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Subject: Re: Approximate # of Unique chess positions = (current estimated #)/4

Author: Chris Hull

Date: 22:39:45 01/17/02

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On January 18, 2002 at 01:25:18, Les Fernandez wrote:

>Every so often this subject gets brought up and I have a thought.  Lets say that
>the current estimated number of unique positions is designated by N.  When we
>say number of positions we need also to take into account some symmetries that
>exist.  If symmetry is found in more then 1 position then we actually do not
>have N number of unique positions.  With this is mind examine the following 4
>diagrams which were generated by my Permutator application:
>
>[D]k1K4R/8/8/8/8/8/8/8 w - -
>acd 4; acn 91; acs 1; ce 32762; pv Rg8; id "-XTDa.1-";
>
>[D]R4K1k/8/8/8/8/8/8/8 w - -
>acd 4; acn 91; acs 1; ce 32762; pv Rb8; id "-XTDb.1-";
>
>[D]8/8/8/8/8/8/8/K1k4r b - -
>acd 4; acn 91; acs 1; ce 32762; pv Rg1; id "-XTDc.1-";
>
>[D]8/8/8/8/8/8/8/r4k1K b - -
>acd 4; acn 91; acs 1; ce 32762; pv Rb1; id "-XTDd.1-";
>
>These 4 diagrams, although symmetrically different, are in fact the identical
>position and contain the identical solution.  This being true implies that when
>estimates for N are performed we need to apply the following 2 rules for
>arriving at a much closer estimate.  #1 N/2 applies for positions that have any
>type of castling rights and #2 N/4 for positions that have no castling rights.
>
>Although N remains still fairly large we are much closer to the right number
>then how it has been estimated in the past.  Whether dividing N by 2 or 4 will
>atleast cut the estimate by 1/2 and I suspect there are many more positions with
>no castling rights then with.  Perhaps other symmetries exist that we have not
>found yet,  and yes I do have some in mind <s>.
>
>Les


Actually there is 8-fold symmetry in this positions (can you find the other 4?).
Also, the estimated number of unique chess position already take into account
this reduction due to symmetry.

Chris

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