.299.71999.90.10.100000
KQQKB32100.00.00.0150.00.199.9-100.00.10.000000
KQQKN32100.00.00.0190.00.199.9-100.00.10.000000
KQQKP32100.00.00.0220.00.799.31399.70.40.000007
KQRKQ3297.02.80.26724.421.254.43875.712.012.3110036
KQRKR3299.80.10.0340.317.182.12091.08.60.20000132
KQRKB32100.00.00.0290.011.688.4-94.25.80.0000012
KQRKN3299.90.10.010.07.792.3196.13.90.000002
KQRKP32100.00.00.0400.31.498.34399.20.70.2000025
KQBKQ3255.744.00.33330.562.37.22431.553.215.425250028
KQBKR3299.30.60.0400.727.571.83085.614.10.4000021
KQBKB3299.70.30.0170.019.880.2290.010.10.000002
KQBKN3299.50.50.0210.016.783.3191.48.60.000002
KQBKP32100.00.00.0321.014.184.92492.57.10.5000025
KQNKQ3250.149.60.34133.562.24.32427.255.916.938380074
KQNKR3299.20.70.0383.027.269.84184.514.01.5000012
KQNKB3299.80.20.0170.020.979.1189.510.60.000007
KQNKN3299.40.60.0210.017.882.2190.89.20.0000013
KQNKP3299.90.10.0301.915.083.12991.57.61.0000046
KQPKQ3268.431.20.412435.251.213.62941.041.217.8640640001179
KQPKR3299.60.30.13819.76.174.13386.93.29.91100216
KQPKB3299.90.10.0280.016.783.3291.68.40.0000016
KQPKN3299.70.30.0300.012.587.5893.66.40.0000041
KQPKP32100.00.00.01053.37.389.43494.73.71.70000622
KRRKQ3258.236.85.12952.037.011.04934.636.928.61010008
KRRKR3299.20.70.0310.433.466.22082.717.10.2000038
KRRKB3299.30.70.0290.022.477.6-88.511.60.011008
KRRKN3299.70.30.0400.015.085.0192.47.70.000008
KRRKP32100.00.00.0331.05.793.35096.72.90.500003
KRBKQ3238.748.013.42171.225.63.27021.036.842.33723720023
KRBKR3241.358.70.0650.894.15.13023.276.40.4171700649
KRBKB3298.21.80.0300.031.168.9283.616.50.0000020
KRBKN3298.91.10.0400.024.076.0187.512.60.000005
KRBKP3299.10.90.0282.417.180.57089.89.01.2110033
KRNKQ3235.441.123.42078.219.72.16918.830.450.84550455015
KRNKR3236.763.30.1373.293.63.24120.078.51.7101000430
KRNKB3297.72.30.0310.032.467.6182.717.40.000007
KRNKN3299.01.00.0370.024.675.4187.212.80.0330012
KRNKP3298.51.50.0294.517.178.46888.59.32.3000032
KRPKQ3237.711.850.56891.07.11.810419.89.570.824322410367
KRPKR3266.633.00.47420.154.425.53346.143.710.3209209009184
KRPKB3296.43.60.0730.032.667.4281.918.10.022522500626
KRPKN3297.52.50.0540.024.775.3886.413.60.041341300397
KRPKP3299.40.40.35610.06.683.510391.53.55.230211092
KBBKQ3215.320.264.52196.52.90.6818.011.680.510103
KBBKR3216.583.40.1231.397.21.5319.090.30.7330013
KBBKB3215.684.30.0220.098.61.428.591.50.0000035
KBBKN3248.251.80.0780.066.133.9141.159.00.0110028
KBBKP3248.050.21.87411.454.134.58341.352.26.6110023
KBNKQ3225.06.468.63697.61.70.75312.94.183.1101013
KBNKR3226.073.80.2363.894.61.64113.884.22.0862064
KBNKB3225.574.50.0390.098.81.2213.486.70.045450054
KBNKN3232.267.80.01070.096.13.9118.182.00.09229220036
KBNKP3291.45.53.210414.723.062.45576.914.39.0626110165
KBPKQ3221.311.567.23596.82.80.45010.97.282.0160160117
KBPKR3230.967.31.84523.473.13.53917.270.212.63064302451
KBPKB3241.358.70.0510.086.913.1327.272.80.016016000570
KBPKN3253.746.30.01050.076.423.6838.761.40.021252112130497
KBPKP3286.49.54.16716.724.159.25172.816.810.4406403211443
KNNKQ320.042.857.1194.06.00.0720.024.475.6229022905
KNNKR320.099.60.436.393.70.0410.096.73.425025015
KNNKB320.0100.00.040.0100.00.010.0100.00.000002
KNNKN320.199.90.070.0100.00.010.1100.00.0362362008
KNNKP3231.366.42.311512.873.613.67422.570.07.63143312419071
