Author: Sune Fischer
Date: 17:52:40 01/18/02
Go up one level in this thread
On January 18, 2002 at 20:24:52, Uri Blass wrote: >On January 18, 2002 at 19:49:25, Sune Fischer wrote: > >>On January 18, 2002 at 19:26:08, Chris Hull wrote: >> >>>On January 18, 2002 at 19:06:26, Chris Hull wrote: >>> >>>>On January 18, 2002 at 17:13:29, Sune Fischer wrote: >>>> >>>>> >>>>>Hehe, this was the old: >>>>>(64*(64 - 4)*62!)/((62 - 30)!*(8!*2!*2!*2*2)^2)= 1.1*10^42 >>>>>capture a piece and turn a pawn into a queen: >>>>>(64*(64 - 4)*62!)/((62 - 29)!*7!*8!*2!(2!*2!*2*2)^2)= 1.33*10^41 >>>>> >>>>>OK, so its about a factor 10 or maybe about same order of magnitude, they drop >>>>>fairly quickly though. >>>>> >>>>>>Here is the table of numbers Uri's program dumps as it goes (which is the list >>>>>>by category): >>>>> >>>>>I would like to know his approach ;) >>>>> >>>>> >>>>>I'm quite sure it is, unless someone can find a flaw in the product (which there >>>>>could be of cause;) >>>>>A number like 64 squares for the king is very optimistic, most of the squares >>>>>will be attacked, you can't be in check if it isn't your turn, and you can't be >>>>>in check by more than two pieces (right?) and so on, many rules that will >>>>>diminish the final product. In particular the pawn movement, pawns have a rather >>>>>limited number of squares they can go to, if just one pawn has half of 64 >>>>>squares, then that is a factor 2 smaller yet. >>>>> >>>>>-S. >>>>> >>>> >>>>Does this take into account all the possible promotions? It is possible to have >>>>9 queens on one side or 9 rooks or 9 bishops or 9 knights, or any combination >>>>thereof. Each possible promotion will make the number larger. 31 pieces on the >>>>board has more possible positions than 32 because you have to take into account >>>>the promotions. If 31 pieces are remaining there is upto 3 possible promotions. >>>>Makes the calculation a little more difficult. >>>> >>>>Chris >>>> >>> >>>Oops, make that 9 queens or 10 rooks, or 10 bishops or 10 knights. Some days my >>>fingers work faster than my brain. >>> >>>Chris >> >>You do have a point Chris. >>I tried to refine the number a little bit: >> >>All 32 pieces: >>(64*(64 - 4)*23^4*21^4*18^4*14^4*62!)/((62 - 14)!*(8!*2!*2!*2*2)^2) >>= 5.13*10^36 >> >>I put in the factor: 23^4*21^4*18^4*14^4 >>because the d and e pawns has 23 squares, the c and f pawns has 21 squares and >>the b and g has 18, a and h has 14 (unless I miscounted;) > >I do not understand it Actually there are 24 squares for a white d pawn: 8/pppppppp/PPPPPPPP/PPPPPPP1/1PPPPP2/2PPP3/3P4/8 w - - 0 1 with 32 pieces it is stopped by the black pawns. The correct factor would be: 24^4*22^4*19^4*15^4 but actually it doesn't change the result all that much. >if there are 32 pieces you can say that for every file you have exactly 15 >options to put the pawns so you get 15^8 options to put the pawns. why 15? You've lost me. >I do not understand how do you get 23 squares for the d and e pawns. You're right, it should be 24 (see diagram). >It is clear that you can reduce the number of possibilities by a big factor with >32 pieces. > >Things are less clear with less pieces. True enough, but the factorial 62!/(62-30)! decreases rapidly as you lower the number of pieces. You get a factor 4-5 back in front, but it is not enough to make up the loss. >I guess that 10^40 is close to the right number of legal positions and I am not >sure if the real number is lower or bigger. I believe the correct number it is around 10^33, so many upper limits could easily cost a factor 100. >It is going to be hard to prove 10^40 as a bound for the number of legal >position and I believe that it may be possible to get an estimate by the monta >karlo method of generating random positions and finding the number of legal >positions from them. > >I have ideas to improve my program for the upper bound and it can help in >generating random positions but I did not find it as important to waste more >time about it. > >Uri I will look at your code tomorrow, right now it is 2:54 am, so I'm headed off to bed;) So long amigos -S.
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