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From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Reinventing (and some edge-flipping techniques)
Date: 26 Apr 1999 18:03:02 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <7g29om$4ua@gap.cco.caltech.edu>
I skimmed through Singmaster's "Notes on Rubik's Magic Cube" this weekend.
It was rather disappointing to realize that the techniques I had invented
five years ago had all been independently invented twenty years ago!
In any case, I thought I might share my thoughts about monoflips with the
list.
--------------
[I shall use "E" to denote moving the center slice (between the U and D
faces) to the left (so that the F center becomes the L center). E' I
will use to denote its inverse.
In other words, E is the same as U'D followed by rotating the entire
cube clockwise from the point of view of the U face.
Similarly, I will use M to denote moving the L-R slice "up".
Traditionally, a slice is counted as two turns, although it is usually
easier to perform than two orthogonal turns.]
As you may know, a "monoflip" is a sequence of moves that flips one edge
on a given face (usually the U face) while not affecting the rest of the
face. For instance, this is a monoflip:
F E F M FF M' (10 turns)
This flips the FU edge cubie. If you do this sequence of moves, it is
particularly easy to look at the FUL, FU, and FUR cubies and see how they
move around to perform this monoflip.
What makes a monoflip so useful is the fact that it can me commutated with
moves of the face (U) to flip two edges. E.g.,
FEFMFFM' U MFFM'F'E'F' U'
will flip the FU and RU edges. (The first FEFMFFM' flips FU while messing
up the rest of the cube, then the U moves the RU to where the FU was,
then MFFM'F'E'F' flips the "new" FU as well as undoing what was done to
the rest of the cube. U' then restores what was done.)
Commutating with other moves will, of course, allow you to flip other edges.
Using UU instead of U will flip the FU and BU edges, for instance.
If you add conjugation, you can get pretty much any two edge pieces to
flip -- e.g.,
R' FEFMFFM' U MFFM'F'E'F' U' R
will flip the FU and RB edges.
--------
The drawback of FEFMFFM', however, is that it is not that easy to remember,
since you also have to remember its inverse MFFM'F'E'F'. (Although the
embedded conjugate MFFM' is its own inverse, which is nice.)
Therefore, the monoflip I see more often in other people's repertoire is
F E' F E' F E' F E' (12 moves)
which has many advantages:
1. It is easier to remember;
2. It is its own inverse;
3. Its results are "elegant". (flips of FU,FL,BR,BL)
Of course, this move is special enough that you don't NEED to use it as
a monoflip if you rotate and reflect this move. To flip two adjacent
edges, simply do
FE'FE'FE'FE' R'ER'ER'ER'E
and you'll flip the FU and RU edges. (The second part is just a
reflection along the BL - FR plane.) Other variants of this move
can pretty much let you flip any combinations of edge cubies you
want.
--------------
I used to use the previous "monoflip" a lot.
The only drawback with the previous monoflip, however, is that it is
too slow, especially those slice moves. Therefore, when I need
to do edge flipping these days, I use this monoflip:
R F' U R' F (5 moves)
At first this doesn't look like a monoflip, since each face seems to be
pretty messed up. But this is not your normal monoflip -- this is a
SLICE-BASED monoflip, where the E slice is intact except for the FR edge.
In fact, I'd conjecture that this is the shortest monoflip there is. To
flip the FR edge, RUF is pretty much the quickest you can to it in. But
this messes up two diagonally opposite edges on the slice, so at
least two more moves are needed to restore this -- hence, the F' and R'.
You can think of this as RF' conjugated with U -- I'm not sure it
helps much though.
The inverse, F' R U' F R', is just a reflection of the original move and
so is pretty easy to remember.
In any case, this makes edge flipping nice and quick:
RF'UR'F E' F'RU'FR' E (flips FR and FL in 12 moves)
RF'UR'F EE F'RU'FR' EE (flips FR and BL in 16 moves)
But how about the other edge pair combinations? Conjugates, of course.
I use these moves:
1. Adjacent edges, same face:
F'R' RF'UR'F E' F'RU'FR' E RF which reduces a bit to:
FFUR'F E' F'RU'FR' E RF (flips FU and RU in 16 moves)
2. Adjacent edges, same slice:
(done above, 12 moves)
3. Opposite edges:
(done above, 16 moves)
4. Skew edges:
R' RF'UR'F E' F'RU'FR' E R which reduces a bit to:
F'UR'F E' F'RU'FR' E R (flips FL and RU in 14 moves)
These are the current edge flips in my repertoire, and they achieve a
balance between being easy to remember and being fast to do.
----------
I'd appreciate if others could share the moves they use for 2-edge flips, as
well as know of any results known of God's algorithm for 2-edge flips.
--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
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She ran by screaming "No, I run by moving my feet rapidly, you idiot!"