[rc5] Chess - Many possible boards

Christopher Hodson (Consultant) cmh at fpk.hp.com
Thu Oct 30 16:00:05 EST 1997


Tobias Bergmann wrote:
> 
> Something about the problem of saving a list of all possible boards:
> 
> Background:
> One piece has 64 possible positions on the board. Two pieces have 64*63
> possible positions. So 32 pieces have 64*63*...*33 (<- yes it is a 33)
> possible positions. With that in mind the upper limit of all possible
> boards is:
> 
>  32
>  ---
>  \     64!
>   |  -------   = 497275065157795229721406525452498161822005325091061504
>  /   (64-i)!
>  ---
>  i=1
> 
> This is about 2^178!
> Of course there are some boards that aren't possible (e.g. a pawn has no
> chance to get to the first row!) but there are quite *enough* possible
> constellations!!!
> Does someone have a better guess about the total number of boards we
> have
>  to evaluate?
> 

Your calculation assumes that all pieces are unique.  For example,
switching
white rooks is not a new position.

I took you number and removed redundencies for rooks, bishops, knights,
and pawns
and got 4779425123005631574633202829863131024761255.
Still too large to process, but going in the right direction(down). 
However, in
your calculation you forgot a very significant thing: pieces can be
captured.
So you also have all the board combinations where there are two kings(or
two
queens since we're not taking out illegal positions yet).  Quite a lot.


--
Christopher M. Hodson
System Admin (Consultant)
Hewlett Packard Engineering Services Group
cmh at fpk.hp.com
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