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In 3d cubic chess, with an 8x8x8 board (which is true 3d chess), the king would have 26 possible moves (3x3x3 cube around it), the rook 21 possible moves, the knight 24 possible moves (6 2- square directions, then 4 1-square moves after that). The bishop would have 3 times the normal number (3 perpendicular planes of movement with 13 squares each), which would be 39 cubes, and the queen would have 86 moves if u add up the rook moves, bishop moves, and 3D DIAGONAL moves, which the bishop doesn't have=21+39+39-13 for 3d diagonal moves included. The pawns would move up or forward, and could only promote on the top-forward rank on blacks rank at the top (8 squares) Each pawns could capture 5 different forward-up directions, NOT UP-backward. You would need 24 pawns for each side to cover all 1st and 2nd ranks. What's cool is what is now insufficient mating material. the standard rook+king couldn't mate and 2 rooks or 1 queen might not be able to force mate. Really interesting, and I wouldn't even call it a variant. Its just extending chess inot higher dimensions. Imagine combining progressive chess, bughouse, chess960, and this 3D chess into 1 integrated variant. Hard, right!
Personally, my preference for rooks is to have al parity odd directions assigned to rooks. So, that includes the triagonal directions.