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SBlkWlf wrote on Sun, Jul 7, 2002 10:31 AM UTC:
Joe Joyce wrote on Mon, Oct 4, 2010 06:42 PM UTC:I'd been thinking about this particular question for a little while, and happened to run across this, unanswered, from 2002: 2002-07-07 SBlkWlf Unverified None i'm just wondering why in most multi-level chesses the boards alternate their patterns (instead of a constant white in the bottom right), and this is the only place i could think to ask. Is it necessary for some reason? Would colorbound pieces be affected adversely if the boards weren't so arranged ? Anyone...anyone... The answer is that it depends on the exact movement patterns of your pieces through the levels. For the most common extension of pieces from 2D to 3D and 4D, the reverse checkering is required. However, when I designed my first chess variant, Hyperchess, I didn't realize this, so I made every level the same pattern of light and dark. For the simplified 4D movement which Hyperchess uses, this is a much better board design. I suspect that it's possible, even likely, that how you checker your higher-dimensional board levels will affect your game design on that board. The different patterns lend themselves to different ways of looking at how to design piece moves.
The alternating of colours is indeed to do with the two Bishop bindings, as each step of the Bishop moves comprises simultaneous steps in two out of forward/backward, left/right, and up/down. In other words the move is rotationally symmetric, so that a series of levels is the equivalent to a series of vertical planes. Now on the 2d board the colouring of the ranks alternates, and the colouring of the files alternates. A 2d plane with a rank or file entirely of white cells followed by one entirely of black cells, and so on, would look very odd, either as a whole board or a single plane of a 3d board. In that context a column of cells all the same colour would also be very odd.
I seen one 3D variant, where bishop moves as bishop only within one level, to change it, he must move as unicorn. If bishop moves in this way, all cells of same column should be like-colored. It's not logical enough, but, maybe, it's easier to visualize than real bishop move. I don't remember, how other pieces moves in this game. Rook probably had only normal rook's moves, but it would be more logical if rook also were able to move as bishop without changing file and rank (and changing level). And knight should move as knight within it's level and as 2,1,1 leaper to go to next or previous level.
Joe Joyce wrote on Mon, Oct 11, 2010 09:07 PM UTC:Daniil, you are right about the difficulties of visualization. There are two inherent problems of higher-dimensional chess. One is the slippery king; without significant restrictions, the king is exceedingly difficult to mate. The other is the sheer number of directions that a piece can move. On a 2D board, the Q moves in 8 directions; on a 3d, 26 directions, and on a 4D board, a 4D queen can move in 80 different directions, and a standard knight move hits up to 48 cells on a large enough 4D board. How do you project several moves into the future? How do you even predict what your opponent will do next turn? If you limit the moves of all the pieces in higher D games, this can make both problems much more tractable. For 4D, Hyperchess and TessChess provide a nice pair of examples, as they are very well matched. Charles, one observation from playing on the Hyperchess board - the same-color patterning of the 2D 'levels' helps in both 'seeing' the moves and in putting the piece down on the correct location. It's a lot easier to hit the wrong square on a 2D layout of a '4D' board than you might think, odd as that sounds.
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