Author: Dan Ellwein
Date: 16:33:39 01/25/00
Hi
Wanted to bounce this off the group again...
my original post was not too accurate:
Number of Pawn Positions
(0,8)(1,7)(2,6)(3,5)(4,4) 8P 48x47x46x45x44x43x42x41 x5 = _______
(0,7)(1,6)(2,5)(3,4) 7P 48X47X46X45X44X43X42 x4 = _______
(0,6)(1,5)(2,4)(3,3) 6P 48X47X46X45X44X43 x4 = _______
(0,5)(1,4)(2,3) 5P 48x47x46x45x44 x3 = _______
(0,4)(1,3)(2,2) 4P 48x47x46x45 x3 = _______
(0,3)(1,2) 3P 48x47x46 x2 = _______
(0,2)(1,1) 2P 48x47 x2 = _______
(0,1) 1P 48 x1 = _______
Number of (non redundant) pawn positions in chess -- TOTAL: _______
8P - 15,214,711,438,080 (48x47x46x45x44x43x42x41)
7P - 371,090,522,880 (48x47x46x45x44x43x42)
6P - 8,835,488,640 (48x47x46x45x44x43)
5P - 205,476,480 (48x47x46x45x44)
4P - 4,669,920 (48x47x46x45)
3P - 103,776 (48x47x46)
2P - 2,256 (48x47)
1P - 48
this (i think) is more accurate...
Number of Pawn Formations In Chess
(4,4) 48x47x46x45/4! x 44x43x43x41/4! = 26,414,429,580
(3,5) 48x47x46/3! x 45x44x43x42x41/5! = 21,131,543,664
(2,6) 48x47/2! x 46x45x44x43x42x41/6! = 10,565,771,832
(1,7) 48 x 47x46x45x44x43x42x41/7! = 3,018,791,952
(0,8) 48x47x46x45x44x43x42x41/8! = 377,348,994
(3,4) 48x47x46/3! x 45x44x43x42/4! = 2,577,017,520
(2,5) 48x47/2! x 46x45x44x43x42/5! = 1,546,210,512
(1,6) 48 x 47x46x45x44x43x42/6! = 515,403,504
(0,7) 48x47x46x45x44x43x42/7! = 73,629,072
(3,3) 48x47x46/3! x 45x44x43/3! = 245,430,240
(2,4) 48x47/2! x 46x45x44x43/4! = 184,072,680
(1,5) 48 x 47x46x45x44x43/5! = 73,629,072
(0,6) 48x47x46x45x44x43/6! = 12,271,512
(2,3) 48x47/2! x 46x45x44/3! = 17,123,040
(1,4) 48 x 47x46x45x44/4! = 8,561,520
(0,5) 48x47x46x45x44/5! = 1,712,304
(2,2) 48x47/2! x 46x45/2! = 1,167,480
(1,3) 48 x 47x46x45/3! = 778,320
(0,4) 48x47x46x45/4! = 194,580
(1,2) 48 x 47x46/2! = 51,888
(0,3) 48x47x46/3! = 17,296
(1,1) 48 x 47 = 2,256
(0,2) 48x47/2! = 1,128
(0,1) = 48
Number of (non redundant) pawn formations - TOTAL: _____________
this total will be reduced by the number of illegal pawn formations associated
with each of the 24 groups...
ie
[D]4k3/8/8/8/8/6PP/7P/4K3
is one example...
and for every one of the 24 groups (0,1 - 4,4) there will be x amount of pawn
positions associated with it...
with the (0,1) group [which is kpk(kkp)] the number of pawn positions would be:
167,248
4 x 48 x 59 = 11,328 4 x 2 = 8
12 x 48 x 57 = 32,832 16 x 3 = 48
12 x 47 x 57 = 32,148 20 x 5 = 100 2 x 58 = 116
36 x 47 x 54 = 91,368 24 x 8 = 192 12 x 55 = 660
167,676 + 348 - 776 = 167,248
thanks...
PilgrimDan
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