Date: 27 Aug 1980 0128-PDT (Wednesday) From: Dal at UCLA-SECURITY (Doug Landauer) Subject: Notation, transforms I like to use ... To: cube-lovers at mit-mc,alan at mit-mc 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 .... -dalgorf- 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 (ie, undo each twist in reverse order to undo this whole move). twists f clockwise; similarly for ,,,, & l,u,b,r,d twists f counter-clockwise; similarly for ,,,,. Note =, =, and so on. Extended moves (centerslices and cube-turning) cube-turning: I,J,K=front-top-right clockwise move cube's front to top or top to front move top to right side or right side to top move right side to front or front to right side centerslice moves (`Horizontal') move H clockwise as seen from above =

move P clockwise as seen from the front = move S clockwise as seen from the right side = 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., 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: = :: {frd fur fdl flu -> rfu rdf luf lfd} switch those pairs of corners all in the front face C7: = <6(RurFU)M6(rFRuf)m> :: {fdl bld -> dlf dbl} twist fdl cw and bld ccw. C8: = <3(FrfR)3(RUru)> :: {flu bul bru -> bru ufl ulb} cycle three top corners (left two & bru) "cw" C9: = <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: = <3(FRRF)3(ffrr)> :: {fl fr fu fd -> fr fl fd fu} EA: = :: {fu ul ur -> ul ur fu} cycle top (left right and front) sidies clockwise EB: = :: {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: or, call E2 (the doubleswap) `Z', . 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): ; or (again): or if Y := , it's simply ! 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): does ccw and 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 :: or or , ccw :: or or frd cw :: or or , ccw :: or or fdl cw :: or or , ccw :: or or flu cw :: or or , ccw :: or or 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 DalGorf