Date:  9 January 1981 0756-EST (Friday)
From: Dan Hoey at CMU-10A
To: Cube-lovers at MIT-MC
Subject:  The Supergroup -- Part 3: A Super-H and Spoilers
Message-Id: <09Jan81 075610 DH51@CMU-10A>

			IMPROVEMENT

	Jim Saxe (16 December 1980 1841-EST) gave a 16-qtw process
for the H position:  FF LL DD FF BB DD RR FF.  Unfortunately, the
position obtained by this process is not H-symmetric in the
Supergroup:  the F, L, B, and R face centers are rotated 180
degrees, but the U and D face centers are in the home orientation.
I have a 16-qtw process which leaves all face centers in the home
orientation:  (FB UD)^2 (UD FB)^2.  This may look familiar --
conjugation by FBUD yields the 16-qtw "Bridge at Midday" I sent day
before yesterday.

			SPOILERS

	I developed a solution method on my own, but it was a long
one.  The following good methods, which affect only face centers,
are from Singmaster.  It seems simplest to solve the cube before
applying them, since some of the most popular processes (e.g., the
Spratt wrench (but not mono-ops!)) change face center orientation.

	The first method can be used to perform any multiple of
four face-center quarter-twists on the faces in a centerslice.  At
most two applications are necessary to accumulate any remaining
twist in one face center.  The method is given in Singmaster as two
examples, but he doesn't explain how they work.  Hopefully the
following discussion will make them easy to use.

	Choose a face center X that needs to be twisted, and a
centerslice containing X and other face centers to be twisted (call
these FCT's).  Place the cube with X up and with the FCT's in the FB
slice (i.e., among R, D, and L).  The basic move has much of the
flavor of a mono-op:
	1: Move the LR centerslice toward you. (Move a face
	    center from U to F.  X and the FCT's are now in
	    the UD centerslice.
	2: Select an FCT, and move it to the F face with UD
	    centerslice moves.
	3: Move the LR centerslice away from you. (Undo step 1.
	    The U face now contains the selected FCT and all the
	    U edge and corner cubies.
	4: Twist the U face the amount required by the selected
	    FCT.
Repeat the basic move until all FCTs have been selected.  Then
perform the move one last time, but select X instead of an FCT
in step 2 and twist the U face to fix the cube in step 4.

[After sending the previous part, I realized that in the case of
 one FCT, this is another 14-qtw relation in the standard cube:
	RL'FB'UD' R DU'BF'LR' U'
 But the one alluded to is in the next paragraph.]

	Since the total number of 90o face center twists must be
even (see Vanderschel's message of 6 Aug 1980 1909-PDT), the
preceding will solve the cube up to a 180o twist.  The process
(URLUUR'L')^2 twists the U face center 180o.  This is the only short
transform given in Singmaster for twisting face centers by a
nonmultiple of four; I'd be interested in any others you know.

That's it for now.  Happy Supercubing!

				Dan