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From: victor@cs.vu.nl (Victor Allis)
Subject: Re: Chess - exhaustive searching
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Date: Wed, 12 Apr 1995 07:48:54 GMT
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geert@sparc.aie.nl (Geert-Jan van Opdorp) writes:

>I think you ought to do Sum(64!/(64-x)!), {x, 32}, since different
>distributions of the pieces over the same squares constitute
>different positions. Again, of course, we are only talking of an
>upeer bound, since pawns of same color *can* be exchanged.
>Now we get ~ 5* 10^53 (!!).

>Geert-Jan

This is an upper bound for the standard set of pieces, and all positions
after captures. However, you can have minor promotions in chess.

I have written a computer program about two years ago, which determines
the number of promotions you can have, in relation to the number of captures
you must make (for instance, you cannot have 16 promotions without having
the pawns making captures, since they need to pass each other).

For each of the sets of pieces you could get in this way, I calculated
an upper bound on the number of legal positions, where I included that a pawn
is restricted in some ways.

Anyway, the bottom line is that I got 10^53 as an upper bound including
all of the above. Given that quite a few positions will be illegal, I
normally quote 10^50 as a reasonable estimate. Of course, using a Monte
Carlo simulation we could determine a good ratio of legal vs. illegal
positions and make the estimate more accurate. That would take
a lot of computer time, though, and does not seem worth it.

Victor Allis.
