Message #4078

From: Marc Ringuette <ringuette@solarmirror.com>
Subject: Re: [MC4D] 2x2x2x2: List of useful algorithms (please add yours)
Date: Sun, 29 Jul 2018 12:04:54 -0700

I thought I’d re-read Joel Karlsson’s message from 12/21/2017 discussing
the commutators that he used to solve the puzzle, and pondering it led
me to this nice short double pair swap.

2x2 cycle:   Uy2 Lx’ Uy2 Lx Uy2 Rxy2   Uy2 Lx’ Uy2 Lx Uy2 Rxy2   (12
moves canonical)
  (all changes are in the UF quarter of the puzzle, LUFO <–> LUFI, 
RUFO <–> RUFI)

And a related 3-cycle,

3-cycle:  Uy2 Rx’ Uy2 Rx’y2 Uy2 Rx Uy2   Ly   Uy2 Rx’ Uy2 Rx’y2 Uy2 Rx
Uy2  Ly’   (16 moves canonical)
 (all changes are in the LU quarter of the puzzle, LUFO –> LUBI –> LUBO)

Both of these act on a restricted 1/4 of the puzzle, in contrast to the
3-cycle I gave in yesterday’s list, where the pieces involved are widely
distributed around the puzzle, so that there is no half of the puzzle
that does not contain a moved piece.

3-cycle:  Lz’  Fz2 Lx Fz2 Lx’ Fz2 Lz  Ox2   Lz’  Fz2 Lx Fz2 Lx’ Fz2 Lz 
Ox2  (16 moves canonical)
  (moves LUFI –> RUFO –> RDBO)

Hmm, I wonder if there might be a sensible classification scheme for
cycles and swaps based on the number of dimensions that are twisted, the
number of half-puzzles that are unchanged, and the fraction of the
orientations that can be accessed.   At this point I have no idea.  As I
try to develop an efficient 3-cycle-based solution scheme, maybe it’ll
come to me.

On 12/21/2017 2:05 PM, Joel Karlsson joelkarlsson97@gmail.com
[4D_Cubing] wrote:
> The commutators I use are based on one simple idea. Isolate one or two
> pieces (depending on what you want to accomplish) from the bottom half
> on the top half (holding the cube upright), then rotate the top face
> (to accomplish a swap or rotation for example), reverse the first
> step, rotate the bottom face and lastly perform the first three steps
> in reverse. The first step can quite easily be done with 7 and 3 moves
> respectively, resulting in sequences from 7 to 32 moves (sometimes the
> first three steps are enough, depending on how much you want to
> preserve).