Date: 10 Jan 1981 13:19 PST From: McKeeman at PARC-MAXC Subject: Re: nomenclature In-reply-to: DDYER's message of 9 Jan 1981 1648-PST To: Dave Dyer , VaughanW.REFLECS at HI-Multics (Bill Vaughan) cc: cube-lovers at MIT-MC I agree there are two problems: 1. Neat "programs" that allow the recording and carrying out of manipulations. 2. Neat "configurations" that allow the recording of the results of carrying out manipulations. In both cases uniqueness, transparanecy, conciseness and all other notational goodies are appropriate. Generalizing a bit on Dave's suggestion, how about: Manipulation = Macro* Macro = MacroName "=" Move* Move = Move "*" Integer -- power | Move ' -- inverse | Move / Move -- conjugate | MacroName | Face (Near (Middle Far?)?)? -- hand moves | "(" Move* ")" -- parenthesization Face = F | U | R Near = Middle = Far = 0 | 1 | 2 | 3 with considerate use of spaces and carriage control. Face signifies "temporarily move that face into the right hand, do the moves, then move it back where it came from". For compatibility, the digits count QTW clockwise (away). The FLUBRD equivalences are: R=R=R1=R10=R100 RR = R*2 RRR = R*3 = R' L=R003 U=U D=U003 F=F B=F003 Slice=R01 AntiSlice=R103 I=F111 J=R111 X Y X' = Y/X Note this has some chance of generalizing to other slice puzzles. (E.g. 4x4x4) It also subsumes FLUBRD with a few appropriate Macro definitions like those above. ------ For configurations the problem can be attacked by specifying which cubies go to which cubies. Singmaster does some of this. The problem is to find a way to specify patterns of change without just listing all the changes. There are positional change, flipping and rotating to be accounted for. For instance, we would like to say (in some much neater way) EdgeFlip = For all X and Y in FLUBRD, edge XY goes to edge YX. The corners each take one step on a Hamilton path. Each corner is rotated 120o. Each center exchanges with its opposit. Each edge XY goes to UV where U is the opposit of X and V is the opposit of Y. Etcetera, etcetera, etc. BFUDRLy yours, Bill  Date: 10 JAN 1981 1405-PST From: WOODS at PARC-MAXC Subject: Re: nomenclature To: McKeeman, DDYER at USC-ISIB, VaughanW.REFLECS at HI-MULTICS cc: cube-lovers at MIT-MC In response to the message sent 10 Jan 1981 13:19 PST from McKeeman@PARC-MAXC I object! Your proposed notation discriminates against lefthanded cubists! -- Don. -------  Date: 10 Jan 1981 14:16 PST From: McKeeman at PARC-MAXC Subject: Re: nomenclature (discrimination against lefties) In-reply-to: WOODS' message of 10 JAN 1981 1404-PST To: WOODS cc: cube-lovers at MIT-MC Don, The discrimination is actually self-imposed. Merely rename your hands and you will never know you were discriminated against. Bill  Date: 12 Jan 1981 0913-PST From: Isaacs at SRI-KL Subject: Stanford Rubik's Cube Club To: cube-lovers at MIT-MC There is a newly formed Rubik's Cube Club, meeting at Stanford, every Thursday, 7 p.m., Crother Memorial Hall, Conference Room. For information, call Kersten (415)321-7725 or Paul 446-0729. Open to all - beginners and experts. First meeting was 1/6/81. Second will be 1/20. -------  Date: 12 Jan 1981 0929-PST From: Isaacs at SRI-KL Subject: nomenclature To: cube-lovers at MIT-MC I gave my father a cube a year ago (for which my mother may never forgive me), and he has been working with it ever since. He has developed his own notation, based on Angevine: Basically, take a corner as an X-Y-Z co-ordinate system, and call the planes (e.g.) X1, X2, and X3 (where x2 is the slice). L is X1, R is X3, F is Y1, etc. + is a clockwise rotation, looking at the -1 face (thus the X3,Y3, and Z3 faces rotate backwards from bfudlr); - is CCW. Half twists are (on a typewriter) +/- (he doesn't have a computer terminal). Anyway, his feeling on Singmaster nomenclature (with wich I disagree) are as follows: "I do feel that Singmaster's limited cubist vocabulary impairs communication of his knowledge and insights. His general use of only six of the nine groups available for rotation is like using only twenty of the twenty-six letters in our alphabet. With an unabridged dictionary and a thesaurus practically anything could still be said, but not as well as using the whole alphabet. ... Perhaps mathematicians just don't care about communication with ordinary people." My father is a lawyer. -------  Date: 12 Jan 1981 10:05 PST From: McKeeman at PARC-MAXC Subject: Re: nomenclature In-reply-to: Isaacs' message of 12 Jan 1981 0929-PST To: Isaacs at SRI-KL cc: cube-lovers at MIT-MC Well, lawyers have themselves occasionally been subject to some criticism for their "communication with ordinary people". My dictionary says "Angevine" has something to do with the line of Plantagenet Kings. I guess the connection is too subtle for me. More constructively, the Isaacs senior notation seems 1-1 with FLUBRD except that it has additional primitives for the slices. The macro facility some have used fills that hole. The real trick is to find notations with (lots of) formal properties reflecting cubik realities. Partly that is a matter of notation design, but mostly it is a matter of deeper understanding of the subject matter. I do not believe it is an accident that great science and great notations have frequently come from the same hand. Bill  Date: 12 Jan 1981 1353-PST From: Isaacs at SRI-KL Subject: Re: nomenclature To: McKeeman at PARC-MAXC cc: cube-lovers at MIT-MC, ISAACS You'd have to ask him about his relationship to the Plantagenets (or Anjous), but James Angevine (with an 'e') wrote out an early(?) solution the the cube which was then sold by Logical Games, Inc, one of the first distributers of (what they called) the Magic Cube. Singmasters first published solution seemed dificult to communicate, so I sent the 'Angevine Solution' to my father. Logical Games, Inc, incedentally, is currently manufacturing the cube in the U.S.A., in white plastic with (I think) a slightly more pleasent color scheme. You're right that it's essentially the BFUDLR + slice, plus the different use of CW and CCW on the face furthest away. I hope to see you at the Rubik's Cube Club meeting. (Actually, I may have to miss this weeks - maybe I'll see you there the 22nd.) --- Stan -------  Date: 15 January 1981 18:30 cst From: VaughanW.REFLECS at HI-Multics (Bill Vaughan) Subject: Weird Algorithm - spoiler warning? To: Cube-Lovers at MIT-MC On going through the old mail, I was a little surprised that nobody uses the same algorithm that I do to solve the cube. But since mine isn't terribly efficient, that's not much of a wonder. Anyway, here it is. 1. Do bottom edges. Honest to god. I put all the bottom edges on top by random dithering, then for each one, turn it so its attached side facie abuts the color-matching face cubie, then rotate that face 180o. That gets the bottom edges right, but random hacking is almost as easy... 2. Do middle edges. (Getting colors right) I only use two moves here. FR'F'R and R'FRF'. I pick a cubie that's on a top edge but belongs on a middle edge - put its side facie adjacent to the matching color face cubie (deja vu) and use FR'F'R if it has to rotate right-and-down, or R'FRF' if it has to rotate left-and-down. 3. Get top edges in correct places. Essentially as in Singmaster, but I use only one of two moves. Align top edges so either: all are OK (skip rest of this step) or one is OK and 3 are wrong. (if that's impossible, use one of this step's moves at random and restart step - guaranteed to work.) Now either FURU'R' or FRUR'U' can be used to get everything OK. 4. Flip top edges as required. I use two different moves for this according to whether adjacent or opposite edges need to be flipped. Opposite: let Q = "turn body-slicing slice 1 qtw clockwise as seen from right". Then 4(QU) 4(UQ) flips FU and BU. Adjacent: FR'F'R.RU'R'U.UF'U'F flips RF and UF - you must reorient the cube to do this on two U edges. (I like this move because of its symmetry and - somehow - completeness. It also rotates the corner cubies adjacent to the edge cubies that it flips.) 5. Get the corners right. Here I have some fun, but the basic moves are: 3(FR'F'R) = (LFU,RBU) (RFU,FRD); 3(FRF'R') which also does a double interchange - tho' it's asymmetrical and I don't use it much; and B'FR'F'RB which stirs 3 of the top 4 CW or CCW - I never remember because I just use it twice for the "wrong" direction. 6. Tumble any corners that need it. Usually not many because of the nice color flipping properties of 3(FR'F'R) -try it. My tumbling move is a monotwist 2(FR'F'R).L.2(R'FRF').L' -- or replace the L and L' with LL if necessary - sometimes it's nastier and you have to do it twice. I've never counted worst-case moves. The algorithm is based almost entirely on Singmaster's Y commutator, and once you get that into your finger bones, you hardly ever make a mistake. On the other hand, this algorithm is bad enough it hardly deserves a spoiler warning. Bill q  Date: 15 January 1981 19:13 cst From: VaughanW.REFLECS at HI-Multics (Bill Vaughan) Subject: "Swirl Patterns" To: Cube-Lovers at MIT-MC I've been recently investigating a set of patterns that I call Swirl patterns for lack of a better name. In a Swirl, each face looks like one of these: X X X X X X X Y Z (Left-hand swirl) Z Y X (Right-hand swirl) Z Z Z Z Z Z where X and Z are complementary colors, and Y is something else. I've classified them roughly into 6 classes, based on handedness of swirl and relative alignment of faces. If you look at the 3 faces adjacent to a corner, they may have the same handedness, or they may have mixed handedness. In addition, two adjacent faces may have parallel or perpendicular swirls. (Parallel swirls have their XYZ columns parallel; perpendicular swirls have thir XYZ columns perpendicular.) Here are my 6 classes; there are another 6 which are mirror images of these, but I don't count them. Nor do I care (at the moment) about color pairings - though I know they are important - or about the colors of the face cubies, which probably aren't important. 1. Same handedness. Two of the three faces have parallel swirls. 2. Same handedness. All three faces have mutually perpendicular swirls. 3. Mixed handedness. The two same-handed faces are parallel, with thir XYZ columns in contact (i.e. forming a belt around the cube). 