Newsgroups: rec.puzzles
From: hoey@ai.etl.army.mil (Dan Hoey)
Date: 30 Aug 90 20:30:18 GMT
Subject: Re: Rubik Cube Notation for > 3x3x3

bru...@tekgen.BV.TEK.COM (Bruce Cheney) writes:

>Is there a standard notation for Rubik cube moves on cubes that are larger
>than 3x3x3 ?  David Singmaster's notation (in Notes on Rubik's Magic Cube)
>seems to be the standard for the 3x3x3, but there are several possible
>extensions to larger dimensions.

In *Rubik's Cubic Compendium* there is an article by Tamas Varga with
an interesting modification to the language for the 3x3x3.  First, he
changes the "U" (for Up) to a "T" (for Top) so that every face is
named by a consonant.  Then he assigns vowels to the possible motions
of a face: "O" for 90 degrees fOrward (or clOckwise), "A" for 90
degrees bAckward (or Anticlockwise), and "I" for a 180 degree
half-turn (twIce).  Then he makes syllables out of a consonant and a
vowel, and words out of the syllables.  So Pons Asinorum (the simple
6X) can be done with FIBITIDIRILI and Laughter (\/\/) is 3 FOBOROLOs.

But wait, there's more!  He also allows you to append a "C" to the
syllable to indicate that instead of turning the face, you move
the whole cube. the notation is not as parsimonious, since FOC=BAC,
TOC=DAC, and ROC=LAC, but it's better than having to stop in the
middle and say ``then move the cube''.  For instance, Jim Saxe's
28-quarter-turn Plummer's Cross (6+) can be done as "FOLIRIFO BOLIRIFO
ROFIBIRO LOFIBIRO TIDI" but it's easier to remember it as FOLIRIFO
BOLIRIFO TOC FOLIRIFO BOLIRIFO TIDI".

Varga also has a way of talking about the slice moves, where you move
the middle layer of the cube instead of the faces.  For moving the
middle, you append "M" to the syllable for a parallel face.  So the
Spratt wrench (4-flip) is 4 TOROMs, and we can do the Plummer's Cross
as FOLIMBO FOLIMFO TOC FOLIMBO FOLIMFO TIM.  Of course
ROM=LAM=ROCRALO.

Unfortunately, Varga stopped there.  I decided to continue.  For
instance, it can be useful to have a way of talking about slice moves
where you move the faces rather than the middle.  Of course, we could
say RAMROC, but I'd rather say something like ROS.

Another idea is to append "N" to syllables for aNtislice moves.  So
Laughter would be six RONTOCs.

We could also append "P" to allow "deeP" moves, where we do RO and ROM
simultaneously by grabbing two layers of the cube and turning them
while keeping the remaining face fixed.

Finally, I use "G" to denote the way we "wrinG" the cube, as with
ROPRO.  So six ROGTOCs does an 8-flip.

To summarize,

  F,B,T,D,R,L -- faces:      Front, Back, Top, Down, Right, Left.
  O,A,I       -- directions: fOrward, bAckward, twIce.
  C,M,S,N,P,G -- extensions: whole-Cube, Middle, Slice, aNtislice,
                 deeP, wrinG.

All extensions but C are redundant, since

  ROM=ROCRALO
  ROS=ROLA
  RON=ROLO
  ROP=ROCLO
  ROG=ROCROLO

(I sent most of the above to the Cube-Lovers mailing list in February.
This corrects most of the mistakes, I hope.)

m...@oddjob.uchicago.edu (Matt Crawford) writes:

>I never saw one [extension to the larger cubes] except the one I made up when
>I first got a 4x4x4 cube.  In mine, the letters u d r l f b represent
>clockwise rotation of an outer 4x4x1 slab and the letters U D R L F B
>represent clockwise rotation of half the cube....
>For an NxNxN cube, I would suggest elementary moves u1, u2, ... to
>represent rotation of the uppermost 1, 2, ... slabs together.

This is the way I did it at first, choosing a cutting plane and
rotating the part of the cube on one side of the plane wrt the rest of
the cube.  I call this philosophy Cutism, because you cut the cube.
However, I later decided to espouse Slabism, where a primitive move
turns an NxNx1 slab wrt the rest of the cube.  In fact, I go for
Eccentric Slabism, where turning the central slab of a cube of odd
edge is not considered primitive.

The fundamental credo behind Eccentric Slabism is that if you consider
the NxNxN cube to contain an (N-2)x(N-2)x(N-2) cube inside it, then
any two quarter-turns are conjugate.  That is, there is an isomorphism
of the cube that takes one quarter-turn to the other.  (I discussed
Eccentric Slabism and its relation to Qubic in a Cube-Lovers message
back in June, 1983).

We might as well label the slabs with the nearest face.  I think the
outer slabs might best be treated as in the 3x3x3.  Inner slabs can be
done with a diphthong or something.

Dan
Hoey@AIC.NRL.Navy.Mil