KNPKQ3217.911.970.24197.22.30.5559.27.183.7520520130
KNPKR3226.769.34.04429.368.52.26714.568.916.711812311580433
KNPKB3238.861.20.0420.088.111.9925.474.70.064264020728
KNPKN3249.250.80.0970.077.222.8736.064.00.041914128630781
KNPKP3278.313.68.15721.827.650.65864.520.615.0230322811481410
KPPKQ3216.012.671.412498.41.50.1418.17.184.92020726
KPPKR3235.420.144.55475.218.26.64021.019.259.9119189921652
KPPKB3254.445.60.0430.075.424.6439.560.50.021221110519
KPPKN3264.735.30.0500.062.437.61751.248.90.010779201570705
KPPKP3277.110.312.612727.719.153.24365.214.720.2423741795265080
KQQQK41100.00.00.030.04.096.0-98.02.00.000000
KQQRK41100.00.00.040.03.196.9-98.51.60.000000
KQQBK41100.00.00.040.02.797.3-98.71.40.000003
KQQNK41100.00.00.040.02.497.6-98.81.20.000002
KQQPK41100.00.00.040.02.197.9-99.01.10.0000012
KQRRK41100.00.00.040.02.098.0-99.01.00.000000
KQRBK41100.00.00.050.01.798.3-99.20.90.000003
KQRNK41100.00.00.050.01.498.6-99.30.70.000003
KQRPK41100.00.00.070.01.198.9-99.50.60.0000026
KQBBK41100.00.00.060.05.095.0-97.52.50.000003
KQBNK41100.00.00.070.01.198.9-99.50.60.000005
KQBPK41100.00.00.090.01.298.8-99.40.60.0000031
KQNNK41100.00.00.080.09.190.9-95.54.60.000000
KQNPK41100.00.00.090.01.099.0-99.50.50.000000
KQPPK41100.00.00.090.00.799.3-99.70.40.0????64
KRRRK41100.00.00.050.00.999.1-99.60.50.000002
KRRBK41100.00.00.0100.00.899.2-99.60.40.000000
KRRNK41100.00.00.0100.00.699.4-99.70.30.000000
KRRPK41100.00.00.0140.00.399.7-99.90.20.000007
KRBBK41100.00.00.0120.04.395.7-97.92.20.000000
KRBNK41100.00.00.0290.00.599.5-99.80.30.000003
KRBPK41100.00.00.0160.00.699.4-99.70.30.0000023
KRNNK41100.00.00.0150.08.591.5-95.84.30.000000
KRNPK41100.00.00.0170.00.599.5-99.80.30.0000015
KRPPK41100.00.00.0150.00.298.8-99.40.10.00000119
KBBBK4174.026.00.0160.031.668.4-71.228.80.000000
KBBNK41100.00.00.0330.04.195.9-98.02.10.000003
KBBPK4198.31.70.0300.06.893.2-95.84.30.000005
KBNNK41100.00.00.0340.08.491.6-95.84.20.000000
KBNPK41100.00.00.0330.00.899.2-99.60.40.0000026
KBPPK4199.80.20.0250.01.398.7-99.30.80.06600100
KNNNK4198.71.30.0210.025.075.0-86.913.20.000000
KNNPK4198.41.60.0280.012.888.0-93.27.20.000007
KNPPK41100.00.00.0320.01.099.0-99.50.50.093930096
KPPPK4199.90.10.0330.00.699.4-99.70.40.011110097

Legend

A White configuration
B Black configuration
C Number of white pieces
D Number of black pieces
E Percentage of positions white to move wins
F Percentage of positions white to move draws
G Percentage of positions white to move loses
H Longest mate for white to move
I Percentage of positions black to move wins
J Percentage of positions black to move draws
K Percentage of positions black to move loses
L Longest mate for black to move
M Percentage of positions white wins (no matter who moves first)
N Percentage of positions white draws (no matter who moves first)
O Percentage of positions white loses (no matter who moves first)
P Number of mutual zugzwang positions
Q Number of mutual zugzwang positions in which white to move draws and black to move loses
R Number of mutual zugzwang positions in which black to move draws and white to move loses
S Number of mutual zugzwang positions in which whoever moves first loses
T Occurrence of this endgame in games per million