4. Mixed handedness. The two same-handed faces are perpendicular; the opposite-handed face is perpendicular to both. 5. Mixed handedness. The two same-handed faces are perpendicular; the opposite-handed face is parallel to one. 6. Mixed handedness. The two same-handed faces are parallel, with their XYZ columns pointing towards the third face. Four of these classes are in the antislice group and are a short distance (8 qtw) away from SOLVED. They are classes 1, 3, 5 and 6. They are also the antislice group's analogues of the slice group's "6-spot" or "twelve-square" patterns. What got me started on this is a problem that I still have. One day while playing aimlessly in the antislice group (I thought I had remained in it) I ran across a class 2 Swirl, which was (a) quite pretty (when looked at from the correct corner it looks like a pinwheel) and (b) a bear to solve. (Clearly I thought it was one of the "easy" Swirls.) Having solved it, I wanted to get back to it and found I didn't know how. I tried solving to it and came up with an impossibility - that's how I know the color arrangements must be important - and I haven't found it in my searches yet - nor have I found class 4. Questions: what's the fastest way to get to a class 2 Swirl? What color arrangements are permissible? Is it really in the antislice group? (I now believe not.) Is any class 4 Swirl achievable? How quickly? Is there anything else interesting about Swirls? I'm still playing with these - will give more data as I get it. Bill  Date: 16 January 1981 12:09 cst From: VaughanW.REFLECS at HI-Multics (Bill Vaughan) Subject: more on Swirl patterns: the Pinwheel To: Cube-Lovers at MIT-MC I now have a class 2 swirl that looks like this: uud uld udd bbfrrrffbrrr buflfrfdblbr bfflllfbblll uud urd udd It's pretty, and looks like a pinwheel from 4 of the corners, so i call it a Pinwheel. My current algorithm to get from SOLVED to Pinwheel is 84 qtw and not worth publishing - I expect to reduce that substantially in the near future. Bill  Date: 01/16/81 1322-EDT From: PLUMMER at LL Subject: Ad in Popular Science To: Cube-lovers at MIT-MC Seems like a guy will tell you how to solve the cube for only $5. Check the classified ads in current Popular Science! --Bill -------  Date: 20 Jan 1981 1632-PST From: Isaacs at SRI-KL Subject: Rubiks Cube Club meeting To: cube-lovers at MIT-MC The next meeting of the Stanford University Rubiks Cube Club (SURCC??) will be thursday, Jan. 22. Meyer Library Room 145 7:30 Puzzling with the cube, introduction to cube solving algorithms 8:00 Discussion on how to design and build new 3-D puzzles (magic tetrahedron, etc.) 8:30 Cube theory and Pretty Patterns. Until 9:30 or so. Further information: Kersten, (415)321-7725. Also see article in Stanford Daily. See you all there. -- Stan -------  Date: 21 January 1981 1246-EST (Wednesday) From: Guy.Steele at CMU-10A To: Isaacs at SRI-KL Subject: Re: Rubiks Cube Club meeting CC: cube-lovers at MIT-MC In-Reply-To: Isaacs@SRI-KL's message of 20 Jan 81 19:32-EST Message-Id: <21Jan81 124647 GS70@CMU-10A> Stanford University Cube Kludge Society?? (Sorry, just kidding.) --Guy  Date: 21 Jan 1981 10:07 PST From: McKeeman at PARC-MAXC Subject: Re: Rubiks Cube Club meeting In-reply-to: Guy.Steele's message of 21 January 1981 1246-EST (Wednesday), <21Jan81 124647 GS70@CMU-10A> To: Guy.Steele at CMU-10A cc: Isaacs at SRI-KL, cube-lovers at MIT-MC No. Actually Stanford University Rubik Environment For University Nuts. Bill  Date: 22 January 1981 0010-EST (Thursday) From: Dan Hoey at CMU-10A, James Saxe at CMU-10A To: Cube-Lovers at MIT-MC Subject: Correction to "Symmetry and Local Maxima" Reply-To: Dan Hoey at CMU-10A Message-Id: <22Jan81 001000 DH51@CMU-10A> In our message "Symmetry and Local Maxima" (14 December 1980 1916-EST) we examined local maxima both in the Rubik group and in the Supergroup. David C. Plummer has discovered a flaw in our argument for the Supergroup, which we now correct. Plummer has previously noted (30 DEC 1980 0109-EST) that the T-symmetric position GIRDLE CUBIES EXCHANGED, depicted near the end of section 4, is an odd distance from SOLVED. This is also true of the composition of GIRDLE CUBIES EXCHANGED with GIRDLE EDGES FLIPPED, ALL EDGES FLIPPED, PONS ASINORUM, or any combination of the three, for a total of eight positions. In addition, there are four different T groups, each corresponding to a choice of opposite corners of the cube. Thus 32 of the 72 positions with Q-transitive symmetry groups are an odd distance from SOLVED. The discussion of the Supergroup in S&LM noted that the only face-center orientations which yield Q-transitive symmetry groups are the home orientation and all face centers twisted 180o (called NOON in Hoey's message of 7 January 1981 1615-EST). Any position with either of these face center orientations must be an even distance from SOLVED, so that any reachable position which is T-symmetric in the Supergroup must be an even distance from SOLVED. In our earlier note, we erroneously calculated the number of Supergroup positions with Q-transitive symmetry groups by simply doubling the number of such positions in the Rubik group to allow for the two allowable face-center orientations. What we failed to notice--until Plummer pointed it out--is that neither of the allowable face-center orientations can occur in conjunction with an odd position. The corrected count of known Supergroup local maxima is determined by counting the 40 *even* symmetric positions, multiplying by two, and subtracting 1 for the identity, yielding 79. As Plummer notes, this is surprisingly close to the number of known local maxima in the Rubik group, which stands at 71. The number of known local maxima modulo M-conjugacy is 25 for the Rubik group and 35 ( = 2*(26-8) - 1 ) for the Supergroup.  Date: 21 Jan 1981 14:42:58-PST From: microsoft!zibo at Berkeley Gentlepeople: I have moved and would like to have my CUBE-LOVERS mail correctly sent: My former destination was: ZBIKOWSKI@MARKET. I now reside in: CSVAX.MICROSOFT!ZIBO. Could you make the necessary changes? Gracias.  Date: 27 January 1981 0102-EST (Tuesday) From: Jim Saxe, Dan Hoey To: Cube-Lovers at mit-mc Subject: Pretty Patterns and Solutions Sender: Dan Hoey at CMU-10A Reply-To: Dan Hoey at CMU-10A Message-Id: <27Jan81 010221 DH51@CMU-10A> We are disappointed at Chris C. Worrell's use of the term "Baseball" for the position known in the literature as the "Worm". Worrell's term propagates the apparently popular misconception that baseballs are covered with three-lobed pieces of leather. The position which *we* call "Baseball" reflects the construction much more accurately: D D D U U U F F F U U U L R B R B L B L R F F F L R B R B L B L R D D D L R B R B L B L R F F F D D D U U U Currently, our best process for this position is 34 qtw. The corners are fixed with FRLUDB, two edge four-cycles are inserted in the middle, and a Spratt wrench is conjugated inside that: FRL (RRLL UUDD F' U (F' LUD' BUD' RUD' FUD' F) UDD RRLL F) UDB. The class of patterns which Bill Vaughan calls "Swirl Patterns" (15 January 1981 19:13 cst) are also known as "6-2L" patterns in Singmaster, and the particular one he calls the "Pinwheel" (on 16 January 1981 12:09 cst) is an M-conjugate of the AC-symmetric "Twelve-L's" mentioned in our message on Symmetry and Local Maxima (14 December 1980 1916-EST, Section 6). [Incidentally, the diagram he displays in the message of 16 January is in error; the left and right face centers have been swapped. This is made less obvious by the unusual orientation of the cube in that diagram.] We have found a totally magical 12 qtw process for the Pinwheel: FB LR F'B' U'D' LR UD. Vaughan's definition of Swirl Patterns seems unduly restrictive to us on one count: he requires the two L's on each face to be of "complementary" (evidently meaning opposite) colors. This is not necessary for an L pattern. According to our analysis, however, at least two of the faces of any L pattern must have L's of opposite colors, and five is easily seen to be impossible. We know of no patterns having three or four such pairs. But there are several with two pairs. Our favorite example is a relative of the Baseball which we name for Linda Lue Leiserson, who has the appropriate initials. D D D F U D F F F U U U L B B R L L B R R D F U L R B R B L B L R D D D L L B R R L B B R F F F U D F U U U We have a 24 qtw process for Linda Lue's L: F'BB L (B U'LR' F'LR' D'LR' B'LR' B') R UD B'FF This has a Spratt wrench, conjugated by B', embedded in magic. David C. Plummer (3 SEP 1980 2123-EDT) reported that it is possible for each of the six faces of the cube to show a capital "T". Our analysis indicates that there are two sorts of T patterns: D U D D D U D U D U U U U U U D D U L L L F F F R R R L R R F F F R L L R L R B F B L R L L L L B F B R R R R L R B F B L R L L R R B F B R L L D D D D U U U D U D D D U D U D U U B B B B B B F B F F B F F B F F B F Tanya's T Plummer's T Tanya's T is named for Tanya Sienko (who inspired the problem) and for euphony. Plummer's T is named for Plummer's Cross (which has the same symmetry group) and for homophony. There are 24 M-conjugates of Tanya's T, while Plummer's T has 8 M-conjugates. By adapting a process due to David C. Plummer, we have developed a 16-qtw process for Tanya's T: (FF UU)^3 (UU LR')^2. The first part swaps two pairs of edge cubies, and the second part is magic. We have found a 28-qtw process for Plummer's T, which is entirely magical: FF UD' F'B' RR F'B U'D RL FF RL' UD' RL FF R'L U'D'. A position which is not so visually striking, but which is important in the symmetry theory we have discussed earlier, is "All Corners Twisted": B U B U U U F U F U L U L F R U R U R B L L L L F F F R R R B B B D L D L F R D R D R B L F D F D D D B D B This can be achieved in 30 qtw with FLU (LRRFFB')^4 U'L'F'. The process is a conjugated from a 24 qtw process invented by Thistlethwaite. Unfortunately, Thistlethwaite's process twists the wrong corners, and no cancellation can be performed in the conjugation. If any process can be found which twists four corners clockwise and four counterclockwise, leaving the rest of the cube fixed, then any such pattern can be made by adding at most 6 qtw.  