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From: roger.broadie@iclweb.com (Roger Broadie)
To: <Cube-Lovers@ai.mit.edu>
Cc: "Frederick W. Wheeler" <wheeler@cipr.rpi.edu>
Subject: Re: Inventing your own techniques
Date: Mon, 3 May 1999 23:40:04 +0100

I'd like to return to Fred Wheeler's interesting question (30 March
1999), for its own sake and partly as a prelude to my next posting.

The processes I originally used to solve the cube - before I had been
introduced to commutators by David Singmaster's little blue book -
were based on a principle that formed itself in my head as I explored
the cube.  I called it "Out and back by a different route".  If pieces
were moved away from their original position by one process and then
restored by a different one, the result would be a process that would
only move pieces other than those moved out and back.  Then it struck
me that the different routes could be based on the obvious processes
used in a bottom-up algorithm to move a bottom corner piece into
position from the top corner vertically above it, depending on the
orientation of the piece. Thus, taking the DLF piece as the one
subject to the out-and-back movement, the out trip would take the
piece to FLU using front face turns, and the return route would bring
it back using left face turns:

F U' F'. U' L' U L  (1)

This process, I discovered, moved an edge piece out of the top layer
into the middle, UB > FL, but had no other effect on the middle layer.
So, at once, it formed the basis for solving the middle layer.

Besides inverses and reflections there is one other essentially
different process of this type involving the DLF corner.  It takes the
piece to the far corner RUB on the out trip:

F U2 F' .  U2 L' U' L  (2)

And it too moves just one edge piece out of the top into the middle
layer, from a different source position but to the same target
position as (1), UF > FL.

We can now repeat this approach, using these two processes and their
inverses and reflections to take a piece out of the top layer and then
return it.  That generates a set of upper-layer processes that led me
to my first solution.  It was not very efficient, but one of these
process is very attractive.  It results from following (2) above with
the reflection of (1) in the diagonal plane FL-BR:

F U2 F' U2 L' U' L  . L' U  L U F U' F'    =  F U2 F' U' F U' F' (3)

It is short, because of the cancellation, very easy to do, because all
the turning can be done with one hand, and has a very useful cycle
pattern, with an untwisted 3-cycle of edge pieces and a twisting pair
of swaps of corner pieces.

That leads on to another basic technique: looking at the cycle pattern
of a process and seeing what can be done to suppress or simplify some
of the cycles.

If we take (3) above, by combining it with its reflection suitably
applied, we can suppress either the movement of the edge pieces to
give a double twist of the corners (4), or the movement of the corner
pieces to give an untwisted edge 3-cycle (5):

F U2 F' U' F U' F' B' U2 B U B' U B UF(L-, R+) (4)
F U2 F' U' F U' F' L' U2 L U L' U L U(F, L, R) (5)

There are shorter alternatives, of course, but (4) remains my
favourite for the purpose, done as

  F U2 F' U' F U' F' [twist whole cube parallel to U'] L' U2 L U L' U
L

Of course, taking powers of a process is one way selectively to
eliminate constituent cycles, and (1) above, if done four times, is a
triple corner twist, because the edge pieces move through a 4-cycle
which is eliminated, one corner piece undergo a twist with no change
of position, which is preserved, and the other two undergo a twisted
2-cycle that leaves them in position but twisted.  But powers tend to
be rather lengthy, as this example illustrates.

Then one can look for patterns in processes and apply them elsewhere.
(3) above can be thought of as taking DLF up to the top and then round
and home again, with the + and - turns of the F face cancelling out.
One other process I discovered on the cube using this principle is the
following, where M is the turn of the centre slice parallel to R:

M2 U M2 U2 M U M2  (UF,UB) (UR, UL) (6)

And the structure of (3) transfers directly to the tetrahedron as

F U F' U F U F' 3-cycle round the vertical axis

and the dodecahedron as

F U2 F' U' F U' F' and  F U2 F' U F U2 F' edge 3-cycles + corner
double-swaps

In these I am taking the puzzle to be sitting on a table, with U being
the vertex (for the tetrahedron) or face (for the dodecahedron) at the
top and F being a vertex or face adjacent to the top pointing towards
you.

For a discussion of designing 3-cycles using commutators and
conjugates, see the message I posted in November 1997, which looked at
processes of the type

[P, TQT']

where P and Q are turns of layers that are parallel to one another,
and T is a turn of a layer transverse to P and Q.   These processes
yield a result that can be expressed as second-level commutation, as
mentioned by David Singmaster.

Roger Broadie