Author: KarinsDad
Date: 11:17:22 09/03/99
Go up one level in this thread
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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