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VR Parton's Alice Chess is a classic abstract strategy board game, one
with a very unique twist. It's played on 2 adjacent boards, and pieces,
when they start a move on one board, must involuntarily transfer to the
other board, into the corresponding square they stopped on, which must be
empty, or the move can't be made. 

Definition: Parton Destination - the square on an adjacent board a piece
making a move must end its move. The piece makes a full, normal move on
its starting board, and must then shift one board 'sideways' to the
corresponding square on an adjacent board, which is its final, or Parton,
destination square. The Parton destination square must be empty at the
beginning of the turn for the move to occur.

In the original version, the Parton destination square is forced, or
involuntary, because there are only 2 boards. And the knights become oddly
colorbound, because the forced transfer allows any particular knight to
attack squares of only one color [different] on each board. You can change
this by adding a third board which is mutually connected with both other
boards. Now, a piece may have a free choice of which other board to make
its Parton destination board. And the odd total number of boards allows
the knights to attack all the squares on each board.

I don't know if this has been discussed before. As this post is getting
longish, I'll develop the idea more in a later post. ;-) Enjoy