Author: Dan Ellwein
Date: 09:26:03 01/14/00
Go up one level in this thread
On January 14, 2000 at 04:10:33, Michael Neish wrote: >On January 13, 2000 at 02:54:31, Timothy J. Frohlick wrote: > >>I don't think the "a" pawn is going to end up on the e,f,g,or h files and don't >>forget that the black pawns prevent occupation except by capture of the white >>pawns. You also have to account for en passant which usually occurs only once >>per game at most. >> >>[D] 8/pppppppp/8/8/8/8/PPPPPPPP/8 >> >>So the progression should be 20x19x18....etc and be extended out for a total of >>sixteen pawns. 20x19x18x17x16x15x14x13x12x11x10x9x8x7x6x5x4= 8.109673360589e+17 >>which is a pretty big number of possibilities. Even if we limited the combos >>to (Whitemoves x 10x9x8x7x6x5x4x3) + (Blackmoves x 10x9x8x7x6x5x4x3)=3,628,800 >>possibilities you can see that we will not run out of moves soon. If the >>average game is 75 moves or 150 ply we would have to play at least 24,192 games >>to exhaust the possibilities. Again, since not every move is a pawn move we >>would have a variable number of pawn moves and positions. So if only one >>quarter of our moves are pawn moves we would have to play about 100,000 games to >>come to a correct estimation of the number of different pawn structures. >> >>This is just my opinion and is probably wrong. >> >>Tim Frohlick > >Very interesting comments. I never thought of that. The best I can >do -- for sixteen Pawns, which seems to be the one giving the most >possibilities -- is 15^8, which is 2.563e9, which is a whole lot less >than the "naive" calculation I slogged through in another post of >about 2.9e16 (for 16 Pawns). For sixteen Pawns my assumption is >that all Pawns are on their original files, although there are >surely many more possiblities if pieces have been captured. > >Okay, let's look at some possibilities. If White has captured just >one piece, creating doubled Pawns on one file, the number of >combinations becomes 3.35e10, by treating each file individually >and multiplying the combinations for each of them. For one doubled >Black Pawn, it's the same, so just multiply this figure by two to >get the combinations for White and Black. > >For one set of doubled Pawns for each side, well, get the computer out. > >For doubled White A-Pawns, Black can have Doubled Pawns on files >A to F, and in addition, for files B-G the file to the left or >right of the doubled Pawns can be empty. > >For the case of doubled White A-Pawns, we get > >White Doubled A, Black Doubled A = 1.025e9 >White Doubled A, Black Doubled B, no B-Pawn on A-file = 2.567e9 >White Doubled A, Black Doubled B, no B-Pawn on C-file = 4.101e9 >White Doubled A, Black Doubled C, no B-Pawn on B-file = 2.734e9 >White Doubled A, Black Doubled C, no B-Pawn on D-file = 6.561e9 >White Doubled A, Black Doubled D, no B-Pawn on C-file = 6.561e9 >White Doubled A, Black Doubled D, no B-Pawn on E-file = 6.561e9 >White Doubled A, Black Doubled E, no B-Pawn on D-file = 6.561e9 >White Doubled A, Black Doubled E, no B-Pawn on F-file = 6.561e9 >White Doubled A, Black Doubled F, no B-Pawn on E-file = 6.561e9 >White Doubled A, Black Doubled F, no B-Pawn on G-file = 6.561e9 >White Doubled A, Black Doubled G, no B-Pawn on F-file = 6.561e9 >White Doubled A, Black Doubled G, no B-Pawn on H-file = 6.561e9 >White Doubled A, Black Doubled H, no B-Pawn on G-file = 6.561e9 > >Which comes to 7.5012e10. > >And this is for a doubled White A-Pawns only. I couldn't >be bothered to write a program to do it for all possibilities >for one set of doubled Pawns for each colour, so if someone >does it could you post the results on here? Thanks. > >By the way, why are we doing this? Academic reasons or is >someone thinking of setting up a Pawn database? Michael... I was doing this just to get a 'feel' for what's involved in chess... obviously... chess is not for the faint of heart... i see... :) and yes... setting up a pawn database is in the works... PilgrimDan > >This is a tricky business! > >Cheers, > >Mike.
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