I've been doing some analysis to see if standard chess endgame theory
holds up in three dimensions. The first test is the white king and
queen vs. the black king. This is an easy win for white normally. Is
it so easy here?

I have the following three pieces:

White king on 1f3
Black king on 2e5
White queen on 3a3

White to move. White will dominate the center board through use of
the king, and force black to the outside. Can black's king escape?
Note that the white king, placed in the center, controls a cube of
squares around him. This is very difficult for the black king to
circumvent, no matter which board he is placed on.

1. K2e3 K3f5
2. Q3e3 K2e5
3. Q2e4+ K1d6
4. K2d4 K1c6
5. Q1d5+

My strategy is to use the queen in the more traditional role of roping
off squares for the black king. It is easiest to keep her off of the
center board, so she doesn't hinder the white king. The white king
moves his cube of influence closer and closer using the center board,
pinning in the black king.

5. ... K2c7
6. K2d5 K3b6
7. Q2c5+ K3c7
8. Q3c6+ K2d8
9. K2e6 K2c8

Black's only other legal moves are to 1d8 or 3d8. Both are met by 10.
Q2d7++. The white squares are all guarded by white's queen.

10. K2d6 K1b8
11. K2c6 K1a7
12. Q2b7++


King vs. King and Queen can be forced to checkmate.