Date: 5 June 1982 01:07-EDT
From: Alan Bawden <ALAN at MIT-MC>
Subject: 4x4x4 mechanics
To: CUBE-LOVERS at MIT-MC

I had a small accident the other day.  I dropped Dave Plummer's 4x4x4 cube and
broke one of the center cubies (sorry about that Dave).  This gave me an
opportunity to closely examine the insides of the beastie.  I can't possibly
describe it through the mail (I can barely describe it in person), but there is
an interesting problem raised by the insides:

Inside of a 4x4x4 cube is a 57th piece.  It is not permanently connected to any
of the cubies.  (The center cubies are free to slide back and forth in slots
cut in the center piece.  The rest of the cubies are held in by the centers.)
It is impossible to determine by examining the outside of a cube exactly what
orientation the center piece has.  However, it IS deterministic how the center
piece will move under a certain twist.  When twisting a face, the center piece
stays fixed with respect to the other 3 layers of the cube.  When doing an
"equatorial" twist, the center piece can follow only one of the halves of the
cube (as determined by it's orientation).  (I believe I have just constrained
things enough so that you can figure out exactly how the thing moves on your
own.) 

Thus there is an even larger permutation group to the 4x4x4 (beyond the
supergroup problem where the identities of the center cubies are considered)
that includes the center piece.  Call this the "hypergroup".  And since the
center piece has a 3 element symmetry group there is another group beyond that
("superhypergroup"?) that takes that into account.

Now the first question to consider about the hypergroup is:  Is it really
larger than the original group or supergroup?  In other words:  When you have
solved the 4x4x4, does the center piece necessarily return to its original
position?  How about if you solve the cube in the hypergroup?

A problem with this problem, is that you cannot learn how to manipulate the
orientation of the center piece without taking your cube apart to look at the
thing.  (I DON'T recommend that.  My experience with Plummer's cube has taught
me that those center pieces are fragile.)  

Anybody have any insights into the problem?

				-Alan