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Date: Tue, 15 Sep 1998 12:21:58 -0400 (EDT)
From: der Mouse  <mouse@rodents.montreal.qc.ca>
Message-Id: <199809151621.MAA19286@Twig.Rodents.Montreal.QC.CA>
To: cube-lovers@ai.mit.edu
Subject: Two-face and three-face subgroups

I've been playing with the two-face subgroup [%] of the 3-Cube and got
to wondering - how much work has been done on the two-face and
three-face subgroups?  Certainly the two-face subgroup "feels" like a
much smaller object than even the 2-Cube (though perhaps more tedious
for human solution), perhaps about the size of the Pyraminx.

[%] Okay, strictly speaking there are two different two-face subgroups,
    but one of them is not even the least bit interesting.

And what about the three-face subgroups?  Certainly the three- and
four-face subgroups are smaller than the whole Cube group, though ISTR
that the five-face (sub)group is actually the whole thing.  But how
much smaller, and how difficult of human solution?  I'd expect one of
the three-face groups (the one involving two opposite faces - call it
the L-F-R one) to be more tedious but no more difficult than the
two-face group, whereas the other one (involving one face from each
pair of opposite faces - U-F-R, say) should have more interest.

In particular, the two-face subgroup is smaller than the set of all
position that leave unchanged the cubies that the two-face subgroup
never touches.  (To put it another way, I'm saying that the subgroup
generated by {R,F} is smaller than the set of positions of the full
group that leaves unmoved the 11 cubies that don't touch either of
those two faces - 7 if you don't count face cubies.)  I can see a
factor of 128 smaller, since it's not possible to flip edge cubies in
the two-face group, but I haven't thought about the corners, so it may
be even smaller than that.

What about the three-face subgroups?  The L-F-R subgroup is also
smaller, if for no other reason than an inability to flip edge cubies,
like the two-face group.  But is the U-F-R subgroup the same as the
subset of the full group that leaves untouched the 7 (4 if you don't
count face centers) cubies in the DBL corner?

What about human solvability?  I've taught myself to solve the two-face
group, and with the tools I developed (largely powers, reorientations,
and inverses of F' R' F R) I feel confident I can handle the L-F-R
three-face group or even the L-F-R-B four-face group.  Can anyone
comment on how humanly difficult the U-F-R group, or for that matter
the U-F-R-L four-face group, is?

					der Mouse

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