Author: Marc Bourzutschky
Date: 19:20:23 05/16/04
Go up one level in this thread
On May 16, 2004 at 21:24:16, Norm Pollock wrote: >On May 16, 2004 at 20:51:49, Marc Bourzutschky wrote: > >>On May 16, 2004 at 20:40:35, Norm Pollock wrote: >> >>>On May 16, 2004 at 20:21:38, Marc Bourzutschky wrote: >>> >>>>On May 16, 2004 at 19:56:08, Norm Pollock wrote: >>>> >>>>>On May 16, 2004 at 17:27:10, Marc Bourzutschky wrote: >>>>> >>>>>>On May 16, 2004 at 17:12:12, Dieter Buerssner wrote: >>>>>> >>>>>>>On May 16, 2004 at 14:28:48, Marc Bourzutschky wrote: >>>>>>> >>>>>>>>Max Euwe: 4,147,200 >>>>>>>>Noam Elkies: 8,294,400 >>>>>>>>Paul Epstein: 5,317,600 >>>>>>>>Marc Bourzutschky: 5,149,368 >>>>>>> >>>>>>>Dieter Bürßner: 4,665,582 >>>>>>> >>>>>>>Idea: 2880 positions per side, of which 2694 have no castling possibilities. >>>>>>> >>>>>>>x = 2880^2-2694^2/2 >>>>>>> >>>>>>>I fear, I thought too simple, >>>>>>>Dieter >>>>>> >>>>>>If instead of 2694 in your formula you use 2508 you get the Bourzutschky result. >>>>>> The difference is that 2508 is the number of positions where neither the >>>>>>position itself, nor the mirrored position, has castling rights... >>>>> >>>>>Fwiw, I disagree with the explanation of 2508. >>>>> >>>>>I think the 2508 is just the number of positions that do NOT have castling >>>>>positions. I calculate 372 castling positions of the 2880 possible positions, >>>>>therefore 2508 in my calculations is the number of positions without castling >>>>>rights. >>>>> >>>>>x = 2880^2 - (2508^2/2) = 5,149,368 is still correct. >>>>> >>>>>It says take all the positions of both sides then remove the duplicate of those >>>>>positions where neither side had castling rights. >>>>> >>>>>-- Norm >>>> >>>>I only calculate 186 castling positions for white, assuming the white king has >>>>to be on e1. In addition, your argument does not seem quite right, because even >>>>if one side has castling rights the symmetry is broken, even if the other side >>>>does not have castling rights... >>> >>>Would you show me your analysis for the 186 castling positions you calculated >>>for white, assuming the white king to be on e1? I want to see where I differ >>>from you. It's a factor of 2 so it should be easy to spot. I'm putting my >>>analysis for 372 castling positions below. >>> >>>For ----K--R, there are 3*3 ways for 2 bishops of opp color, 1 way for the king, >>>2 ways for the castle at h1, 4*3/2 ways for 2 knights, 2 ways for the queen and >>>1 way for the 2nd castle. >>>Sub-result= 3*3*1*2*(4*3/2)*2*1 = 216 >>> >> >>There is only 1 way for the rook on h1, otherwise you will be double counting >>positions where only the two rooks are swapped. It looks like you have a >>similar issue with your other calculation, which gives a factor of 2 overall. >> >>>For R---K---, there are 4*2 ways for 2 bishops of opp color, 1 way for the king, >>>2 ways for castle at a1, 4*3/2 ways for 2 knights, 2 ways for the queen and 1 >>>way for the 2nd castle. >>>Sub-result= 4*2*1*2*(4*3/2)*2*1 = 192 >>> >>>For R---K--R (to remove duplicates from 2 sub-results), 2 ways for castle at a1, >>>1 way for castle at h1, 1 way for king at e1, 3*2 ways for 2 bishops of opp >>>color, 3*2/2 ways for 2 knights, and 1 way for the queen. >>>Sub result= 2*1*1*3*2*(3*2/2)*1 = 36 >>> >>>Result = 216 + 192 - 36 = 372 >>> >>>-Norm > >OK. So there are 186 white castling positions. 186 white mirrored positions. >These 372 white positions are not logically equal to their respective mirror >positions. The similar 372 black positions are not logically equal to their >respective mirror positions. Therefore instead of excluding half of the possible >full positions (=2880 * 2880)/2, we exclude half of the possible full positions >that are logically equal to their mirrors (=2508^2/2). Am I seeing it right? > >x = 2880^2 - (2508^2/2) = 5,149,368 Sounds right to me.
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