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A few thoughts:
	There are 2 equally valid ways of moving chess to higher dimensions:
'extending' or 'expanding', with extending being linear and expanding
being planar. Linear pieces are weaker, and planar pieces very much take
on the character of the 'Mad Queen', in being so much more powerful than
their 2D brethern. However, 'planar pieces' may be inherently too
powerful as a concept - consider the extension to 4D, where a rook would
move in a 3D volume of its' own choice, there being 4 different 3D
volumes this hypothetical rook could move in. Even in 3D, the actual
planar pieces used are the most limited and weak version. The strongest
planar piece could get to a location if any one of all its' possible
paths were open. An average planar piece could get to a location if at
least half the possible paths are open. The weakest planar piece is one
that is blocked by merely a single piece. (This is somewhat like damming a
river by throwing a stone into it.) Now it's true that using the weakest
possible planar pieces makes it easier to visualize moves, but using more
powerful versions of planar pieces should shorten the games and make
checkmate noticably easier. This approach may eliminate the 'need' for
restricting the king to get a reasonable chance to mate, if the 'weak'
planars still have trouble. You might also consider 3D movers, in a
'weakened' piece form, as, for example, a 'baby 3D rook' that could
move to any open square within a 5x5 cube centered on the piece, just as
the king can move to any square in a 3x3 cube centered on it. Of course,
this 'baby 3D rook' would require at least an 8x8x8 board to dilute
its' power. This rook, guarded by a friendly piece should be able to move
next to the enemy king and mate it.
	The 6x6x6 shatranj 3D sounds interesting. I, too, have been looking at
shatranj pieces for the past few months, and, with C. Bagley-Jones, have
made a 'Shatranj Capablanca' variant, with a knight-alfil and a
knight-dababba; and have also, for Modern Shatranj, tried out 2 'new'
pieces, the knight-ferz and knight-wazir, which seem to work quite nicely.
 I'd expect that short-range pieces in a 3D shatranj game would find it
useful to have those little 'extra' moves. And I think 6x6x6 is probably
about a perfect size for a 3D board. I'd be interested in playtesting such
a game.
	Finally (I hear echoes of 'about time!'), the not-so-elusive king.
Designing a sitting-duck king, as has been done, may not be the best way
to increase mates. I will, once again, float the idea of a 'king hold'
rule. I did a 4D game, 4x4x4x4, using linear pieces and an extremely low
piece density; and the king was all but uncatchable, even though it could
move to only 16 locations. The game could not be won by a king and 3
queens vs a bare king. Effectively, I had no game until the king hold
rule. The board is laid out as 16 4x4 2D boards, and the king hold rule
states that a king, moving onto the same 2D board as the opponent's king,
prevents the opponent's king from moving off that board, until the
'holding' king either moves or is chased off that board. This allows a
standard 2D checkmate, with no 'extra' pieces required. While this is a
restriction of the king's move, it is only temporary, and to maintain the
restriction on the opponent's, one must move one's own king into
potential danger and restrict its' move in the same way. This method
seems easily adaptable to 3D.