Date: Tue, 30 Dec 80 01:09:00 -0500 (EST)
From: David C. Plummer <DCP@MIT-MC >
~~~ ~~~ Subject: Hackery (92 line message) (first of three)

Greetings...I'm back from vacation. I took a copy the old mail
home and read it. It was an interesting experience.

Existence proofs:
Saxe and Hoey described (14 December 1980, 19:16 EST)
several local maximums that were previously not described. I have
algorithms for a couple that are 28 moves long, showing that
THESE PARTICULAR local maximums are no further than 28 away from
SOLVED.

==========
The edges flipped along the girdle (hereafter refered to as
GIRDLE EDGES FLIPPED [until somebody else thinks of a better
name]) can be done as follows:
(F D'U L D'U B D'U R D'U ) (U'L'F)
(L'R F' L'R U' L'R B' L'R D') (F'L U)
[This is a Sprat Wrench, get-ready, another Sprat Wrench,
un-get-ready.] At first glance this is 30 moves, but notice that
the U and U' near the end of the first line cancel, making it 28.

==========
The edges not along the girdle flipped (hereafter refered to as
OFF GIRDLE EDGES FLIPPED) (NOTE: This is GIRDLE EDGES FLIPPED
compounded with ALL EDGES FLIPPED), can be done as follows:
(L B'F U B'F R B'F D B'F ) (F F U)
(R'L U' R'L F' R'L D' R'L B') (U'F F)
[Again, Sprat-mung-Sprat-unmung] And again, 30 at first glance,
but this time the FFF near the end of the first line becomes F',
so 30-2 is again 28.

==========
==========
I noticed something peculiar about the second kind of girdle
configuration the mentioned. Hoey and Saxe describe it as "each
edge on the girdle may be swapped with the diametrically opposite
edge, provided that the corners on the girdle are swapped with
their opposites as well" and they give a picture. It is indeed
reachable, but the most interesting thing (in my opinion) is that
it is an odd distance away!! (Of course so are the related
configurations obtained by performing on it: PONS, ALL EDGES
FLIPPED, and PONS WITH ALL EDGES FLIPPED.) ODD!! Most of (if not
all) the previously described "nifty patterns" were even. And on
top of that, this odd permutation is a local max. Most of the
tools people seem to use are even. Several of them fall along
the lines of MUNG/DO SOMETHING/UNMUNG, and DO SOMETHING is often
even, and two MUNGS is even. Now the question is: HOW FAR AWAY
ARE THESE, AND HOW DO YOU (THE HUMAN) TRY TO FIND THE ALGORITHMS
TO DO THEM? My suggestion: Do something, Mung, Do something
(maybe different). I also suggest that Do somethingbe of length
12, and that mung be 3 or 5. I would really apprecitate it if it
were less than 28 -- I happen to like 28 as the maximal length.

==========
The local max which I described (Plummer 10 December 1980, 08:34
EST) which is all corners rotated such that adjacent corners are
rotated in opposite directions (shall we call this ALL CORNERS
ROTATED? I know it isn't explicit, but it is descriptive enough
for our purposes) can be done in 40 by a Christman Cross (16,
thanks to Saxe 10 December 1980, 18:41 EST), two half twists (4),
another Christman Cross (16), and undoing the two half twists
(4). I have been trying for the better part of a week to get this
down to 28 (my favorite number), but have not yet succeeded. What
I want is a 14 move algorithm to exchange all corners with their
diametrically opposite corner in such a way that the top forms a
cross of the top as the cross and the bottom at the corners. Two
of these with a whole cube manipulation in between will do the
trick. Alternately, if somebody could come up with a 12 mover to
move FDL to FUR (and do this so that it sould look this way if
viewed from F,L,R or B), you could do U <transform> U' and the
opposite corners would be swapped as desired. I have an almost
algorithm, but it falls short in that the four edges on the
center horizontal slice are swapped with their diametric
opposites. The algorithm is
R'L' F'B' R'L' FB RL FB
If anybody can find the desired algorithms, please publish.

==========
is best executed by a right handed person." The way I do the
Wrench does not descriminate, and I think it is worth it to
describe it:
(0 QTW) Holding the right face in the right hand,
(0 QTW) holding the left face in the left hand,
(2 QTW) rotate the center verticle slice toward you,
(1 QTW) rotate the NEW front face clockwise (either hand will do)
Do this a total of four times, giving 4*3=12 QTW. Try it, and
also try turning the NEW front face counterclockwise.

==========
That's all I can think of on this subject. Coming soon:
Design of the higher order cubes
Thoughts on manipulating the higher order cubes.