Author: blass uri
Date: 17:05:06 05/21/99
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On May 21, 1999 at 19:33:51, blass uri wrote: >This seems absurd also to me. > >I think that we can prove practically that it is impossible. > >I believe that we needs more than 100 bits only to represent the positions when >every side has exactly 2 queens,2 rooks,2 bishops,2 knights,2 pawns. > >The first step to prove it is to create 1000 random positions with this material >configuration when the pawns are at legal squares. > >It is not hard to create the random positions by a program. > >we have 48*47/2=1128 cases for white pawns. >46*45/2=1035 cases for black pawns, >60*59=3540 cases for kings > >58*57/2=1653 cases for white queens, 56*55/2=1540 cases for black queens >54*53/2=1431 cases for white rooks,52*51/2=1326 cases for black rooks >50*49/2=1225 cases for white bishops,48*47/2=1128 cases for black bishops >46*45/2=1035 cases for white knights,44*43/2=946 cases for black knights > >If we multiply these 11 numbers we get more than 2^110 > >The second step is to check how many random position out of 1000 are legal. > >If we assume that there are less then 2^100 positions then we expect probability >of less than 1/1024 for a legal position by putting the pieces at random >squares. > >I believe that more than 1/1000 of these random positions are legal but I am not >sure because I did not do this experiment. > >Uri I have a better idea to prove that it is impossible. do not put the kings and only prove for every position that we can choose squares for the kings. we will have 1128*1035*1770*1653*1540*1431*1326*1225*1128*1035 clearly bigger than 2^101 and I believe that in most of the cases we can put the kings such that the position is legal. If we take 100 cases and see that in 90 out of them we can put the kings such that the position is legal then we have a probabilistic proof that we cannot represent all the positions of chess by 100 bits. for choosing random positions we only have to choose 24 random different integers(4 for pawns and 20 for the other pieces). Uri
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