Date: 29 JAN 1981 0344-EST From: BSG at MIT-AI (Bernard S. Greenberg) Mailed-by: BSG @ MIT-Multics Subject: Lisp Machine Cubesys Improvements To: CUBE-LOVERS at MIT-MC Largely due to the newfound coincidence of relentless Lisp Machine hacking with my job, I have added a slew of features to Lisp Machine Cubesys, viz., all kinds of New Window System and flavor hacking. It is now completely mouse-oriented, all moves are made by mousing menu items, OR by mousing at cube-sides on the display (either display) to turn faces, left mouse button for left (ccw) turn, etc.! To find out more about it, load it in the usual way, (load "bsg;cubpkg >"), invoke it in the usual way ((cube)), and type the HELP key (or mouse the HELP menu item). Have fun, -bsg  Date: 1 February 1981 0539-EST (Sunday) From: Dan Hoey at CMU-10A To: Cube-lovers at MIT-MC Subject: Algorithm for finding cube group orders Message-Id: <01Feb81 053933 DH51@CMU-10A> David C. Plummer (31 Dec 1980 1210-EST) gave a preliminary analysis of the 5x5x5 cube. I complete it here. Let a move consist of twisting any of the six faces, at a depth of 1 or 2. It will be necessary to consider the two depths as distinct; M1P will refer to the number of depth 1 moves (mod 2), while M2P will refer to the number of depth 2 moves (mod two). It is important to realize that the two parities vary independently. The tabs on each face are assigned types C L E R C R D A D L E A X A E L D A D R C R E L C as in Plummer's analysis. Let COP ("C" Orientation Parity) and CPP ("C" Permutation) parity be defined as before. As before, COP=0 (mod 3). We must be explicit about the CPP this time: Since either kind of move is an odd permutation of the "C" faces, CPP=M1P+M2P. As in the 4x4x4 case, "R"'s may be ignored and "L"'s have no orientation. The permutation parity (LPP) is important, however. Depth 1 moves are an even permutation of the "L"'s (two 4-cycles), so they do not affect the LPP, but Depth 2 moves are an odd permutation of the "L"'s (three 4-cycles). Therefore LPP=M2P. Note that while LPP and CPP may vary independently, they together determine both M1P(=LPP+CPP) and M2P(=LPP). The "E" faces act as in the 3x3x3 case, with orientation and permutation parity. Orientation changes on four "E"'s with every move, so EOP=0 (mod 2). Permutation parity changes with every move, so EPP=M1P+M2P. This has already been determined by CPP, so only half of the "E" permutations are possible. Every move is an odd permutation of the "D" faces, so DPP=M1P+M2P. Since M1P+M2P=CPP is determined, only half of the "D" face permutations are possible. Moves work differently on "A" faces depending on depth: Depth 1 moves are odd permutations of the "A"'s, and depth 2 moves are even. Thus APP=M1P, which is determined by CPP+LPP, so only half of the "A" permutations are possible. Finally, the "X" faces have orientation which changes on every move, so XOP=M1P+M2P, and only half of the "X" orientations are possible. Thus there are 96 orbits, corresponding to COP (mod 3) and EOP, EPP+CPP, DPP+CPP, APP+CPP+LPP, and XOP+CPP (mod 2). The basic combinatoric is as Plummer described: 8! C Permutations 3^8 C Orientations 24! L Permutations 1 R Permutation 12! E Permutations 2^12 E Orientations 24! D Permutations 24! A Permutations 4^6 X Orientations which when multiplied together and divided by 96 yields about 5.289*10^93. [This differs from Plummer's result by a factor of 4096 because (4^6/2) he didn't count X Orientations, and (2) he did not realize that LPP and CPP are independent.] My implementation of Furst's algorithm claims that all of these are reachable. To count the number of reachable color patterns, divide this note that there are by (4!)^6/2 invisible D permutations, (4!)^6/2 invisible A permutations, and 4^6/2 invisible X orientations that satisfy the invariants. While there are pairs of L/R edges that look the same, they cannot be interchanged, for that would entail putting an L tab into an R position. So there are 2.829*10^74 different color patterns achievable. ----------------------------------------------------------------  Date: 1 February 1981 0612-EST (Sunday) From: Dan Hoey at CMU-10A To: Cube-lovers at MIT-MC Subject: Algorithm for computing cube group orders Message-Id: <01Feb81 061255 DH51@CMU-10A> Oops... you have just received part 3. This is part 1.... This note is in somewhat delayed response to the note by David C. Plummer (31 DEC 1980 1115-EST) regarding the 5x5x5 Rubik cube, and some related ideas. In that note he tried to calculate the size of that cube's Rubik group, but left several of the values open to conjecture. I will complete the answer, and answer a few others that haven't been addressed here. Computing the size of a Rubik group is a special case of computing the size of a permutation group, given generators for that group. The technique we have already seen in these pages is in two parts. The first part seems relatively easy: certain invariants must be observed in the generators, such as "Corner Orientation Parity" and "Total Permutation Parity." [In this general setting, such invariants as "Colortabs on the same cubie move together" must also be considered.] It may take some thought to dig out the invariants, but once you have seen them demonstrated for Rubik's Cube, you have an idea of what to look for. The second part is the devil: it must be demonstrated that every permutation satisfying those invariants is actually generated. This involves developing a solution method for the puzzle. Given the days or weeks (or eternity) it takes most people to develop such a method--with cube in hand!--it is hardly surprising that few answers have been developed. Well, the second part is no longer a hard problem. The answer lies in a paper by Merrick Furst, John Hopcroft, and Eugene Luks, entitled "Polynomial-Time Algorithms for Permutation Groups," which was presented at the 21st Annual Symposium on Foundations of Computer Science, October 1980. Among the results is an algorithm which takes as input a set of permutations on n letters, and reports the size of the group G which is generated by those permutations. The algorithm operates by decomposing G into a tower of groups I=G[0], G[1], ..., G[n]=G, where G[i] contains those permutations p in G for which p(k) = k whenever i < k <= n. The index of G[i-1] in G[i] is developed explicitly by the algorithm; in fact, a representative g[i,j] of every coset of G[i-1] in G[i] is exhibited. These coset representatives generate G; in fact, every element of G is representable as a product of the form (g[1,j1])(g[2,j2])...(g[n,jn]). For this reason the coset representatives are called "strong generators" for G. There is a good deal of structure that can be learned from the strong generators, in addition to the size of G. I have coded this algorithm in Pascal, and offer the program for the use of anyone who needs to find group orders. The relevant files are on CMU-10A, from which other sites may FTP without an account. The relevant files are all:group.pas[c410dh51] The source all:rubik.gen[c410dh51] A sample input -- the supergroup all:rubik.lst[c410dh51] Sample output. Of course, CMU Pascal is probably slightly different from yours, and OS-dependent stuff like filenames is likely to be wrong. I'll be glad to help out in cases of transportability problems. The other problem you may run into is resource availability. The running time of the algorithm is proportional to (nm)^2, where m is the total number of strong generators; the supergroup (n=72, m=279) takes 639 cpu seconds on a KL-10, and bigger problems grow rapidly. The program also requires 47000+47m words. It might seem that the problem has been answered, but I find that simply knowing the size of a group is not very satisfying. There doesn't seem to be a better way of demonstrating lower bounds, but the upper bounds that come from invariants are much more elegant than a simple numerical answer. Unfortunately, I know of no mechanical way of finding the invariants. Furthermore, using group theory does not help much when we ignore the Supergroup. Consider the 4x4x4 cube. If we are only concerned with the color pattern on the cube, then a twist may or may not affect the four face centers--it depends on whether they are the same color or not. In summary, the algorithm has inverted the hard and easy parts of cube analysis. The size of the group is now easy to determine, making invariant-finding the hard part. Further, the algorithm works on the Supergroup, making counting distinct color patterns the part which requires further analysis. Two messages follow, supplying these answers for the 4x4x4 and 5x5x5 cubes.  Date: 1 February 1981 0651-EST (Sunday) From: Dan Hoey at CMU-10A To: Cube-lovers at MIT-MC Subject: Analysis of the 4x4x4 cube Message-Id: <01Feb81 065108 DH51@CMU-10A> The first problem for the 4x4x4 cube is in eliminating positions that arise from whole-cube moves. This was done with the 3x3x3 by keeping the face-center positions fixed, but there are no face centers on the 4x4x4--or there are, but they don't maintain a fixed orientation relative to each other. I standardize the spatial orientation by keeping the DBR corner in a fixed position and orientation. A move then consists of twisting one, two, or three layers parallel to the U, F, or L face. Thus F3' is equivalent to twisting the B face. I will refer to the number of layers twisted as the "depth" of the move. Following David C. Plummer's notation (31 DEC 1980 1210-EST), organize each face of the cube as C L R C R X X L L X X R C R L C. I will assume familiarity with David Vanderschel's analysis of the 3x3x3 case, which was presented in his message of 6 August 1980. "C" faces act as they do in the 3x3x3 case, except that one of them does not move. Corner Orientation Parity (COP) is preserved and Corner Permutation Parity (CPP) changed by every quarter-twist. Depending on the depth, a quarter twist can permute the "L" faces in an odd or an even permutation. Also, "L" faces do not change orientation (or move to "R" positions). Every "R" face is determined by the "L" face (on an adjacent side of the cube) with which it shares a cubie. Thus the arguments for EOP and EPP do not apply. Every quarter-twist is an odd permutation of the "X" faces: either one, three, or five four-cycles, depending on the depth. Letting XPP be the permutation parity of the "X" faces, the Total Permutation Parity TPP=XPP+CPP (mod 2) is preserved by every quarter-twist. Thus the 4x4x4 cube group has at least six orbits, according to COP (mod 3) and TPP (mod 2). The basic upper bound of 7! Corner Permutations 3^7 Corner Orientations 24! L Permutations (which determine the R permutations), and 24! X Permutations, divided by six, yields an upper bound (of about 7.072*10^53). I have run Furst's algorithm on the problem, and my program claims that all these positions are reachable. To calculate the number of reachable color patterns, note that there are 4! permutations of each quadruple of "X" faces which are indistinguishable. However, the TPP constrains the XPP so as to reduce this by a factor of two. Dividing 7.072*10^53 by (4!)