Re: FYI & my annoyment

[Ray Tayek <rtayek@xxxxxxxxxx>]

At 05:24 PM 4/21/99 +0300, Antti Huima wrote:
>On Tue, 20 Apr 1999, Mousheng Xu wrote:
>...
>It seems that the claim 4!->9! is based on a model where every
>intersection is filled exactly once and no captures are made.
>
>With this model, 9! = 362880.
>
>However, a more careful calculation under the same model yields a
>different result:
>
>Let `P[b,w]' denote the number of distinct positions with `b' black and
>`w' white stones.
>
>We have
>           
>  P[b,w] = C(9, b) * C(9-b, w)
>
>where C(n,k) denotes the binomial coefficient, i.e. the number of
>k-subsets from a set of size n. The total number of positions to examine
>(in the model given) is
>
>  P[1,0] + P[1,1] + P[2,1] + P[2,2] + P[3,2] + P[3,3]
>         + P[4,3] + P[4,4] + P[5,4]
>
>which has the numerical value of
>
>  6045
>
>Comparing this with 9! = 362880 shows that, under this model, using a
>transposition table would immediately save 98.33% of the work on the 3 x 
>3 board.
>

since you can have captures and or passes, i think its a little bigger
(19683).

size=9
c(9,0)=1, c(9,0)=1, p[b,w]=p[0,0]=1
c(9,0)=1, c(9,1)=9, p[b,w]=p[0,1]=9
c(9,0)=1, c(9,2)=36, p[b,w]=p[0,2]=36
c(9,0)=1, c(9,3)=84, p[b,w]=p[0,3]=84
c(9,0)=1, c(9,4)=126, p[b,w]=p[0,4]=126
c(9,0)=1, c(9,5)=126, p[b,w]=p[0,5]=126
c(9,0)=1, c(9,6)=84, p[b,w]=p[0,6]=84
c(9,0)=1, c(9,7)=36, p[b,w]=p[0,7]=36
c(9,0)=1, c(9,8)=9, p[b,w]=p[0,8]=9
c(9,0)=1, c(9,9)=1, p[b,w]=p[0,9]=1
c(9,1)=9, c(8,0)=1, p[b,w]=p[1,0]=9
c(9,1)=9, c(8,1)=8, p[b,w]=p[1,1]=72
c(9,1)=9, c(8,2)=28, p[b,w]=p[1,2]=252
c(9,1)=9, c(8,3)=56, p[b,w]=p[1,3]=504
c(9,1)=9, c(8,4)=70, p[b,w]=p[1,4]=630
c(9,1)=9, c(8,5)=56, p[b,w]=p[1,5]=504
c(9,1)=9, c(8,6)=28, p[b,w]=p[1,6]=252
c(9,1)=9, c(8,7)=8, p[b,w]=p[1,7]=72
c(9,1)=9, c(8,8)=1, p[b,w]=p[1,8]=9
c(9,2)=36, c(7,0)=1, p[b,w]=p[2,0]=36
c(9,2)=36, c(7,1)=7, p[b,w]=p[2,1]=252
c(9,2)=36, c(7,2)=21, p[b,w]=p[2,2]=756
c(9,2)=36, c(7,3)=35, p[b,w]=p[2,3]=1260
c(9,2)=36, c(7,4)=35, p[b,w]=p[2,4]=1260
c(9,2)=36, c(7,5)=21, p[b,w]=p[2,5]=756
c(9,2)=36, c(7,6)=7, p[b,w]=p[2,6]=252
c(9,2)=36, c(7,7)=1, p[b,w]=p[2,7]=36
c(9,3)=84, c(6,0)=1, p[b,w]=p[3,0]=84
c(9,3)=84, c(6,1)=6, p[b,w]=p[3,1]=504
c(9,3)=84, c(6,2)=15, p[b,w]=p[3,2]=1260
c(9,3)=84, c(6,3)=20, p[b,w]=p[3,3]=1680
c(9,3)=84, c(6,4)=15, p[b,w]=p[3,4]=1260
c(9,3)=84, c(6,5)=6, p[b,w]=p[3,5]=504
c(9,3)=84, c(6,6)=1, p[b,w]=p[3,6]=84
c(9,4)=126, c(5,0)=1, p[b,w]=p[4,0]=126
c(9,4)=126, c(5,1)=5, p[b,w]=p[4,1]=630
c(9,4)=126, c(5,2)=10, p[b,w]=p[4,2]=1260
c(9,4)=126, c(5,3)=10, p[b,w]=p[4,3]=1260
c(9,4)=126, c(5,4)=5, p[b,w]=p[4,4]=630
c(9,4)=126, c(5,5)=1, p[b,w]=p[4,5]=126
c(9,5)=126, c(4,0)=1, p[b,w]=p[5,0]=126
c(9,5)=126, c(4,1)=4, p[b,w]=p[5,1]=504
c(9,5)=126, c(4,2)=6, p[b,w]=p[5,2]=756
c(9,5)=126, c(4,3)=4, p[b,w]=p[5,3]=504
c(9,5)=126, c(4,4)=1, p[b,w]=p[5,4]=126
c(9,6)=84, c(3,0)=1, p[b,w]=p[6,0]=84
c(9,6)=84, c(3,1)=3, p[b,w]=p[6,1]=252
c(9,6)=84, c(3,2)=3, p[b,w]=p[6,2]=252
c(9,6)=84, c(3,3)=1, p[b,w]=p[6,3]=84
c(9,7)=36, c(2,0)=1, p[b,w]=p[7,0]=36
c(9,7)=36, c(2,1)=2, p[b,w]=p[7,1]=72
c(9,7)=36, c(2,2)=1, p[b,w]=p[7,2]=36
c(9,8)=9, c(1,0)=1, p[b,w]=p[8,0]=9
c(9,8)=9, c(1,1)=1, p[b,w]=p[8,1]=9
c(9,9)=1, c(0,0)=1, p[b,w]=p[9,0]=1
positions=19683
9!=362880

Ray (will hack java for food) http://home.pacbell.net/rtayek/
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