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Subject: Total of positions of the Checker game - smaller in compared to chess ??

Author: Tania Devora

Date: 23:14:20 01/23/01



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.

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