Message #34

From: mahdeltaphi <mark.hennings@ntlworld.com>
Subject: Orientations of the centre cubes
Date: Sun, 07 Sep 2003 22:46:27 -0000

I have recently discovered the Rubik’s tesseract, and found that a
large number of the sequences of moves that I used to use to solve
the basic 3x3x3 cube were useful, or could be extended to aid the
solution of the 4D cube.

The current implementation of the tesseract does not keep track of
the orientations of the centre cubes of the 8 3x3x3 faces of the
tesseract. Accounting for all permutations of pieces and their
orientations, each rotation move on the tesseract seems to permit
only even permutations. Consequently, in theory, it should be
possible to have all the other pieces in place, and yet have the
centre cubes of each of the 8 cubical faces in a variety of
orientations. This was certainly the case of the original 3x3x3 cube
(I used to mark the centre square of each 3x3 face to indicate in
which direction it should be pointing, and doing so reduced the
number of possible solutions down to 1 from a total of (4^6)/2). If
the same result applied to the tesseract, marking the centre cubes
would reduce the number of correct solutions from (6^8)/2 to 1.

I think it should be possible to change the orientations of
the "centre cubes", without changing the positions of the pieces
that share 2,3 and 4 cubical faces. Has anybody thought of moves
that would do this? I suspect that it is likely that the moves that
did similar things for the orientations of the centre pieces of the
3x3 sides of the 3D cube could be extended to do this.

The current implementation of the cube makes such investigations
difficult, since the orientations of the "centre cubes" are not
marked. Perhaps the 6 faces of each "centre cube" could be marked
with the colours of the 3x3x3 faces towards which they should be
pointing.

Any ideas?

Mark