Stefan Pochmann wrote:

> > > > ZBLL: z' U L' U' L R U2 L' U L U L' U R' (L-L')
> > >
> > > P.S. study the last 9 moves - you don't need ZBLL :-)
>
> Hmm, I can't follow. What last 9 moves, U2L'ULUL'UR'L or
> RU2L'ULUL'UR'?

The latter. See below:

http://vanderblonk.com/cube/cubeapplet.asp?alg=DR2U'RURU'RD'

> And what does it mean you don't need ZBLL?

Sorry, that was a reference to earlier discussions I had with
Johannes, but I will try to explain what that was about.

Most methods, including ZBLL and Fridrich, were designed on the
premise that symmetry is a "good thing". Symmetry means that in a
great many cases we can simply rotate the cube around, and then solve
the case using a rotation of an existing algorithm instead of an
entirely new one, for example. In other words, symmetry reduces the
number of cases that we need to memorise.

However, the position that I will argue is that symmetry is what makes
cases difficult in the first place, and is what leads us to memorisation.

There are three reasons why I think symmetry is "bad":

1. A system in which the solver only thinks about symetric positions
does not have closure. That is, the solver cannot make a single move
on its own without breaking out of the system and becoming lost. This
is why it is possible for Fridrich solvers can become lost in the
middle of an algorithm from cube amnesia. Simply, their system only
recognises the beginning and end of a sequence, and they are trapped
in a mindset that prevents them from being able to see the middle on
the same level as the ends.

2. Symmetric positions tend to make the shortest paths non-obvious.
e.g. when all of the unsolved pieces are on the last layer, and we
have the typical symmetry of U rotations, there are less hints
provided by that cube state as to how to start to solve that position
intuitively. Each move will appear to move the cube pieces further
away from where they should be (breaking the symmetry that is formed).

3. Symmetric positions tend to require longer, more complicated
solutions than asymetric positions. In a perfectly symmetric position,
the only way to proceed is to go out the way you came in (or a way
that is "equivalent" to the way you came in). If your system (i.e. the
set of moves/operations you are restricting yourself to) is such that
the way in is 1 move, then entering a symetric position is a waste of
2 moves. However, if your system is such that the way in is 2 moves,
then you are wasting 4 moves.

My method is fundamentally asymmetric. It recognises that while
asymmetric steps result in far too many cases to memorise, they also
make the cases easier to solve so that you do not need to rely on
memorisation.

Now back to the case given:

http://vanderblonk.com/cube/cubeapplet.asp?alg=DR2U'RURU'RD'

This is a typical position within the Heise system: the front F2L
column is free, the edges have been oriented, and the two corner/edge
pairs have been formed. The design of this step is such that it has
the following properties:

- Maneuvers from one position to another are very short (typically
between 3 and 7 moves).
- You never need to move too far away from a recognisable position.

It is possible to move between different states in this step using
only a limited number of strategies, one of which is demonstrated by
this sequence. In this particular case, we are dealing with strategies
involving three correct edges on top. We always deal with three
oriented edges on top, with the following permutations:

- 3 edges in correct order (as above)
- 2 opposite edges correct
- 2 adjacent correct + 1 incorrect (2 varieties)
- no correct adjacent edges (2 varieties)

.. along with one or two corner/edge pairs.

In each case, the edge permutations are fundamental and dictate the
kind of strategy that is needed.

The overall point was that if you make yourself familiar and
comfortable with the asymmetric positions, then you will find you
don't need to memorise large numbers of algorithms such as ZBLL. The
operations will be short enough that you will be able to see them.

Finally, to clarify how the above strategy fits into my system when
the steps are done in the normal order, here is a description of how
my method works in the advanced case:

1. 4 (potentially non-matching) squares
2. Edge orientation
3. 1 corner/edge pair
4. Edges + 1 corner
5. Last 3 corners

The relevant step is (4) in which we manipulate the edge permutation
while manipulating 1 of the corners. It sometimes (often!) happens
that you end up with more corner/edge pairs by this stage just by
chance, or because you have preserved them, and then you are able to
skip step 5 entirely. I'm sure Johannes will be familiar with this
experience.

The webpage describing this approach is still a work in progress, however:

http://www.ryanheise.com/cube/method/

I will hopefully have more time soon.

P.S. I should point out that my interest in designing this method is
in fact for speedcubing. That is, my goal is to find techniques that
enable few moves but ALSO a framework for allowing the solver to do
this at high speeds. I would still say it is in its infancy in this
respect. My fastest average with this method is 28.36 seconds and so
it has a long way to go.

--
Ryan Heise
http://www.ryanheise.com/cube/