^6/2 yields 7.401*10^45. [At this point, you may find it instructive to view the message before last, which analyzes the 5x5x5 cube in the context of this message and the one immediately preceding. I regret the accidental disorder. These three are all for now, although I have results on tetrahedra, octahedra, and a dodecahedron which I am in the process of writing up.]  Date: 6 Feb 1981 at 1330-CST From: korner at UTEXAS-11 Subject: cube lube To: cube-lovers at mit-mc After trying almost all the cube lubricants suggested (with the notable exceptions of the plastic eating varieties)- I would like to suggest that a local maxima seems to be silicon gel (of the sort used to lubricate SCUBA O rings or food processors- not the spray, the gel). To use this stuff, one must disassemble the cube. As long as it's apart, take a fine flat file to the cubies and remove any seams from the molding process and any imperfections from the glue job (cement beads or protruding plates). Cubus hungarius finishes well with just a file, cubus americanus (the white one) may need work with wet fine sandpaper to restore a smooth surface after filing. If you're really fanatic, adjust the screw tension (ala Singmaster). Clean off the debris and apply liberal coats of the gel to all tab faces. Reassemble the cube and enjoy- one handed cubing not guaranteed but definitely possible. -KMK -------  Date: 9 FEB 1981 2345-EST From: JURGEN at MIT-MC (Jonathan David Callas) Subject: True Stories of Cubism To: CUBE-LOVERS at MIT-MC I was at the Hirshhorn (Smithsonian Modern Art Museum) last Sunday to see the exhibit of Avant-garde Russians, and lo, I saw in the museum shop what could only be Cubus Albus! I played with it for awhile (I solved the top 2 tiers before my girlfriend said "That's enough! You can do that at home!") and it worked more smoothly than my C. Americanus! So now, I guess, not only is the Cube a source of mathematical inspiration, but an objet d'art as well. Happy Cubing, Jurgen at MIT-MC  Date: 11 FEB 1981 1600-EST From: RP at MIT-MC (Richard Pavelle) To: CUBE-LOVERS at MIT-MC The March issue of Scientific American is out. Guess what is on the cover as well as in the interior?  Date: 12 Feb 1981 0816-PST Sender: OLE at DARCOM-KA Subject: The England Scene From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Message-ID: <[DARCOM-KA]12-Feb-81 08:16:37.OLE> Yes, cubes are indeed very big here in England due mostly to the fact that they have been featured on television several times recently. About 3 weeks ago a kid solved a cube in 37 secs on the Saturday mor- ning BBC1 show "Multicoloured Swap Shop" (very appropriate name). In a follow up a group of people challenged him, but "only" managed it in 57 secs. Nothing was however said about local maxima etc, so it wasn't a very scientific exercise. The recordholder's solution se- quence was shown in slow motion (his hands still seemed to move very fast) and as far as I could determine he uses Kertezs's Algorithm, i.e layer-by-layer, but with some clever shortcuts rather than just using the macros blindly. At the moment cubes are impossible to get but we are hoping for a new shipment to arrive soon. A cube club will probably be formed here at Newcastle University, Newcastle upon Tyne and I would be surprised if other universities won't be doing the same. By the way, has anyone ever tried turning cube when the temperature is 5-6 degrees C? I have, because that is the temperature my room is at when I come home at night. English houses are VERY cold. Ole  Date: 16 February 1981 1229-EST (Monday) From: Guy.Steele at CMU-10A To: bug-lispm at MIT-AI, cube-lovers at MIT-MC Subject: Scientific American Message-Id: <16Feb81 122922 GS70@CMU-10A> Congratulations to cubemeisters, LISP Machinists, and Symbolicists alike for making *Scientific American*. Now that the LISP Machine has been used to serve the cause of cubing, has any thought been given to the converse? For example, perhaps a mouse/joystick-like device could be built based on cube technology? Also, anyone thought about the limiting case of odd-shaped polyhedra: the continuous cube (or, Rubik's sphere)? There are three possible places to introduce continuity. For a given twist, one must choose an axis, choose a depth of slice, and choose an angle of twist. For the cube all three are quantized. What are the geometric/topological properties of an object where some subset of these three choices are given a continuous domain? (I haven't the mathematics undert my belt to attack this problem -- sorry.) --Guy  Date: 16 February 1981 2327-EST (Monday) From: Jim Saxe, Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Four colors suffice CC: Mary Shaw at CMU-10A, Paul Hilfinger at CMU-10A, Bill Wulf at CMU-10A, Dorothea Haken at CMU-10A Sender: Dan Hoey at CMU-10A Reply-To: Dan Hoey at CMU-10A Message-Id: <16Feb81 232721 DH51@CMU-10A> Douglas Hofstadter, in the Metamagical Themas column in Scientific American this month, shows two alternate ways of coloring a cube. Both suffer from two drawbacks: They fail to distinguish all cube positions, and they use more than six colors. This seems inefficient to us, since there is a coloring of the cube which distinguishes all elements of the Supergroup and uses only four colors (and which, like Hofstadter's colorings and the standard coloring, satisfies the restriction that every whole-cube move is a color permutation, as discussed in point 2 below). Our coloring, called the Tartan, is formed by assigning the colors blue, green, red, and yellow to the four pairs of antipodal corners of the cube. Thus for each face of the cube, the four corners of the face are assigned four different colors. We use the term ``plaid'' to denote such an assignment of colors to the corners of a square. To color the cube, divide each facelet of each cubie into four squares, and color the squares so all facelets on a side of the cube display the plaid associated with that face. The result is shown below, with the initial assignment of colors to corners in lower case. (r)---------------(y) | R Y R Y R Y | | B G B G B G | | | | R Y R Y R Y | | B G B G B G | | | | R Y R Y R Y | | B G B G B G | (r)---------------(b)---------------(g)---------------(y) | R B R B R B | B G B G B G | G Y G Y G Y | | G Y G Y G Y | Y R Y R Y R | R B R B R B | | | | | | R B R B R B | B G B G B G | G Y G Y G Y | | G Y G Y G Y | Y R Y R Y R | R B R B R B | | | | | | R B R B R B | B G B G B G | G Y G Y G Y | | G Y G Y G Y | Y R Y R Y R | R B R B R B | (g)---------------(y)---------------(r)---------------(b) | Y R Y R Y R | | G B G B G B | | | | Y R Y R Y R | | G B G B G B | | | | Y R Y R Y R | | G B G B G B | (g)---------------(b) | G B G B G B | | R Y R Y R Y | | | | G B G B G B | | R Y R Y R Y | | | | G B G B G B | | R Y R Y R Y | (r)---------------(y) To understand the importance of the Tartan, there are several points to consider: 1. By reading off the four colors of a plaid in clockwise order, starting at an arbitrary point, we obtain four permutations of the four colors. Quadruples read from different faces are disjoint, so all 24 permutations of the four colors appear on the Tartan, once each. 2. Every motion in the group C of whole-cube rotations is a permutation of the pairs of antipodal corners, and so corresponds to a recoloring of the Tartan. Some restriction of this sort is necessary to prevent us from simply drawing a different black-and-white picture on each facelet and calling that a two-coloring. 3. Point 2 implies that C is isomorphic to a subgroup of S4, the group of permutations on the four colors. But both C and S4 have 24 elements, so C is isomorphic to S4 itself (a fact well-known to crystallographers). 4. Since every color permutation is realizable by a whole-cube move, there is only one Tartan (up to whole-cube moves). This is why we use colors as labels, rather than some FLUBRDoid positional scheme. [The actual choice of colors and the name ``Tartan'' arise from the DoD Ironman project.] 5. Every reflection of the Tartan is color-equivalent to a rotation. In particular, the identity is color-equivalent to a reflection through the center of the cube. If you were to lend your Tartan to someone who ran it through a looking-glass, you could not discover the fact except by removing the face-center caps and examining the screw threads! We have constructed a Tartan from a Rubik's cube and colored tape. Due to the similar appearance of the plaids, it takes us several times as long to solve the Tartan as it takes to solve Rubik's cube. Our search for pretty patterns has not been particularly rewarding. Part of the reason seems to be that the cube's appearance is strongly constrained by the Tartan's coloring. On Rubik's cube one may make a particular face pattern (e.g. orange T on white background) using any of several identically colored facelets. On the Tartan, however, the plaid on any facelet of a cubie, together with the orientation of the plaid relative to the cubie, determines the plaid and orientation of the other facelet(s) of the cubie. The one nice pattern we have is in fact the conceptual precursor to the Tartan. It is Pons Asinorum (FFBBUUDDLLRR) applied to the position shown in the diagram above. In this position, the plaids of adjacent facelets line up with each other to display the same arrangement of plaids, magnified by a factor of two. Each face looks like the following, for some assignment of colors to the numbers 1 through 4: (1)---------------(2) | 1 2 2 1 1 2 | | 4 3 3 4 4 3 | | | | 4 3 3 4 4 3 | | 1 2 2 1 1 2 | | | | 1 2 2 1 1 2 | | 4 3 3 4 4 3 | (4)---------------(3)  Date: 17 Feb 1981 07:54:00-PST From: microsoft!zibo at Berkeley Is it possible??? In the SciAm article they mentioned 4x4x4 cubes... Has anyone seen them??  