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Great concept! The planar pieces seem a bit too powerful for a 3D version. However, I think they would be more useful in a 4D game. I think the dimensions of movement of a piece should not excede half the number of dimensions of the field.
<p>I have been playing around with the idea of a '4D8L' type game, but it now occurs to me that a 4D field would be too 'open' for a family of 'knight-hoppers' and 'line-movers' alone, even if there were 512 pieces and 512 pawns per side. So the concept of 'planar' pieces my help balance the 4D game.