Author: KarinsDad
Date: 13:49:33 03/20/00
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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 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 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 :)
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