Date: 17 Feb 1981 10:49 PST From: McKeeman at PARC-MAXC Subject: Re: 4x4x4 cube In-reply-to: Your message of 17 Feb 1981 07:54:00-PST To: microsoft!zibo at Berkeley cc: cube-lovers at MIT-MC Zibo, All things are possible in the computer. My undertanding is that a 4x4x4 is being built, but I have not heard that it is yet complete. The real mind-bender is the continuous Rubik Sphere. Take a sphere, slice it arbitrarily many times. Now each slice makes a plane of rotation and gives a degree of freedom. The Rubik Cube is a special case for which there is a mechanical implementation. Bill  Date: 17 Feb 1981 1445-PST (Tuesday) From: Mike at UCLA-SECURITY (Michael Urban) Subject: Rubik's Sphere To: cube-lovers at mit-mc While it shouldn't be TOO hard to program an arbitrarily-fine simulation of a Rubik sphere, one wonders how it's colored. do the latitute/longitude coordinates on the sphere correspond to hue/intensity? While you can settle for any coordinate labelling of points, different coloring schemes will dictate what constitutes a "pretty" pattern, yes? Mike -------  Date: 17 February 1981 18:03 cst From: VaughanW at HI-Multics (Bill Vaughan) Subject: Rubik's Sphere Sender: VaughanW.REFLECS at HI-Multics To: Mike at UCLA-Security, cube-lovers at MIT-MC Do you color a Rubik's Sphere discretely or continuously? And if continuously, it's probably too easy to solve - seems as though the discontinuities would show you where to turn it; and by always turning along a discontinuity until it vanishes, you can get to SOLVED by what seems to be God's Algorithm. Bill  Date: 17 Feb 1981 16:12 PST From: McKeeman at PARC-MAXC Subject: Re: Rubik's Sphere In-reply-to: Mike's message of 17 Feb 1981 1445-PST (Tuesday) To: Mike at UCLA-SECURITY (Michael Urban) cc: cube-lovers at mit-mc (I wish Hofstadter were on the net) Mike, Are you proposing a truly continuous Rubik sphere with an infinite, nay uncountable, number of slicings with continuously varying hue to distinguish "slices"? Such cubes could differ in the "function" that connects the motion of "neighboring slices". We could have linear, quadratic, and even hyperexponential axes of rotation. Then giving the cube a spin about each of its (many) axes, we would have a continuously shifting pattern of color. Maybe would should leak this idea to George Lucas for the visuals of StarWars III? Or maybe one of the LISP machine folks can whip up a simulation overnite? Bill  Date: 17 Feb 1981 1622-PST From: Steve Saunders Subject: Rubik-like sphere To: Cube-Lovers at MIT-MC And if you color it continuously, why not have continuous moves, too? For instance, a smooth twist about an axis (like twisting a rubber ball that's glued to sticks at its poles -- carries meridians into spirals), or a smooth bending (like pushing one of those poles sideways while holding the other fixed -- makes parallels not parallel). I suspect that the groups resulting from some sets of smooth motions would be very simple, but some might have interesting interactions. A problem with all this smoothness (a feature?) is that it would enable approximate solutions, iterative converging infinite "solutions", and disputes about whether SOLVED has in fact been reached -- none of which occur with the real Rubik's. Steve -------  Date: 17 Feb 1981 1716-PST (Tuesday) From: Lauren at UCLA-SECURITY (Lauren Weinstein) Subject: Sphere carrying case To: CUBE-LOVERS at MC I know where I'm going to keep my Rubik's Sphere when I get one: inside my Klein Bottle! --Lauren-- -------  Date: 17 Feb 1981 18:00 PST From: McKeeman at PARC-MAXC Subject: Re: Rubik's Sphere In-reply-to: VaughanW.REFLECS's message of 17 February 1981 18:03 cst To: VaughanW at HI-Multics (Bill Vaughan) cc: Mike at UCLA-Security, cube-lovers at MIT-MC, SAUNDERS at USC-ISIB Bill, Well, My idea is that the continuous slicing is constrained by some function so that you basically only have one degree of freedom per axis. The trick is to scramble on several axes, and then try to get back. If the coloring is continuous before any twisting, then it is always continuous. (I think) Another Bill  Date: 20 February 1981 20:17 est From: Greenberg.Symbolics at MIT-Multics Subject: A lighter note To: CUBE-HACKERS at MIT-AI in this increasingly hirsute forum: A man called us at Symbolics today, having seen our name in the Scientific American article. He was having trouble getting cubes in the Chicago area, and wanted to know if we could sell him some.... (a true story)...  Date: 20 February 1981 20:19 est From: Greenberg.Symbolics at MIT-Multics Subject: A lighter note To: cube-hackers at MIT-MC in this increasingly hirsute forum: A man called us at Symbolics today, having seen our name in the Scientific American article. He was having trouble getting cubes in the Chicago area, and wanted to know if we could sell him some.... (a true story)...  Date: 20 FEB 1981 2108-EST From: RP at MIT-MC (Richard Pavelle) Subject: Rubik To: CUBE-HACKERS at MIT-MC When speaking to the editor of Scientific American yesterday, the subject of the cube came up. I mentioned that Rubik did not get dollar much less a forint for his effort. Guess what! A subscription to Sci. Am. is on the way to him (lifetime I suppose).  Date: 21 February 1981 00:14-EST From: Ed Schwalenberg Subject: A lighter note To: Greenberg.Symbolics at MIT-MULTICS cc: CUBE-HACKERS at MIT-AI Isn't this the C-Machine that you all are working on?  Date: 03/05/81 0839-EDT From: PLUMMER at LL Subject: another article To: CUBE-LOVERS at MIT-AI Check today's Wall Street Journal: front page, center. --Bill -------  DAN@MIT-ML 03/05/81 20:58:59 Re: Rubiks Cube info needed To: cube-lovers at MIT-AI I have a few questions which you may be able to help me with... 1. Could you please add me to this mailing list 2. I am looking for a rubiks cube solver to play with on my microcomputer, and would like to know if such a program exists. Would prefer Pascal or "C" (as I most readily hack these), but Lisp, et.c would be fine. 3. Is there an archive of "Cube" info, back letters, documents, bibliographies, etc. lying around on one of the ITS machines? Thanks - Dan  Date: 6 MAR 1981 0849-EST From: JURGEN at MIT-MC (Jonathan David Callas) Subject: Cube Solver To: DAN at MIT-ML CC: CUBE-LOVERS at MIT-MC I have a program written in pascal that won't *SOLVE* the cube but will manipulate it. It will also find the order of a given move. It was written by Tom Davis (of this list) and modified by me. It should run on any USCD system with no hassles, and very minor ones for any other sort of pascal. Since Tom was giving the program out before, I shall assume that there is no problem with distributing this version. If you (or anyone else) wants acopy, write me (Jurgen at MC) & I'll send you a copy. -- Happy cubing, -- Jurgen at MC  Date: 7 Mar 1981 0224-PST From: Alan R. Katz Subject: how about... To: cube-lovers at MIT-MC cc: katz at USC-ISIF How about a Braile cube (with dots instead of colors) for the blind, or so one could solve it with both eyes closed??? (dont ask me why you would want to solve it with both eyes closed). Alan -------  Date: 8 Mar 1981 1834-EST From: JURGEN at MIT-DMS (Jonathan David Callas) To: KATZ at USC-ISIF, CUBE-LOVERS at MIT-MC Subject: Braille Cube Message-id: <[MIT-DMS].189113> I'm sure that it's all ready been done. (Blind people are very clever that way) There are the equivalents of dymo label-makers that print in Braille, and any random sighted person could label a cube (even randomized). I'd bet that it would very very slow, though. The eyes have a much greater informational bandwidth than the fingers. -Jurgen at MC  Date: 9 MAR 1981 0855-EST From: JURGEN at MIT-MC (Jonathan David Callas) To: CUBE-LOVERS at MIT-MC I have sent out copies of the cube program I mentioned earlier to (I think) everyone who asked for it. If you didn't get it, or it was munged, or you would like it, I saved a copy of the msg in: DM:USERS1;JURGEN CUBE --Happy Cubing --Jon  Date: 9 Mar 1981 10:02 PST From: McKeeman at PARC-MAXC Subject: Re: how about a Braille cube... In-reply-to: KATZ's message of 7 Mar 1981 0224-PST To: Alan R. Katz cc: cube-lovers at MIT-MC Wonderful idea! Probably even fundable by some gov't agency for the handicapped. As to why do it with your eyes closed, that was in some sense the original intent of the cube: spatial visualization. Besides, it confuses me to look at during a macro. Bill  Date: 9 Mar 1981 at 1721-CST From: korner at UTEXAS-11 Subject: edge cubie rotation To: cube-lovers at mit-mc does anyone have a nifty edge cubie rotation algorithm that doesn't do a di flip in the process. I'm getting tired of f r 3(F R R F) R 3(U R R U) F There must be something better- I just haven't found it. -Kim Korner -------  Date: 10 MAR 1981 0556-EST From: ACW at MIT-AI (Allan C. Wechsler) Subject: edge cubie rotation To: korner at UTEXAS-11 CC: CUBE-LOVERS at MIT-AI My basic triple-edge tool is FFRL'UUR'L. It rotates three edges that all lie in one equator. Manipulation hint: move that equator instead of doing RL' and R'L. Something like Kim's tool can be obtained by setting up with RL'U and finishing with U'LR'. All together: RL'U FF RL' UU R'L U'LR'. ---Wechsler  Date: 10 Mar 1981 1910-PST From: CSL.JHC.DAVIS at SU-SCORE Subject: Edge Cubie Rotation To: korner at UTEXAS-11 cc: cube-lovers at MIT-AI I have been using an even shorter tool to do Kim Korner's transformation. In the Befuddler notation, it is: R' L B L' R D D R' L B L' R It is much easier to do than this notation makes it seem. I think of it as pushing a center cubie down to the bottom, turning it off to the side, and bringing back the old top. Then I move it around to the other side of the bottom, and go back down to pick it up. If you begin the transformation with RR LL instead of R' L, and end it with RR LL instead of L' R, it does the same thing except with no flipping of center cubies. -- Tom Davis PS. As in Wechler's tool, think of center-slice moves. -------  Date: 12 MAR 1981 2317-EST From: ATTILA at MIT-MC (Sean N Levy) Sent-by: ATTIL0 at MIT-MC Subject: Re: Rubik's Sphere carrying case To: CUBE-LOVERS at MIT-MC CC: Lauren at UCLA-SECURITY Instead of putting it in a klien bottle, how about using one of Escher's impossible boxes (decorated with an Escher on the outside, of course...) -- Attila  Date: Monday, 16 March 1981 19:28-EST From: Pat O'Donnell To: cube-lovers at mc Subject: other orbits Has anyone investigated what kinds of patterns exist in the other 11 orbits?  