From: der Mouse  <mouse@Rodents.Montreal.QC.CA>
To: C.McCaig@queens-belfast.ac.uk
Cc: cube-lovers@ai.mit.edu
Subject: Re: 4x4x4 solution

> i recently borrowed a friends 4x4x4, and i know the basic method for
> solving it.  [...]  however, about half the time i end up with a
> single edge pair inverted and cant figure out a move for
> reorientating the single edge pair.

Make a single 90-degree inner-slice turn, then solve as before.  This
introduces an odd permutation on the edge pairs, which gets you back
into easily solvable space.  (It's usually easiest if you make sure
that the two swapped edge cubies are part of the slice turn, by placing
on the same slice beforehand if necessary.)

I'm not sure quite what the parity constraint here is.  There is some
kind of even-parity constraint on the edge cubies, it appears, with a
linked constraint on the face centres, but it's not as simple as the
parity of the edge and face permutations being both even or both odd,
because the single slice turn introduces two nonoverlapping 4-cycles on
the face centre cubies - which is, overall, an even permutation on
them.

I do notice, though, that a slice turn produces a 4-cycle on the edges
and two 4-cycles on the face centres; a face turn produces a 4-cycle on
the face centres and two 4-cycles on the edges (and a 4-cycle on the
corners, which may or may not be relevant).  I wonder if there's a
multiple-of-three constraint lurking.

Doubtless some group theorist has long ago worked out exactly what the
constraints are, but I haven't heard.  (I tried to work through a
group-theory text recently, got stalled along about the time it got to
cosets, quotient groups, normal subgroups, etc.)

					der Mouse

			       mouse@rodents.montreal.qc.ca
		     7D C8 61 52 5D E7 2D 39  4E F1 31 3E E8 B3 27 4B

------------------------------

Date: Tue, 30 Sep 1997 14:11:28 -0400 (EDT)