[next] [prev] [up] Date: Wed, 27 Aug 80 01:28:00 -0700 (PDT)
[next] ~~~ [up] From: Doug Landauer <Dal@UCLA-SECURITY >
~~~ ~~~ [up] Subject: Notation, transforms I like to use ...

After 5 beers and two Drambuies, finally, here we go ...

Here is all of my knowledge and
notation regarding the cube ...
SPOILER warning
these transformations are enough
to solve any cube. I look forward
to anyone publishing a compendium;
please go ahead and use anything
in here.

Two notes about my particular cube ...
it worked fine until I decided to lubricate it (as is recommended in
one of these cube-lovers messages ... they say to use a graphite lubricant
(I believe) ... do so.) DO NOT use "dri-slide". It is a Molybdenum
Disulfide lubricant for use with metals and it tends to eat plastic, making
it harder to turn. Also (this may be unrelated), three weeks or so after I
lubricated mine (an Ideal from Toys-R-Us in the San Fernando Valley (in LA))
one of the facies popped off and now it's out of commission until I try to
glue it back together.

I found that the descriptions of its inner workings do not do it justice.
It is indeed a work of 3-dimensional mechanical genius. To elaborate on
the descriptions ... the face cubies are about 7 mm thick, and they are
what holds the other pieces in. The corner cubies have a growth on their
"opposite" corners which is roughly cubical but the "inside" corner is
rounded to allow them to rotate.
The side cubies (sidies?) have a 7-mm thick disk-like thing sticking out
of the center of their least visible edge (the one opposite the one that
the sidie's two colors border on). It's actually more like a thick
square with one rounded corner (the inside one).

Anyway, here's a description of my notation followed by a set of
transformations I use; this is indeed sufficient to fix a cube and
if you wish to remain self-sufficient in this matter, read no further;
but then what're you doing on this list anyway?

happy cubing ....

flubrd Notation: (pronounce it `flubberd') Object names:
Faces: f l u b r d stand for the front, left, up, back, right
and down faces respectively. (up==top and down==bottom)
Little cubies
The 6 face cubies (one at the center of each face)
are seldom referred to; use the face name (e.g the f-face cubie)
Side (edge) cubies: fl fu fr fd ul dl
bl bu br bd ur dr
Corner cubies: flu fur bul bru fdl frd bld bdr
For side and corner cubies, the order of its faces only matters
in permutation descriptions (see below), but by convention, the
corner cubies are all named clockwise starting in front or back.
Center Slices
H (`Horizontal'==`floor-parallel') runs through faces f,r,b,l
P (`Parallel' for `body-Parallel') runs through faces u,r,d,l
S (`Slicing' for `body-Slicing') runs through faces u,f,d,b
Moves: The ambiguity between object names and moves is cleared up
by always surrounding a move in angle brackets (<>).
A move can be any of the following:
- a single letter, meaning twist a single face or centerslice, or
rotate the entire cube one quarter-turn (see below);
- a move followed by a move indicates those two moves in sequence;
- a number n followed by '(', a move, and a ')' is the same as
repeating the move n times;
- a move followed by a prime ("'") indicates the inverse of the
move --- by convention also all one-letter moves are defined
so that changing the letter's case also inverts the move.
A move may be grouped by parentheses, so that e.g. <(FUDRB)'> is
equal to <B'R'D'U'F'> (ie, undo each twist in reverse order
to undo this whole move).
<F> twists f clockwise; similarly for <L>,<U>,<B>,<R>,<D> & l,u,b,r,d
<f> twists f counter-clockwise; similarly for <l>,<u>,<b>,<r>,<d>.
Note <F'>=<f>, <f'>=<F>, and so on.
Extended moves (centerslices and cube-turning)
cube-turning: I,J,K=front-top-right clockwise
<I> move cube's front to top <I'> or <i> top to front
<J> move top to right side <J'> or <j> right side to top
<K> move right side to front <K'> or <k> front to right side
centerslice moves
<H> (`Horizontal') move H clockwise as seen from above = <uDk>
<P> move P clockwise as seen from the front = <fBJ>
<S> move S clockwise as seen from the right side = <rLi>
Move combinations are two-letter names beginning with C, E or A;
C1-C9 and (CA-CZ, Ca-Cz if needed) describe corner-combinations;
E1-E9 and EA-EZ (Ea-Ez if needed) describe edge-combinations;
A1-A9, AA-AZ & Aa-Az describe combinations that mess with both.
I also use V,W,X,Y,Z and v,w,x,y,z as temporary macros
Permutations = { list of cubies -> list of new spots }. The permutation
described puts the first list into the new LOCATIONS described by the
second list, moving each face of each cubie in the order specified.
