Author: Paul
Date: 16:46:49 03/21/00
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Hi Dad :), Thanks for the reply, hope I didn't steal to much time away from Karin :) Some notes below. On March 20, 2000 at 16:49:33, KarinsDad wrote: >On March 19, 2000 at 19:37:04, Paul wrote: > >>The last couple of hours I've spent trying to find the number of >>possible positions with 3 pieces on the board. I've come up with >>the number 3891550, using a little program I just wrote. >>This number is lower than Hans Havermann's, but higher than >>KarinsDad's. Maybe someone can straighten this out :) >> > >It's possible that you have very little or no error in your calculations. My >calculations were not meant to come up with the exact number. They were meant >to show that Hans' calculations were off by quite a bit. I did my calculations >by hand and figured that they were incorrect (but close). In fact, I realized >that my castling calculations were off a little after I posted it, but it >wasn't enough to worry about. > >In fact, it is not too hard to show that your calculation is probably a lot >closer than mine. > >For your calculation, 3891550 / 7244 results in an average of 537 possibilities >of a single piece for each king/king position. > >For my calculation, 3761048 / 7244 results in an average of 519 possibilities >of a single piece for each king/king position. > >I could "pretend" that the king often gets in the way of a check for a piece >(i.e. for a given king/king position, the side to move king blocks out some >illegal positions, but not all of them). The number of positions should >theoretically fall below this number. > >I could "pretend" that the side to move king never gets in the way of a check >for a piece (i.e. for a given king/king position, the side to move king never >blocks out some illegal positions) and the pawn for side to move cannot check >the king, so it is limited to only 46 squares. The number of positions should >theoretically fall above this number (since we are making a bad assumption >here). > >King often in the way: > >62 * 4 = 248 piece moves for side to move does not own piece (no illegals) >48 * 1 = 48 pawn move for side to move does not own piece (no illegals) >48 * 1 = 48 queen (14 queen squares cannot be blocked) >55 * 1 = 55 rook (7 rook squares cannot be blocked) >62 * 1 = 62 bishop (all bishop squares can be blocked, king in corner) >58 * 1 = 58 knight (high average) >47 * 1 = 47 pawn (high average) >Total = 566 I think that in the above the queen and bishop numbers can be incremented by one, since there's always a square between the kings that can be filled by the piece. Not that it matters much of course (564 iso 566). >King never in the way: > >62 * 4 = 248 piece moves for side to move does not own piece (no illegals) >48 * 1 = 48 pawn move for side to move does not own piece (no illegals) >41 * 1 = 41 queen (21 queen squares cannot be blocked) >48 * 1 = 48 rook (14 rook squares cannot be blocked) >48 * 1 = 48 bishop (14 bishop squares cannot be blocked) >54 * 1 = 54 knight (low average) >46 * 1 = 46 pawn (low average) >Total = 533 Here I don't really understand the queen and bishop numbers. The queen one seems very high (21 squares is only at the edge of the board, in the centre it's 6 more), and the bishop one is low (at most 13 cannot be blocked, I think). So I guess the total should be adjusted downward, but again it doesn't matter much, it's the thought that counts :) >Your 537 falls within the range (and actually comes close to what I would guess >would be the proper value based on the charts here), my 519 does not. So, you >are probably much closer than I was. I expected there to be some error in my >calculations since I did them by hand. > >KarinsDad :) Yes, I know, I cheated by using a computer to calculate :), but I was curious for the exact number, and it was only a minor addition to my chess engine. Further more I'm an applied mathematician, not a 'pure' one ;) The difficult thing I find in calculating the numbers is, what is the definition of a legal position ? Eg. the following 4-piece position is legal, but not reachable in a game, and therefore illegal (not to be counted)? [D]7k/5K1P/7Q/8/8/8/8/8 w - - Eliminating these kind of positions is not something I would want to have to think about when writing a calc-routine. Oh, the number of positions excluding castling/ep/etc I've found sofar are: F[2] = 7 224 F[3] = 3 891 168 (538) F[4] = 1 035 174 640 (266) F[5] ~ 200 000 000 000 (193) Didn't have time to complete the 5-piece one, have to speed up my code :) Thanks again for the response, was appreciated. cheers, Paul
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