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I was wondering whether anyone has ever designed chess rules for more than 3 dimensions. At that point you wouldn't be able to visualize the play using one board. And you'd instead need to to use 8 3-D boards (in the case of 4-D). When searching for multidimensional chess, the only thing I can find is 3-D chess. So has anyone ever designed a multidimensional chess game where (N > 3)?

If you think this question should be posted elsewhere, please inform me.

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    I thought about this for a little bit myself. One of the fundamental issues with n-d chess is that the volume of the board increases with the dimension. So essentially this means that you have to pile on a lot more pieces into your game for the armies to ever actually even clash. This would lead to a decreasingly strategic game as n->infinity unless the game was fundamentally changed. – NoseKnowsAll Sep 8 '18 at 23:49
  • That comment doesn't answer your question, but may be another reason you haven't seen higher dimensional chess. If someone has made it and it's somewhat interesting, it's probably not "chess" anymore. – NoseKnowsAll Sep 8 '18 at 23:50
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Yes, there are a few four-dimensional chess variants: https://www.chessvariants.com/index/mainquery.php?type=Any&category=4d&usethisheading=Four+(or+more)+Dimensions

Sphinx Chess, by V.R. Parton, is probably the first, c.1970. Timeline is also described there (it's actually a full 4 spatial dimensions, being 4x4x4x4, plus the "timelines"), as well as my own variant TessChess.

Most 3D chess (at least online) is played by splaying the levels out next to each other, since seeing and reaching between levels may be difficult. Once you get used to that, it's not so unnatural to put the levels in a grid, and you've got four dimensions.

Also, most 3D chess is on a smaller board than 8x8x8 (8x8x3 or 5x5x5 are common), and higher dimensions similarly shrink the edge length. This, together with appropriately extending piece movements, get around NoseKnowsAll's density argument. But, of course they are also right that the game becomes rather different from chess.

V.R. Parton also created a 6-dimensional game, Ecila. I and others have toyed around with even more dimensions, but it obviously becomes more difficult for the "array of array of arrays" board simulation to work out.

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There's a vaguely chess-based game called "Timeline" that was described as 4-dimensional chess. The pieces left 'trails' as they move over a 4x4x4 grid, and you could capture a piece "before" it reached its current square.

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Not sure if this qualifies for exactly what you're asking, but I have an idea that could fulfil your requiruments

In the "doubles" varient of normal chess, where two players play on a team with two 2d boards, that can be considered three dimensional.

If such a game were played with 3d boards, I guess that could be considered 4d, albeit the 4th dimension has a cardinality of only 2

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