Author: Chris Hull
Date: 09:50:43 01/18/02
Go up one level in this thread
On January 18, 2002 at 01:52:23, Les Fernandez wrote: >On January 18, 2002 at 01:39:45, Chris Hull wrote: > >>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?). > >I think I can but do those other 4 symmetries exist for all chess positions >including pawns?? <S>. I was not aware that the estimates took symmetries into >account. Thanks for the info. BTW what is the current estimate out of >curiosity? > For positions with pawns, you only have 2-fold symmetry. I haven't looked at the estimates in a while but the last ones I recall were on th order of 10^65 unique positions. Basically, a very large number. Chris > > >>Also, the estimated number of unique chess position already take into account >>this reduction due to symmetry. >> >>Chris
This page took 0.01 seconds to execute
Last modified: Thu, 15 Apr 21 08:11:13 -0700
Current Computer Chess Club Forums at Talkchess. This site by Sean Mintz.