NULP

(Number of Unique Legal Positions in chess endgames)

Number of generalized chess endgames

This table shows the number of endgames and ESMs (endgame + side to move) in generalized chess (unlimited number of pieces and pawns for promotion), for up to 64 pieces.

PiecesEndgamesEndgames
Cumulative		ESMsESMs
Cumulative
2 1 1 1 1
3 5 6 10 11
4 30 36 55 66
5 110 146 220 286
6 365 511 715 1,001
7 1,001 1,512 2,002 3,003
8 2,520 4,032 5,005 8,008
9 5,720 9,752 11,440 19,448
10 12,190 21,942 24,310 43,758
11 24,310 46,252 48,620 92,378
12 46,252 92,504 92,378 184,756
13 83,980 176,484 167,960 352,716
14 147,070 323,554 293,930 646,646
15 248,710 572,264 497,420 1,144,066
16 408,760 981,024 817,190 1,961,256
17 653,752 1,634,776 1,307,504 3,268,760
18 1,021,735 2,656,511 2,042,975 5,311,735
19 1,562,275 4,218,786 3,124,550 8,436,285
20 2,343,770 6,562,556 4,686,825 13,123,110
21 3,453,450 10,016,006 6,906,900 20,030,010
22 5,008,003 15,024,009 10,015,005 30,045,015
23 7,153,575 22,177,584 14,307,150 44,352,165
24 10,080,720 32,258,304 20,160,075 64,512,240
25 14,024,400 46,282,704 28,048,800 92,561,040
26 19,284,460 65,567,164 38,567,100 131,128,140
27 26,225,628 91,792,792 52,451,256 183,579,396
28 35,304,920 127,097,712 70,607,460 254,186,856
29 47,071,640 174,169,352 94,143,280 348,330,136
30 62,203,340 236,372,692 124,403,620 472,733,756
31 81,505,820 317,878,512 163,011,640 635,745,396
32 105,959,504 423,838,016 211,915,132 847,660,528
33 136,719,440 560,557,456 273,438,880 1,121,099,408
34 175,174,205 735,731,661 350,343,565 1,471,442,973
35 222,945,905 958,677,566 445,891,810 1,917,334,783
36 281,963,990 1,240,641,556 563,921,995 2,481,256,778
37 354,465,254 1,595,106,810 708,930,508 3,190,187,286
38 443,085,225 2,038,192,035 886,163,135 4,076,350,421
39 550,858,165 2,589,050,200 1,101,716,330 5,178,066,751
40 681,329,000 3,270,379,200 1,362,649,145 6,540,715,896
41 838,553,320 4,108,932,520 1,677,106,640 8,217,822,536
42 1,027,233,130 5,136,165,650 2,054,455,634 10,272,278,170
43 1,252,716,850 6,388,882,500 2,505,433,700 12,777,711,870
44 1,521,162,500 7,910,045,000 3,042,312,350 15,820,024,220
45 1,839,537,700 9,749,582,700 3,679,075,400 19,499,099,620
46 2,215,814,250 11,965,396,950 4,431,613,550 23,930,713,170
47 2,658,968,130 14,624,365,080 5,317,936,260 29,248,649,430
48 3,179,209,800 17,803,574,880 6,358,402,050 35,607,051,480
49 3,787,984,200 21,591,559,080 7,575,968,400 43,183,019,880
50 4,498,241,475 26,089,800,555 8,996,462,475 52,179,482,355
51 5,324,436,975 31,414,237,53010,648,873,950 62,828,356,305
52 6,282,847,506 37,697,085,03612,565,671,261 75,394,027,566
53 7,391,571,330 45,088,656,36614,783,142,660 90,177,170,226
54 8,670,895,455 53,759,551,82117,341,763,505107,518,933,731
5510,143,295,635 63,902,847,45620,286,591,270127,805,525,001
5611,833,860,640 75,736,708,09623,667,689,815151,473,214,816
5713,770,292,256 89,507,000,35227,540,584,512179,013,799,328
5815,983,392,920105,490,393,27231,966,749,880210,980,549,208
5918,507,065,720123,997,458,99237,014,131,440247,994,680,648
6021,378,872,240145,376,331,23242,757,703,560290,752,384,208
6124,640,032,560170,016,363,79249,280,065,120340,032,449,328
6228,336,060,632198,352,424,42456,672,074,888396,704,524,216
6332,516,764,280230,869,188,70465,033,528,560461,738,052,776
6437,236,965,920268,106,154,62474,473,879,480536,211,932,256
© 2013–2014 Kirill Kryukov
Available under the CC BY 3.0 License