E.g., <F> does {flu fu fur fr frd fd fdl -> fur fr frd fd fdl fu fdl}

Useful basic corner combinations:
C1: <6(RurFU)> :: {fdl flu bul -> dlf luf ulb} twist corners clockwise
C2: <6(rFRuf)> :: {fdl flu bul -> lfd ufl lbu} " " ccw (lf2 + bul)
C3: <3(RUru)> :: {frd fur bul bru -> rfu rdf ubr ulb}
switch those pairs of corners, twisting each
C4: <3(fuFU)> :: {frd fur flu bul -> rfu rdf bul flu} (ditto)
C5: <3(FrfR)> :: {frd fur flu bru -> rfu rdf bru flu} (ditto)

compound ones:
C6: <lC4L> = <l3(fuFU)L> :: {frd fur fdl flu -> rfu rdf luf lfd}
     switch those pairs of corners all in the front face
C7: <C1MC2m> = <6(RurFU)M6(rFRuf)m> :: {fdl bld -> dlf dbl}
     twist fdl cw and bld ccw.
C8: <C5C3> = <3(FrfR)3(RUru)> :: {flu bul bru -> bru ufl ulb}
     cycle three top corners (left two & bru) "cw"
C9: <C5C4> = <3(FrfR)3(fuFU)> :: {flu bul bru -> bul bru flu}
     cycle three top corners (left two & bru) "ccw"

Useful basic edge (side) combinations:
E1: <4(FH)4(HF)> :: {fl fr -> lf rf} twist two sides in place
E2: <2(ssdd)> :: {fu bu fd bd -> bu fu bd fd}
     "doubleswap" swaps front for back sides on top and bottom
E3: <3(FRRF)> :: {fl fr ru rd -> fr fl rd ru}
E4: <3(ffrr)> :: {fu fd ru rd -> fd fu rd ru}
E5: <3(URRU)> :: {ul ur rf rb -> ur ul rb rf}
E6: <3(ffuu)> :: {ul ur fl fr -> ur ul fr fl}
E7: <3(FLLF)> :: {lu ld fl fr -> ld lu fr fl}
E8: <3(ULLU)> :: {lf lb ul ur -> lb lf ur ul}
E9: <E3E4> = <3(FRRF)3(ffrr)> :: {fl fr fu fd -> fr fl fd fu}
EA: <f(rE3R)E5F> = <fr3(FRRF)R3(URRU)F> :: {fu ul ur -> ul ur fu}
     cycle top (left right and front) sidies clockwise
EB: <F(LE7l)E8F> = <FL3(FLLF)l3(ULLU)F> :: {fu ul ur -> ur ul fu}
     cycle top (left right and front) sidies counterclockwise
Pons Asinorum: HHSSPP (180 degrees on each centerslice)
Laughter: <4(lrfb)> = <6(lrK)> (but in the latter the cube is KK'd)...
     Another laugh resolves the cube; a K first produces the 4 crosses.
Crux Christmani:  <SSHiE2fE2FBE2bE2iE2KjE2> or, call E2 (the doubleswap) `Z',
 <SSHiZfZFBZbZiZKjZ>.  This is 6 crosses in 3 pairs:  colors go top with
 bottom, front with right side and back with left side.
Crux Plummeri (the fancy one): <SHiE2fE2FBE2biE2fE2FBE2bE2jE2>; or (again):
 <SHiZfZFBZbiZfZFBZbZjZ> or if Y := <iZfZFBZb>, it's simply <SHYYjZ>!
 This is 6 crosses in 2 sets of three each:  Front, Right and Top each with
 each other; and Back, Left and Bottom each with each other.
Greenberg's "well-known center-cubie rotation algorithm" has 12 varieties:
 Pick two centerslices (say X and Y) and a direction (cw or ccw):
 <xYXy> does ccw and <YxyX> does cw.	Details:
     (xyz cw) means the center (face) cubies of the faces x y and z are
     rotate clockwise; ... ,ccw means counterclockwise.
     fur cw :: <SpsP> or <sHSh> or <PhpH>, ccw :: <pSPs> or <HshS> or <hPHp>
     frd cw :: <PspS> or <SHsh> or <PHph>, ccw :: <sPSp> or <HShs> or <HPhp>
     fdl cw :: <hSHs> or <spSP> or <hpHP>, ccw :: <ShsH> or <psPS> or <phPH>
     flu cw :: <HphP> or <PSps> or <shSH>, ccw :: <pHPh> or <SPsp> or <hsHS>
Outstanding questions mentioned in the note about Singmaster's talk:
(I.e., I ain't seen these yet ... )
1. Conway's Monoswop
2. Rubik's Duotwist

(ignore the two lines above beginning with Pick ... and ending with
... Details: they're wrong. I'd go edit them but my Unix is out
of space on /tmp and on my filesystem and I only have 300-baud access to
it anyway).

Have fun

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