Author: Dann Corbit
Date: 13:44:37 01/21/05
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On January 21, 2005 at 05:23:55, Uri Blass wrote: >On January 21, 2005 at 04:21:24, Dann Corbit wrote: > >>On January 21, 2005 at 03:33:21, Uri Blass wrote: >> >>>On January 20, 2005 at 20:45:23, Dann Corbit wrote: >>> >>>>On January 20, 2005 at 20:04:22, Louis Fagliano wrote: >>>>[snip] >>>>>Actually 10^43rd power does not shrink at all. You started out "shrinking" by >>>>>throwing out idiotic moves when considering all possible chess games which is >>>>>something like 10^120th power. That number can be "shrunk" by throwing out >>>>>idiotic games. But 10^43rd power is the number is the number of legal positions >>>>>in chess, not the number of different possible games since there are an >>>>>inumberable ways of reaching any particular legal position by an inumberable >>>>>number of different move orders. The number of legal positions can never be >>>>>"shrunken" because every legal position must be considered in order to solve >>>>>chess regardless of whether or not the moves that preceeded it in order to reach >>>>>that position were idiotic or masterful. >>>> >>>>10^43rd power can be shrunken by a factor of 4 through simple reflections of the >>>>board. Perhaps there are additional symmetry arguments that can reduce it >>>>further. >>> >>> >>>Where is the proof that 10^43 is correct? >>> >>>I read that Vincent claimed that it is correct but I saw no proof for it. >>>I do not say that it is wrong but we need a link to some proof before claiming >>>that it is at most 10^43 >> >>That is based upon 162 bits to encode the position. >>I believe that you did a counter program which came up with a different figure. >>Do you remember what it was? >162 bits means 2^162>10^48 so I do not see how you get 10^43 > >I remember that I got something smaller than 10^47 but not 10^43 Yes, your program said 3.70106e+046 Probably, the quote comes from Shannon. I. J. Good came up with estimates for this sort of thing in 1968 "Shannon states that the number of possible chess positions is of the general order of 64!/(32!*8!^2*2!^6) or roughly 10^43. The formula indicates that he was assuming that no pawn had been promoted. In Good (1951), I calculated that the number of positions in which no pawn has been promoted, and there are no doubled pawns, is less than 2*10^39. The number of positions in which no capture has occurred is about 10^32." So the 10^43 is probably an underestimation (did not consider promotions). I am guessing that your figure of 3.7e46 is as good as any. So a factor of 3000 larger than 10^43.
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