Author: KarinsDad
Date: 11:17:22 09/03/99
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On September 03, 1999 at 12:53:23, Michel Langeveld wrote: [snip] > >4 = 60 other places for blackking >6 = 58 other places for blackking >9 = 55 other places for blackking > >So there are >1 x 4 = 1 x 60 = 60 >3 x 6 = 3 x 58 = 174 >6 x 9 = 6 x 55 = 330+ > --- > 564 positions for KK. [snip] > >So there are >1 x 4 = 1 x 60 = 60 >6 x 6 = 6 x 58 = 348 >9 x 9 = 9 x 55 = 495+ > --- > 903 positions for KK (in a pawn endgame). [snip] I do not understand this at all. I can understand left and right transpositions (left and right side of the board), but I cannot understand how any other directional transposition would work to minimize this once you start putting on pawns. With no pawns, you can effectively divide by 4 for the corner squares and 8 for the edge squares and 6 for the center squares (it is not really a divide by 6, but rather a divide by 4 directions to get 9 squares where these can be represented in 2 directions by 6 of 9 squares). Another way of saying this is that you divide by 4 for each square, but there are duplicate edge squares and duplicate diagonal squares, hence, those repeats can be removed. So there are 4 / 4 = 1 unique square x 60 = 60 24 / 8 = 3 unique squares x 58 = 174 36 / 6 = 6 unique squares x 55 = 330+ --- 564 positions for KK (as per your method) Effectively, without pawns, you have transpositions due to side to side, 1st rank to 8th rank, and along both diagonals. 1 corner square 3 edge squares 3 interior main diagonal squares 3 interior non-main diagonal squares But with pawns, you can only divide by 2 for all squares since pushing direction is important (i.e. in effect only allowing you to flip the board around the central axis between the d and e files, and not also along both diagonals, the axis between the the 4th and 5th rank). So there are 4 / 2 = 2 x 60 = 120 24 / 2 = 12 x 58 = 696 36 / 2 = 18 x 55 = 990+ --- 1806 positions for KK (in a pawn endgame). Effectively, with pawns, you only have a side to side transposition. 2 corner squares 6 edge squares (3 on the file, 3 on the rank) 6 interior main diagonal squares 12 interior non-main diagonal squares Please explain the other direction the board can be flipped in order to drop 1806 down to 903. I cannot imagine how that would be done. Thanks. KarinsDad :)
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