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Date: Wed, 19 Nov 1997 17:41:50 -0500
From: Jim Mahoney <mahoney@marlboro.edu>
Organization: Marlboro College
To: Tenie Remmel <tenie1@juno.com>
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: Rubiks Revenge moves
References: <19971118.204255.7126.1.tenie1@juno.com>

Tenie Remmel wrote:

> Is there an easy way to cycle three adjacent top edges on the
> Rubiks Revenge?  I can't find one shorter than 62 moves, but if
> there was a short one I could simplify my solution greatly.
>
> . b c .    . a b .
> a . . . => c . . .
> . . . .    . . . .
> . . . .    . . . .

You can cycle these three edges on the 4x4x4 in 14 quarter turns
without disturbing the corners.

With the "up" and "front" faces like this (in a kind of projection view;
the corners are given by "*"; the "right" face is not shown),

       *  .  .  *
      .  .  .  .
     .  .  .  C
    *  A  B  *
    .  .  .  .
    .  .  .  .
    *  .  .  *

a procedure to cyle A,B,C is as follows:
 (1) 2 preparation moves which put C on "down" slice and B on "up/back"
 (2) 3 moves to get A off top slice and replace with C.
 (3) 2 moves (1/2 rotate) the top slice to put B where C (orginally A) was.
 (4) undo (2), restoring bottom layers and bring A back to top, in new spot.
 (5) undo (3)
 (6) undo (1), the prep moves.

(In each case "undo" means to do the inverse of the same moves in the
opposite order; that is,  "undo" ABC means C'B'A' where C' is the inverse
of move C.)

The hardest part I have of describing the specifics of each of
these is the notation; each of the 6 steps is only a couple of moves.

Let me define U,R,F as the Up, Right, and Front faces, and
number the slices by integers 1 to 4, so for example (F1,F2,F3,F4)
are clockwise quarter turns on the 4 slices (front to back)
parallel to the front face.  Counterclockwise turns are indicated
with either a ' (indicating "inverse") or lower case, so
F1' = a counterclockwise quarter turn of the front face.

In pictures,

        *  -  -  *                      *  -  -  *
      .  1  2  .                      .  3  1  .
     .  3  4  .                      .  4  2  .
    *  -  -  *     ==>  U1 ==>      *  -  -  *
    |  a  b  |                      |  a  b  |
    |  c  d  |                      |  c  d  |
    *  -  -  *                      *  -  -  *


        *  -  -  *                      *  -  -  *
      .  1  2  .                      .  1  2  .
     .  3  4  .                      .  3  4  .
    *  -  -  *     ==>  F1' ==>     *  -  -  *
    |  a  b  |                      |  b  d  |
    |  c  d  |                      |  a  c  |
    *  -  -  *                      *  -  -  *

where the letters are on the "front" face
and the numbers are on "up" face.

Then with this notation, steps (1) through (6) of this procedure are
  (1) F2  R2
  (2) F3' U4' F3
  (3) U1  U1
  (4) F3' U4  F3'
  (5) U1'  U1'
  (6) R2' F2'
which is 14 moves.

[Moderator's note: Certainly (4) should be F3' U4 F3, but that still
 cycles the wrong edges.  With (2)=R2 U4' R2', (4)=R2 U4 U2' we cycle
 the correct triple of edges, but in inverse order. ]

I confess that I don't have a 4x4x4 anymore and so can't try this -
I may have visualized one of the details wrong.  Hope not.

> I'd also like to see some short moves for the following 3-cycles:
>
> . * * .    . . * .    . . . .    . . * .
> . . . .    * . . *    * . . *    . . . *
> * . . .    . . . .    . . . *    * . . .
> . . . .    . . . .    . . . .    . . . .

You can do any of these as variations of the method I give above
with different "preperation" moves to get the edges into the
proper positions, namely two on the same slice which can be turned
into one another (the top slice in these cases),
and the third on another slice (usually the bottom slice) which
can replace one of the edges from the top slice.

I have a discussion of the NxNxN cube which includes in section (VI)
this recipe for 3-cycles of any kind of edge, corner, or face
piece; you can read it at
http://www.marlboro.edu/~mahoney/cube/NxN.txt if you're interested.


Regards,

 Jim Mahoney   (mahoney@marlboro.edu)
 Marlboro College