Date: 18 March 1981 22:34-EST From: Alan Bawden To: CUBE-HACKERS at MIT-MC Check out this week's TIME magazine. (The "Living" section, I believe.)  ISRAEL@MIT-AI 03/20/81 15:55:21 Re: two-person games using the cube To: CUBE-LOVERS at MIT-AI Folks, I was examining my cube the other day and I noticed that each side looks like a tic-tac-toe board and I realized that we've never considered the idea of two-person games using the cube. Here are some games that I've come up with. Some of these may be trivial and uninteresting (i.e. obvious wins for the first or second player) and some may be too easy to draw with, but I'll throw them out anyway. The first game I thought of was Rubik's tic-tac-toe. This is just regular tic-tac-toe with a twist (pun intended). Each person takes turns first writing his symbol on one of the 54 facelets on the cube. After doing that he twists one face and passes the cube to his opponent. There are a number of different variations of this game. 1) The first person to win any side of the cube wins. This seems to be a very easy game so to make it more interesting we add the rule that for a person to win, he must do it before executing a twist to the cube. 2) To win, a person must win a majority of the faces on the cube. This game has the interesting property that if the cube is full (a draw in normal tic-tac-toe) twists can continually be made until a win is reached, both people agree on a draw, or some arbitrary upper limit on the number of moves beyond a full cube is passed. 3) One person must fill up all nine facelets of any face with his symbol. This game may be too difficult to win. 4) Each person has pattern of X's, O's, and don't cares which his opponent doesn't know and has to get one face to look like that pattern. Each of these games can be modified by adding restrictions on the twist such as; a) only quarter turns CW and CCW are allowed; b) a player cannot turn the same face his opponent just turned; c) a player cannot turn the same face that he turned last turn; d) if a player made a quarter turn last turn he must make a half turn this turn and vice versa; or any combinations of the above restrictions or others. Does anyone know of good erasable writing utensil to use on your cubes or have a metal cube that can be used for these games with magnetic X's and O's Another version of these games could be played without writing on your cube by allocating one, two or three colors to each person and starting from a randomized cube, try to play any of the above games with each turn being taken by twisting a side and using the players set of colors as his symbol. - Bruce ^_  Date: 21 MAR 1981 1454-EST From: LSH at MIT-MC (Lars S. Hornfeldt) To: CUBE-LOVERS at MIT-MC, RP at MIT-MC What are the best CUBE-times nowadays? A young guy Kimmo Eriksson in Stockholm yesterday solved 10 (ten) cubes in a series, with an AVERAGE of 52 seconds, and with individual times varying between 47 sec and 61 sec. -lsh  Date: 22 March 1981 0829-EST (Sunday) From: Dan Hoey at CMU-10A To: Cube-Lovers at mit-mc Subject: No short relations and a new local maximum Message-Id: <22Mar81 082919 DH51@CMU-10A> Well, the gigabyte (well, 300Mb) came in, and brute force is having its day. I have a little program that generates all positions accessible from a given position in a given number of quarter-twists. With the increased storage available here, I was able to run it to five quarter-twists. The first important fact to emerge is that there are exactly 105046 different positions at a distance of at most 5 qtw from START. This has two consequences to the argument given in my message on the Supergroup, part 2 (9 January 1981 0629-EST). Note that the results here pertain to the usual group of the cube, rather than the Supergroup, since the program does not keep track of face-center orientations. The first consequence is that there are exactly 93840 positions exactly 5 qtw from START. The message cited above proved the inequality P[5] <= 93840; this is now known to be an equality. The second consequence is that there are no relations (sequences that lead back to START) of length 10, with the exception of those that follow from the relations FFFF = FBF'B' = I (and their M-conjugates). This is because relations of length 10 would reduce P[5], which is not the case. There are, however, relations of length 12; the only known ones are FR'F'R UF'U'F RU'R'U [given in Singmaster] and its M-conjugates. These results can be extended to the Supergroup, by noting that the set of observed positions places a lower bound on the number of Supergroup positions at a distance of 5 qtw, while the upper bound given in the cited message relies on the relations FFFF = FBF'B' = I, which are relations in the Supergroup. A particular result which may be of greater interest to readers of this list concerns the relation between symmetry and local maxima. In our message on the subject (14 December 1980 1916-EST) Jim Saxe and I mentioned that the six-spot pattern is not a local maximum, as verified by computer. [The same program was used, but only four-qtw searches were needed.] With five-qtw searches, it became possible to check another conjecture, using an approach that Jim suggested. The four-spot pattern U U U U U U U U U R R R B B B L L L F F F R L R B F B L R L F B F R R R B B B L L L F F F D D D D D D D D D is solvable in twelve qtw, either by (FFBB)(UD')(LLRR)(UD') or by its inverse, (DU')(LLRR)(DU')(FFBB). It is immediate that a twelve qtw path from this pattern to START can begin with a twist of any face in either direction. The program was used to verify that there are no ten qtw paths. (It generated the set of positions attainable at most five qtw from START and the set of positions obtainable from the four-spot in at most five qtw, and verified that the intersection of the two sets is empty.) Thus the four-spot is exactly twelve qtw from START and all its neighbors are exactly eleven qtw from START, proving that the four-spot is a local maximum. (Worried that there might be an eleven qtw solution to the four-spot? Send me a note.) This is the first example of a local maximum which cannot be shown to be a local maximum on the basis of its symmetry. To be more precise, let us define a "Q-symmetric" position to be a position whose symmetry group is Q-transitive. This extends the terminology developed in "Symmetry and Local Maxima". In that message, we showed that all Q-symmetric positions, except the identity, are local maxima. Until now, these were the only local maxima known. The four-spot, however, is not Q-symmetric; the position obtained by twisting the U or D face of the four-spot is not M-conjugate to the position obtained by twisting any of the other faces. This lays to rest the old speculation that one might find all local maxima, and thereby bound the maximum distance from START, by examining Q-symmetric positions.  Date: 28 MAR 1981 1259-EST From: DCP at MIT-MC (David C. Plummer) Subject: New toy (long message, but read it anyway!!) To: CUBE-LOVERS at MIT-MC Tanya Sienko is visiting me, and she says that the cube is the craze of Japan. She also presented me with a new toy, given to her by some Japanese. (I don't know if is in this counrty -- yet.) The thing is shaped like a barrel mounted on a supporting structure. The barrel can move one UNIT up or down in the structure. Around the circumference of the barrel there are five equally distributed columns. Two of the columns have four rows, and three of them have five. The ones with five have a plunger on the associated part of both the top and bottom (or left and right) parts of the supporting structure. Two plungers are next to each other, and the third is opposite their midpoint. There are 23 balls in the device: four each of green, yellow, blue, red, orange (one for each column) and three black balls. (in a minute you will see where these black balls go). The barrel is divided into four parts. The left- and right-most parts are fixed with respect to the supporting structure. Each has three cavities either to hold a ball or one of the plungers. The barrel moves, so either the left has balls in the cavity and the right has the plungers, or vice versa. The middle two sections of the barrel have two cavities in each row, and these rotate around the circumference, taking balls with them. I have been trying to say left and right, because I think the corect way to thing of this devices is as follows: Hold it horizontally, with the barrel centered in the supporting structure. This means that each plunger is half way into its cavity. A MOVE consists of moving the barrel one half unit right or left, then moving one of the rotating middle sections forward or backward one unit, and then returning the barrel to center position. This creates four generators: move barrel [left,right], then move middle section-[left,right] forward (or backward, which is the inverse). Visually: | | | | A A A A A B B B B B \ \ / / A A A A B B B B / / \ \ A A A A A B B B B B \ \ / / A A A A B B B B / / \ \ A A A A A B B B B B | | | | | | | | C C C C C D D D D D \ \ / / C C C C D D D D / / \ \ C C C C C D D D D D \ \ / / C C C C D D D D / / \ \ C C C C C D D D D D | | | | Where A is move barrel left , move left section B is move barrel right, move left section C left , right D right, right The top and bottom of these drawings are connected, cavities (filled with the balls) move along the lines. All balls move in the same direction the same number of units (i.e., the middle sections are rigid). I hope this is a good enough description, if not send me mail and I will send an addendum. The object, so I hear, is to get each column (row in these pictures) a single color, and if there are five slots (of which there are three), the fifth has a black ball in it, when the barrel is pushed all the way to one side, the plungers take up three of the outside-barrel-sections, and the black balls take up the opposite three. from a symmetric point of view, I think it would be more general to SOLVE it so that the black ball is in the middle of the five balls (this may not be solvable though).. If we ignore the obvoius left-right symmetry of the above pictures, the first assumption of the combinatorics of this beast is simply P(23;4,4,4,4,4,3)=numbers of ways to permute 4 balls of each of 5 colors and 3 balls of another color= 23! ------------------- = 541111756185000 = 541 trillion 4! 