Author: Marc Bourzutschky
Date: 13:24:34 08/08/01
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On August 08, 2001 at 15:37:02, Ian Kennedy wrote: >Is it always possible to predict what level of symmetry exists in a given eg db >given only knowledge of how the pieces move? > >I initially thought yes but after mucking around with generating KBNK I'm now >less sure. I thought it reasonable that this was 4-way symmetric* until it >failed to get the 'right' results (as measured against other dbs, books) and >more importantly I reread Reuben Fine where he says that the king must first be >driven to a corner with opposite color to the bishop before mating in a >same-coloured corner. Even then, how do you know that some solutions do not >cross midway into a third quadrant, or whether 2-way symmetry will now suffice? > >Or is there some simple ruleset that always applies eg any pawn eg will always >have horizontal symmetry only, any with Rook, Queen, Knight will have horizontal >and vertical, and with Bishop(s) ...? > >* eg if a dark squared bishop is mating the bk on a1 then that will mirror to h8 >and a *light squared* bishop mate will mirror at a8 and h1 > >Ian Kennedy The chessboard actually has 8 symetries: Horizontal (b3 <-> g3 etc) Vertical (b3 <-> b6 etc) Diagonal (b3 <-> c2 etc), and all possible combination of those, which will give you rotations by 90, 180, 270 degrees, and diagonal reflection along the h1-a8 diagonal (one symmetry is the trivial "do nothing" symmetry). Since all the piece movements except for pawns are symmetric under all these transformations, symmetry alone should therefore allow a reduction of a factor of eight in the number of positions necessary for endgames without pawns. Most people approximately achieve this factor of eight by restricting one of the kings to the a1-d4-d1 triangle, and the other king to squares on or below the a1-h8 diagonal if the first king is on that diagonal. Some of the symmetry operations clearly map white squares into black squares, and to utilize those symmetries to achieve all positions you therefore must allow bishop positions on both colors. Alternatively, you can restrict the bishop to one color if you exclude all the symmetry operations that change the color, but that is more cumbersome. -Marc
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