consider a large cube made of congruent small cubes. consider just the small cubes on the outer shell of the large cube, those touching the outside. consider playing chess in those outer cubes.
it is mostly 2D chess among 6 boards, the 6 faces of the large cube. we restrict movement during a turn to a face: no traveling unimpeded over an edge like a true 2D manifold. if a piece starts in an edge cube, it is in 2 faces simultaneously: it can choose which face to move in. (for a computer UI with 6 2D boards, if a piece moves from an edge to an internal square of one board, it disappears from the other board. likewise, if a piece moves to an edge, it appears on another board. the overlapping edges need to be identified, maybe by color.) corner cubes are simultaneously part of 3 faces. because moves are restricted to a face, moves which hop from face to face without stopping on an edge are forbidden. (without this restriction, we might imagine a bishop or knight hopping from face to face bypassing an edge.)
you can only move one piece per turn: pick which board is most urgent.
conventional wisdom for chess, central squares are important, edges are weak, likely no longer holds.
size 8 cube = 296 surface cubes, or 4.625 larger than a standard chessboard
size 6 = 152 = 2.375x
size 4 = 56, but perhaps a little cramped?
generally, n^3 - (n-2)^3.
initial position of pieces and details of piece movement (especially pawns) remain to be specified. checkmates become more difficult because you cannot use the sides or corners of a board to restrict movement of the opposing king.
bishops can change color.
more generally, we can glue together many boards through overlapping edge squares. (previously, not overlapping.) many boards, not just two, can overlap at an edge. similar to abstract polyhedra. a piece in an overlap region occupies many boards simultaneously. tesseract topology might be interesting: 24 boards, 3 boards per edge, 6 boards per corner.
four boards:
W X
Y Z
bottom edge of W and top edge of Y overlap, populated with white's pieces in initial position. bottom edge of X and top edge of Z overlap, similarly more white pieces. top edge of W and bottom edge of Y overlap, black's pieces in initial position. top edge of X and bottom edge of Z overlap, more black pieces.
right edges of W and Y overlap with left edges of X and Z, respectively. left edges of W and Y overlap with right edges of X and Z, respectively.
approximately flat torus topology.
white king starts at bottom right corner of W, equivalently bottom left corner of X, top right corner of Y and top left corner of Z. there is only one white king. similarly, black king at top right corner of W, also occupying all 4 boards simultaneously. whatever pieces (perhaps queens) that start at the top left corner and bottom left corner of W are also shared between all 4 boards.
assume 8x8 boards. pawns start on 2nd and 7th ranks of all 4 boards. white pawn a2 is also on h2 of a horizontally adjacent board, similarly a7 and h7. because of sharing, each player starts with 28 pawns. white pawns in W and X, starting on the second rank, advance upward. white pawns in Y and Z, starting on the seventh rank, advance downward. black pawns the other way. pawn promotion is the other end of the board. promotion is particularly powerful because promotion zones are simultaneously part of 2 boards (or 4 if corner).
4 boards, each n-by-n, has (2n - 2)^2 distinct squares. 196 squares for n=8. intriguing is n=5, 64 squares, which starts with the same number of non-pawn pieces as orthodox chess but twice as many pawns. opposing pawns start separated by only one rank. lengthening each board to 5x8 restores 4 rank initial separation. (2n-1)*(2m-1) distinct squares: 135 squares for 5x8.
even more generalization: each board can be any shape composed of squares (polyominoes, polyplets). an overlap region between a pair of boards is a bijection between subsets of squares (not necessarily edge squares).
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