Author: Uri Blass
Date: 23:53:55 01/23/01
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On January 24, 2001 at 02:14:20, Tania Devora wrote: > > >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 > > >Seems to be a smaller number compared to all chess positions. > >There are any formula like this to know how many position exist in chess? > > > > >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 3: 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 > > >Thanks! >Tanya,D. I think that these numbers are not correct and they are only upper bounds. It is clear that there are many illegal positions that cannot be reached from the initial position and it seems that you counted them(otherwise the C vs V number seems too big). It is clear that chess is a more complicated game. Uri
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