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I was reading over the Tetrahedral Chess page a while back, and decided to
understand the statement that certain (skew) planes form hexagonal
chessboards.  I can confirm now that this is true, and Gilman's M&B even
enumerates the number of such planes through each cell.

Now note that the 4-coloring of Tetrahedral Chess lends all four of its
colors to these hex-boards.  I cannot recall finding a hex chess that uses
a four-colored board.

However, a longer period of time ago, I noticed that the hex board can be
thought of as a certain quotient (that's a technical math term in this
context) of a 3d cubic board along a single unicorn line.  (For those not
used to the math lingo, think of it as an optical illusion: you look at the
cubic board so as to line up opposite corners, and all cells along your
line of sight are treated as being equivalent.)  This quotienting does some
weird things with the pieces, but what about the colors?  The only obvious
coloring that could be maintained by this quotient is the unicorn's
4-binding!  It probably comes as no shock that this coloring is the same as
in Tetrahedral Chess (how many 4-colorings can there be of a hex board?)

I think furthermore that some of the pieces in Tetrahedral chess, when
restricted to one of the hex-planes, turn out to be very similar to the
cubic pieces modulo the unicorn's diagonal.  (I had worked some of this
out, but don't have the notes handy.)