Total Number of Positions
Here are the total number of checker positions sorted according to the number of pieces on the board.
# PIECES # POSITIONS
1 120
2 6,972
3 261,224
4 7,092,774
5 148,688,232
6 2,503,611,964
7 34,779,531,480
8 406,309,208,481
9 4,048,627,642,976
10 34,778,882,769,216
11 259,669,578,902,016
12 1,695,618,078,654,976
13 9,726,900,031,328,256
14 49,134,911,067,979,776
15 218,511,510,918,189,056
16 852,888,183,557,922,816
17 2,905,162,728,973,680,640
18 8,568,043,414,939,516,928
19 21,661,954,506,100,113,408
20 46,352,957,062,510,379,008
21 82,459,728,874,435,248,128
22 118,435,747,136,817,856,512
23 129,406,908,049,181,900,800
24 90,072,726,844,888,186,880
Total: 500,995,484,682,338,672,639
Of particular interest are those positions where either the material is even if there is an even number of pieces on the board, or the difference is no more than one when there are an odd number of pieces present (for example, 4 vs 3 and 3 vs 4 for 7 pieces).
1 vs 0: 60 (x 2)
1 vs 1: 3,488
2 vs 1: 98,016 (x 2)
2 vs 2: 2,662,932
3 vs 2: 46,520,744 (x 2)
3 vs 3: 783,806,128
4 vs 3: 9,527,629,380 (x 2)
4 vs 4: 111,378,534,401
5 vs 4: 998,874,699,888 (x 2)
5 vs 5: 8,586,481,972,128
6 vs 5: 58,769,595,279,296 (x 2)
6 vs 6: 384,033,878,250,176
7 vs 6: 2,046,244,120,757,760 (x 2)
7 vs 7: 10,359,927,057,187,840
8 vs 7: 43,428,742,062,013,440 (x 2)
8 vs 8: 171,975,762,422,069,760
9 vs 8: 569,058,493,921,640,448 (x 2)
9 vs 9: 1,765,698,358,650,175,488
A vs 9: 4,596,454,069,579,874,304 (x 2)
A vs A: 11,113,460,838,901,284,864
B vs A: 22,520,313,165,772,750,848 (x 2)
B vs B: 41,842,926,176,229,654,528
C vs B: 64,703,454,024,590,950,400 (x 2)
C vs C: 90,072,726,844,888,186,880
Total number of positions: 329,847,169,676,858,217,781