4! 4! 4! 4! 3! Until I have played with it for a while, I can't even guess on how many orbits there are. Perhaps only one -- I don't know. Super-groups come in a few classes: (1) Each non-black ball gets a second label (1-4) giving size 23!/3! = 4.3*10^21 (2) Each black gets a second label (1-3) giving size 23!/(4!)^5 = 3.25*10^15 (3) (1) and (2), all balls distinct giving size 23! = 25.8*10^21 If anybody sees one in this county, please let me know. Tanya believes they are only in Japan at the moment. She has donated the one I have seen to me/SIPB, so people at MIT and area are free to come to 39-200. PLEASE BE CAREFUL with it. It is plastic and it looks breakable -- especially the outer part of the supporting structure looks like it dould break. I think a better construction would be to have them be plates which are attached to the axis with screws. This might lead to a temptation to disassemble, which may be epsilon below breakage.  Date: 28 March 1981 16:13-EST From: Carl W. Hoffman Subject: Also from Japan ... To: CUBE-LOVERS at MIT-MC Cubes of different sizes and colors. There is one 3 centimeters on an edge sitting in the SIPB office.  Date: 31 Mar 1981 2133-PST From: Gary R. Martins Subject: B E W A R E !! To: cube-lovers at MIT-MC cc: gary at RAND-AI Bought a new cube today. One of Ideal's "Rubik's Cube"s. Same price as first cube, bought about a month ago. Packaging looks same. Ditto cube, except that the center-white face has "Rubik's Cube (tm)" printed on it in various fonts. Also, closer inspection of the package shows that a stick-on stripe acknowledges manufacture in Hong Kong. The cube itself is INFERIOR in various ways. I'd recommend you not buy them, unless the vendor will offer you a refund. The worst and most obvious feature of this cube is that is seems to have NO lubricant in it. The faces seem more vulnerable to fingernail damage etc. and the colors and materials seem shoddier. The cube has a flimsy feel to it, and seems poorly finished in general. Anybody else notice this, or have I just caught a lemon ? Gary -------  Date: 1 APR 1981 0104-PST From: MAXION at PARC-MAXC Subject: Re: B E W A R E !! To: gary at RAND-AI, cube-lovers at MIT-MC cc: MAXION In response to the message sent 31 Mar 1981 2133-PST from gary@RAND-AI I had the exact same experience. The first one was wonderful; the second (just as you described) was awful. Same packaging, same story, same observations as yours. Roy -------  ZEMON@MIT-AI 04/01/81 07:45:10 Re: B E W A R E !! To: CUBE-LOVERS at MIT-AI I have one of the offending cubes -- yes, it \is/ falling apart. The colored faces have developed crinkles, holes and some are even peeling off. This is after only 4 weeks of use. Taking the cube apart and sprinkling its insides liberally with baby powder will effectively lubricate it (although it will smell for a while) and make it essentially noiseless, I have heard. -Landon-  Date: 1 Apr 1981 1459-PST From: Gary R. Martins Subject: Cube Lube To: cube-lovers at MIT-MC cc: gary at RAND-AI Is there a consensus on the best lube for one's cube ? White lithium grease, silicon grease, silicon spray, and baby powder have all been mentioned. Anybody know what's really 'best' ? Gary -------  Date: 2 Apr 1981 0132-CST From: Clive Dawson Subject: Re: Cube Lube To: gary at RAND-AI, cube-lovers at MIT-MC In-Reply-To: Your message of 1-Apr-81 2203-CST I suspect that "best" in this case is probably a matter of personal opinion...Also note that a lot depends on trimming, filing, sanding, etc. Besides the lubes mentioned by Gary, I can also recommend dry graphite powder (I used "Mr. Zip Extra Fine Graphite") which gave me very good results on my cube. Then I finally got a chance to examine Kim Korner's cube (Korner@UTEXAS) and must admit his is much much better. He used silicon gel, of the sort used to lubricate "o" rings in Scuba equipment. See his message to Cube-lovers of 6-Feb-81 for more information. About the only shortcoming I noticed was a very slight "slimy" feeling to the cube which I'm sure will wear off with time... By the way, on the subject of the declining quality of Ideal Toy's version of the cube-- I too was surprised when I examined one which was bought last month by a friend of mine. The first thing I noticed was that the some of the interior faces of each cubie were missing. My first reaction was that they'd found a way to skimp on plastic; then I thought that maybe it was a way to cut down on internal friction. Judging from some of the other recent reports, it sounds like my first hunch was correct. Another annoying characteristic was the shoddy work in attaching the colored faces. Most were not only crooked, but also liberally sprinkled with air bubbles throughout. Happy cubing, Clive -------  Date: 2 Apr 1981 1723-PST From: Hopper at OFFICE Subject: Re: B E W A R E !! To: gary at RAND-AI, cube-lovers at MIT-MC cc: hopper at OFFICE I've bought cubes recently as follows: 1-FEB (approx) , Ideal's with "Rubik's CUBE tm" on the center white cubie. Pakaged in cardboard and cellophane. Hollow edge and corner cubies. Squeekie, but fine after lube with graphite. No problems with facies. Good size tolerances--very smooth operation after lube. 20-FEB (approx) , bought 3 that looked identical to the pevious one, except they were packaged in cylindrical plastic packages. Two of the three turned out the same as the one purchased 1-FEB, except that size tolerences were very poor and operation was very rough, even after lubrication. No problems with the faces. The third cube was packaged the same and has the same "Rubik's CUBE tm" on the center white cubie, but is quite different. Although the cubies are hollow, it is heavier then the others. The plastic isn't so squeeky and seems more like earlier cubes from last year. The edge cubies have casting ridges visible through the middle of the faces like the early, early cube I got before last summer. The shoulders on the edge cubies were flat so the action was very rough. The corner cubies were loose. Filing down the inside surfaces of the edge cubies cured the loose corners, and rounding their shoulders made the action quite acceptable. LASTLY, ONE (just ONE!) of the red facies is inferior, darker in color, and crinkling! 20-MARCH (approx) , bought a cube with no acknowledged manufacturer, with cylindrical plastic pakage very similar to Ideal's, labeled "Made in Taiwan". Cubies are hollow very much like the third one from the 20-FEB batch, but the corner cubies have covers glued in the openings of the inside surfaces. The quality of the facies seems good, but it may be too soon to tell. The orange facies are not brilliant like Ideal's--more of a peach color. The size tolerances seem quite good compared to Ideal's recent cubes. Biggest drawback (and possible overriding factor) is the plastic seems much softer than Ideals's and lubrication (at least with graphite--I haven't tried other recommended lubes such as silica gel) doesn't seem to make it any easier to turn. It remains quite stiff. Also, some of the centers were screwed in much to tightly. I'd be interested to hear any other experiences with recently-bought cubes. I'm curious about their availability in the Bay area and elsewhere. --Dave-- -------  Date: 2 Apr 1981 2325-PST From: Gary R. Martins Subject: New Yorker To: cube-lovers at MIT-MC cc: gary at RAND-AI Current issue of 'New Yorker' magazine has some cubic discussion in the opening 'Talk of the Town' section. Also mentions Marvin Minsky! P. 29, March 30, 1981 issue. Gary -------  Date: 3 April 1981 0500-est From: Allan C. Wechsler Subject: Magic barrel. To: CUBE-LOVERS at AI Haal yawm! I am a barrel-solver this day! ---Wechsler  Date: 3 Apr 1981 0750-PST Sender: OLE at DARCOM-KA Subject: Cube lube (yet again) From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-lovers at MIT-MC Message-ID: <[DARCOM-KA] 3-Apr-81 07:50:47.OLE> The following is a collection of thoughs and experiences on cube lubrication. It only applies to the Ideal cubes which are the only ones I have played with, but should have some applicability to other brands. Getting a smooth turning cube seems to me to be a combination of the right lubrication with the spring/screw tension. If the screws are too lose, the cube will turn easily, but frequently jam since the lose cubies tend to get in each other's way. On the other hand a very high tension without any lubrication would mean a very stiff and fast wearing cube. My solution is simply to use candle wax. I have taken my cube apart and rubbed each cubie with a standard candle. (My friend from the Chemical Eng. Dept. says parafine wax would be even better. This is what we used to rub on our skis back in Norway before all the fancy ski-waxes became available. I don't know how easy it is to get these days.) I also "fine-tune" the cube by adjusting the screws on each face. A couple of strips of double-sided tape stops the caps from acci- dentally falling out during use. You may need to take the cube apart a couple of times after the initial lubrication to allow superfluous wax to fall out. I also found that turning a newly waxed cube under a hot tap seems to make the wax settle nicely. This lubrication has the advantage of not (seemingly) coming out on your hands or otherwise disappear,- one treatment will last you very long indeed. The only slight problem is that the cube needs some "warming up" when it has been left idle for some time especially in cold places. (Ref. my earlier message) But a couple of minutes of random twisting produces a smooth and silent cube. Good luck OLE  Date: 4 Apr 1981 1727-EST From: JURGEN at MIT-DMS (Jonathan David Callas) To: Cube-lovers at MIT-MC Subject: Cube preferences Message-id: <[MIT-DMS].192744> I have two cubes, a C. Americanus ("Rubik's Cube") which I bought last summer, And a C. Albus that I got from Logical Games in Haymarket Va. The white cube came lubricated with something resembling musician's cork grease, and has not needed to be lubricated. The Rubik's cube has