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Date: Tue, 02 Jul 1996 00:20:44 -0700
From: Mark Longridge <cubeman@dis.on.ca>
Organization: Computer Creations
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To: cube-lovers@ai.mit.edu
Subject: Cube Moves
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 > My name is Isidro Costantini, I'm a cube lover since '81.

   Welcome to cube lovers the mailing list.

>  ( Where's a place to check for those formulas? )

Well, I'm not finished yet, but I do archive all the cube formulas
I get a hold of or compose. Some of the work is with the assistance
of computers and/or mathematical insight.

http://www.dis.on.ca/~cubeman

>  Another good example is (xchg 3 edges,noFlip) (12) R2 U1 F1 B3 R2
> F3 B1 U1 R2 (9 moves using your way of counting) and another
> equivalent: B3 U3 R3 U1 R1 B1 followed by F1 R1 U1 R3 U3 F3
> (6+6 moves, same position).
>  Another way of counting could be adding the suffix (1,2 or 3)
> (counting only clockwise moves) which would preserve parity as well.
> I would be pleased if some one can tell me about this subject.

The sequence  X = (B3 U3 R3 U1 R1 B1 F1 R1 U1 R3 U3 F3) is a very
interesting one.

Note that X = B3 [U3 R3] B1 + F1 [R1 U1] F3

The above makes use of conjugates and commutators.

The following is a top view of a megaminx (magic dodecahedron):
      /\
    /    \
  /        \
  \   U    /
L  \      / R
    \____/
      F

Then the very similar sequence   R+ F+ U+ F- U- R-
                                 L- U- F- T+ F+ L+

...suffices to also 3-cycle the edges  (uf, lf, rf) on the megaminx.

In this case I don't like the U3 = U- or U' notation.
Clearly on the megaminx U3 <> U'

Note that each turn of a face is always turned one way and then back.
The 5-period rotation of a face is never used. In special cases like
these cube moves from the standard 3x3x3 are directly transferable
to the megaminx.

I have found that isoflips and isotwists work very well on the
megaminx.

The shortest flip of 2 adjacent edges uses the same 4 sides
(so I say "this sequence has face-index 4), is the following:

Note use of L-- and L++ etc to denote 2 one-fifth turns of a face!

It is of the form P U1 P' U' which is another commutator.

L-- R++ F+  U-  R+  U+  L++ R++ U+
R-- L-- U-  R-  U+  F-  R-- L++ U-

= 18 face turns or 26 one-fifth turns. Perhaps there is some
improvement to this sequence.

-> Mark <-