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What a brilliant concept.
Having looked at this again I have got to grips with your idea. What puit me off was the use of names. To me and, I suspect, to most people into 3d Chess, Rook means something confined to the 6 orthogonal directions, Bishop something confined to the 12 diagonal directions, and Knight a 2:1:0 leaper. Queen may be either Rook+Bishop or Rook+Bishop+Unicorn; I prefer the former, with Duchess for Rook+Unicorn, Governor for Bishop+Unicorn, and Empress for all three, along with Emperor for a King that also moves one square triagonally. For your pieces I would suggest names making a clear break with these linear ones, based on the theme of Plane as a reflection of the scope of the moves and a pun on a far greater flier than a Rook! Plane itself cannot be used as it shares the initial of Pawn, but you might reflect the relative powers of the pieces by calling your first piece a Jetplane and your second a Biplane, which has the advantage of starting with the first two letters as Bishop. The combined piece might be called a Hyperplane, and the ruling piece is the same as my Emperor above. This frees up the initial K to call your fifth piece a Kite, which rhymes with Knight.
Having the second part of the move be 'up to' the same number of squares (rather than exactly the same, as has been done in some two dimensional games) is correct, it's a very good idea. I have raised my rating from good to excellent.
who is they?
Although I haven't playtested your idea, it does seem to me that perhaps it makes the pieces more powerful; and this could be a good thing because the 3D King (as I found in my own examinations of 3d chess) is difficult to checkmate.
This is an interesting idea. Here's a logical extension of it:
<p>In 2d chess (thinking in the abstract), pieces are 0 dimensional (ie. points) that move in a line (1 dimensional movement).
<p>Perhaps in 3d chess, pieces could be 1 dimensional (ie. line segments) that move (as suggested) in a 2 dimensional plane. Their direction of movement would be constant, it would simply be their area of movement that would cover a two dimensions.
<p>In the diagram below, the 3d Rook piece at [a1,d1] could move to [a2,d2] or [a3,d3]. It would be blocked by the 3d Bishop piece at [c4,d4]. It could, however move to [a1,d1] on the next level up, assuming no friendly pieces are blocking it. However for the 3d Rook to move to the other half of the board (ie. files e-h) it might have to rotate. Or then again, perhaps in that direction it only covers a single rank instead of a plane.
<pre>
+---+---+---+---+
| | | | | 6
+---+---+---+---+
| | | | | 5
+---+---+---+---+
| | |---B---| 4
+---+---+---+---+
| | | | | 3
+---+---+---+---+
| | | | | 2
+---+---+---+---+
|-------R-------| 1
+---+---+---+---+
a b c d
</pre>
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