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Thanks for the comment, Kevin. Checkmate is one of the trickier parts of higher-dimensional chess. The standard method for K + Q vs K in 4D is to move the king to the/a middle square, then move your queen between your king and the enemy king, pinning it against the edge of the board. But this only works if your individual "little" boards are no bigger than 5x5. My method, by restricting diagonal moves greatly and introducing the "held king" concept, will work on any size (rectangular) "little board". It allows forced mate with K and any 2 of the Q and pair of Bs vs. a lone K. And it took a little help, as Abdul-Rahman Sibahi gave me the final piece of the hold rule - that it works on the individual matching squares in each little board. A version of the hold rules can be applied to any higher than 2D variant, though it might well have to be tweaked to fit each higher dimension. 

I like this design because it is humanly playable, looks like chess, and gives people the feel of 4D and the pieces the freedom to move through 4D space without overwhelming the players with 3D and 4D diagonals, or very many of the available 2D diagonals, for that matter. Most who design 4D chess variants using the 2D layout of 2D boards to represent 4D space give the pieces moves that are based on a 4D space that is 2D x 2D = 4D. I've found this creates a totally chaotic game, where the state of the board cannot be reasonably projected even 2 or 3 turns into the future. Pieces move so freely that in a couple moves, they can be anywhere on the board, generally by many paths which can't all possibly be guarded in 2 or 3 moves. So I get my 4D by basing movement on a 2D + 2D = 4D concept. It gives you the same 4D game space, but it restricts movements to fairly easily visualizable, fairly simple patterns that players can project 2 - 3 moves into the future. While it is fast-moving, it is not chaotic. 

Almost nobody plays it, or ever has. Grin, that's the common fate of most variants. But getting it to work, finally, taught me something about chess design. For one thing, your pawns are there as much or more to protect your opponent's pieces as yours. And the checkmate problem in higher-D chess is merely a symptom of the chaos problem in higher-D chesses. The attempt to reduce chaos is part of what led me to short-range pieces. Restricting board size only limits the number of chaotic states that can occur, bringing it down from the all-but-infinite to a smaller but still effectively infinite number in terms of human lifetimes, as individuals or as a species.