Author: Les Fernandez
Date: 21:27:39 01/18/02
Go up one level in this thread
On January 18, 2002 at 12:50:43, Chris Hull wrote: >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. Cant we say there is actually 4-fold symmetry?? Take a look, the positions are exactly the same: You can rotate top to bottom with pawns on the board and still get 4-fold symmetry legally you just have to follow a few rules. Rotating top to bottom requires you invert the color of all the pieces, reverse the row number and you must invert the side to move. [D]k7/8/8/8/8/8/1P6/K7 w - - acd 4; acn 7; acs 1; ce 32720; pv Ka2; id "-XTDa.1-"; [D]7k/8/8/8/8/8/6P1/7K w - - acd 4; acn 7; acs 1; ce 32720; pv Kh2; id "-XTDb.1-"; [D]k7/1p6/8/8/8/8/8/K7 b - - acd 4; acn 7; acs 1; ce 32720; pv Ka7; id "-XTDc.1-"; [D]7k/6p1/8/8/8/8/8/7K b - - acd 4; acn 7; acs 1; ce 32720; pv Kh7; id "-XTDd.1-"; FWIW >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
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