Author: Norm Pollock
Date: 18:24:16 05/16/04
Go up one level in this thread
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
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