Author: Dieter Buerssner
Date: 10:53:49 05/17/04
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On May 17, 2004 at 02:25:34, Tony Werten 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, > >Yes, you also forgot that a white position (no castle ) might be mirrored, but >only if the black position doesn't have castle rights either. I didn't forget that, I only forgot positions that mirror into positions with castling rights. To elaborate on the formula, and the wrong idea: Two positions are different, when they are not congruent and don't mirror each other. The castling rights belong to the position. There are different 2694 positions for one side, that have no castling rights, and 186 positions with castling rights for each side. The 2694 white positions can be combined with 2694 black positions, that neither have castling rights. Mirroring each of the resulting positions will give an identical position (which is not true - as Marc pointed out*). So we counted each of those 2694^2 positions twice. There will be 2694*186 positions where white can castle and black not, 2698*186 positions where black can castle and white not, and 186^2 positions where both can castle. This leaves 2694^2/2 + 2*186*2694 + 186^2 = 2880^2-2694^2/2 different positions. *) The flaw of course is, RNBKQBMR mirrors into RNBQKBNR. The former has no castling posibility, the later has. So instead of arguing with positions that have no castling rights above, one has to argue with positions that have no castling rights, and neither has the mirror position. The number of those positions is 2508 for each side, and we get 2880^2-2508^2/2 different positions. It reminds me of one puzzle, that many people almost solve. Just before the end, they have a flaw in their thinking process, and give the wrong anwer. Two brothers decide, to sell their herd of sheep. For each sheep they get as many Euro, as there where sheep alltogether (say, they had 100 sheep, the would get 100*100 Euro). The price is paid in 10 Euro bills, and the remaining amount in coins of 1 Euro. They start to share the money. The older brother gets the first bill, the younger the second bill, and so on. At the end the younger brother complains: "You got the first and the last one, and have more then me". "Ok", the older brother replied, "you get all the coins". "Sorry, but still you got more money of it than me", said the younger. "Well, I will give you a check, so that we both will have earned the same amount". What amount of money was written on the check? Regards, Dieter
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