never been lubricated either, but hasn't seemed to need it. Iprefer the white cube to the black one for some nebulous reason. It is not nearly as smooth-turning as the black one, but in a perverse way, I like that. It seems to be better built, but I can't substantiate that with facts, that's just gut-feeling. I *DO* like the fact that they are uncommon, and now that people don't go "Ooh, what's *THAT*" when they see the cube, and now people do get amazed at the sight of the white-faced cube. Now that it seems that Ideal is going for the bucks (a friend of mine has also gotten one of the cheap cubes, but I thought it was my imagination. I guess now there's real reasons for getting the white cubes. --Happy cubing, --Jurgen  Date: 6 APR 1981 1501-EST From: DCP at MIT-MC (David C. Plummer) Subject: Japan frob revisted (180+ lines) To: CUBE-LOVERS at MIT-MC This is a long overdue re-explanation of the Japanese frob. Hoey and Saxe at CMU gave several comments and suggestions. PART I -- Try again =================== Take a hollow cylinder (like a doubly unlidded coffe can), cut it open and unravel it. We now have something like xxxxxxxxx | | b | where the cut was made along the x o t and top and bottom are where the t o lids used to be t p o | m | | | xxxxxxxxx The supporting structure corresponds roughly to the lids of the can. LL " ' "" ' " RR LLLLLL" B ' B "" B ' B " B RRRRRR LL " ' "" ' " RR LL " ' "" ' " RR LL " ' "" ' " RR LL " B ' B "" B ' B " RR LL " ' "" ' " RR LL " ' "" ' " RR LL " ' "" ' " RR LLLLLL" B ' B "" B ' B " B RRRRRR LL " ' "" ' " RR LL " ' "" ' " RR LL " ' "" ' " RR LL " B ' B "" B ' B " RR LL " ' "" ' " RR LL " ' "" ' " RR LL " ' "" ' " RR LLLLLL" B ' B "" B ' B " B RRRRRR LL " ' "" ' " RR LL " ' "" ' " RR (In the three dimensional case, the top and bottom of this picture are connected together.) L is the left part of the supporting structure, and R is the right. They are firmly connected to each other, and are therefore fixed in space with respect to each other. They are really circular, but this is a view of the outside. B are the balls. The balls can move left or right by being PLUNGED. The only allowed plunge in the above diagram is to move the supporting structure to the left (or equivalently the barrel to the right). With respect to the barrel, only the balls in the first, third and fifth rows (refered to as columns in previous message) are affected. The diagram would now look like LLL B B B B B RRR L B B B B R LLL B B B B B RRR L B B B B R LLL B B B B B RRR Balls move vertically by by moving either of the two " ' " sections vertically, and the balls within that section stay fixed in space with respect to each other and the section, but not fixed with respect to the other balls (within the context of one turn) or the supporting structure. Thus the moves are: PLUNGE RIGHT or LEFT, whichever is appropriate (or move barrel LEFT or RIGHT) and MOVE LEFT or RIGHT section UP or DOWN My suggestion was that in the between move state, the barrel was centered in the plungers, so the PLUNGE move is HALF-PLUNGE LEFT or RIGHT (both of which are apporpriate), then do the vertical move, then UN-HALF-PLUNGE. PART II -- Hoey's comments to my original message ================================================= [Hoey 28 March 1981 1500-EST] Is the move you designate by | | A A A A A \ \ A A A A / / A A A A A \ \ A A A A / / A A A A A | | really the permutation that takes | | | | 1 2 3 4 5 6 7 3 4 5 \ \ \ \ 6 7 8 9 10 11 8 9 / / / / 10 11 12 13 14 to 15 16 12 13 14 \ \ \ \ 15 16 17 18 19 20 17 18 / / / / 19 20 21 22 23 1 2 21 22 23 | | | | ? Do you mean to imply that moves of the form | | | | X X X X X X X X X X \ \ / / X X X X X X X X \ \ \ \ X X X X X or X X X X X / / / / X X X X X X X X / / \ \ X X X X X X X X X X | | | | (whatever they mean) are prohibited (as primitives, at least) by the construction of the barrel? [Both answers are YES] PART III -- Comments later that night ===================================== [In response to the updated description Hoey 20 March 1981 1836-EST] First, it should be made clear that in (either) plunged position, the two " ' " sections rotate freely; i. e. it is not necessary to plunge in between. For instance, one could solve by counting plunges, but not rotations. Jim suggested that it might be "neater theoretically", but I think it smells of the half-twist metric. Second, the inclusion of permutation diagrams will make the puzzle clear to anyone who doesn't understand the mechanics. Something like I gave in the last message, but with all permutations given, the note that "\|/" are only comments, and the description of the goal: move Black to 5,14,23, and make the sets 1-4, 6-9, 10-13, 15-18, and 19-22 each a solid color. I ran this through the Furst/Hopcroft/Luks algorithm, and found that in the Supergroup (all balls distinct) you get the alternating group on 23 balls: all even permutations. Thus if any two balls are indistinguishable, you can get all configurations. Saxe remarks that there is only fourfold symmetry: Reflection left-to-right and up-to-down. Their composition is in fact achievable: turn the whole puzzle upside down, while continuing to face the front of it. Strangely enough, this is an ODD permutation: it takes you to the other orbit! [Hoey 28 March 1981 2143-EST Subject: Simpler and harder toy] Try taping the center two rings together. Thus A is always performed with C, and B with D. The same set of permutations is achievable! [I assume the proof is an enumeration of states by the above algorithm.] PART IV -- Developments by Alan Bawden (ALAN@MC), Allan Wechsler (ACW@AI) and myself. ================================================================ Alan Bawden sat down patiently one night (Tuesday March 31 1981, I think) and discovered the necessary TOOL (or concept) (singular !!) that is needed to solve the toy. I will not give a spoiler here. Getting most of it is rather easy. The last few balls take a little extra work. Alan told me the concept, and the next day I successully solved it. Alan solved it later that day, and soon Allan Wechsler solved it a few days later (signified by his yelp to this mailing list). The three of us solve it slightly differently (s)o like the cube, there are personal sovling styles). We now solve it reliably, including the last few balls. Happy what-ever-ing...  Date: 8 Apr 1981 16:07 EST From: Marshall.WBST at PARC-MAXC Subject: Please add me to the distribution list To: Cube-Lovers at MIT-MC cc: Marshall.WBST Please add my name to the rubik's cube distribution list. I have a copy of Kertesz' solution but am interested in better solutions and/or insights into the underlying group. Thank you --Sidney (Marshall.WBST at PARC-MAXC)  Date: 18 April 1981 08:52-EST From: Lars S. Hornfeldt To: CUBE-LOVERS at MIT-MC, gary at RAND-AI Kimmo Eriksson is 14 years old (a good age for cubism), and in his series of 10 consecutive cubes, the average time was 52 sec, and average number of moves was 95, varying between 70 to 120 (half-turns and slices counted as one move). He uses 5 macros with uncountable longer variants. The longest of the macros are 11 moves. -lsh  Date: 20 Apr 1981 0906-PST From: Isaacs at SRI-KL Subject: (Response to message) To: LSH at MIT-MC cc: ISAACS, cube-lovers at MIT-MC Who and where is Kimmo Eriksson? Where was this timing done? What are his macros? In what order does he solve it? The same way each time? etc. ---Stan Isaacs -------  CMB@MIT-ML 04/23/81 13:04:49 To: cube-lovers at MIT-MC From the Boston Globe: Abbie Hoffman, the former Yippie leader who managed to escape the toils of the law for seven years by living incognito in upstate New York, has finally gone to prison, but not for having been a fugitive. Hoffman surrendered in New York yesterday to begin a three-year prison term for selling cocaine and jumping bail. In the photo, he is shown being frisked. In his left hand he holds a magic cube puzzle, which he said he will solve in prison. In his right hand he holds a copy of the book, "Fire in the Minds of Men," that had a bookmark looking suspiciously like a hacksaw blade. This was whisked away from him. Hoffman denied the props, including the hacksaw, were a publicity stunt.  Date: 23 Apr 1981 1105-PST From: Gary R. Martins To: CMB at MIT-ML cc: cube-lovers at MIT-MC, gary at RAND-AI In-Reply-To: Your message of 23-Apr-81 1304-PST He should have offered those narcs a snort of vodka ! G -------  Date: Sunday, 26 April 1981 10:54-EDT From: Pat O'Donnell To: Cube-Lovers at MC cc: PAO at MIT-EECS May issue of Reader's Digest has a (very) short article on the cube. It includes a claim for a French fellow solving the cube in an average of 32 seconds. The article contains almost no technical information--mostly historical.  Date: 27 Apr 1981 0923-PDT From: Isaacs at SRI-KL Subject: cubes, barrels, and stuff To: cube-lovers at MIT-AI I was just at the fourth international puzzle party in L.A. and saw several offshoots of the cube. The barrel, previously mentioned in this digest, is the best. It is called "The Ten Billion Puzzle" (I think). (Note to people in the Palo Alto area - come to the Rubiks Cube Club And Other Puzzles at Stanford on Thursday night if you want to see a couple.) Also there were two small cubes, about 2/3 size, one from Japan, and the other from (I think) Taiwan. There was a 2x2 version (about half the size of Rubiks), with things like hearts, stars, etc on it. There was also a Rubik type, but with figures instead of colors. The Missing Link is now out from Ideal, and should be easily findable (as of this week). But, though they treat it as a follow-up on the cube, it is MUCH simpler, and closer in principle to a sliding block puzzle. Nice, but simple. There were also several other types of cylinders, but mostly related to the Missing Link, or to Instant Insanity type, rather than cube type. By the way, Jerry Slocum, puzzle collector extraordinaire and the puzzle party host, thinks the magic cube will have a real impact on society - that it will lead to a resurgance of interest in puzzles in general, and in thinking-type games. Let us hope he is right. (Send a puzzle to your congressman - make him think!) --- Stan Isaacs -------  Date: 27 April 1981 12:15 cdt From: VaughanW.REFLECS at HI-Multics Subject: Re: cubes, barrels, and stuff To: Isaacs at SRI-KL cc: cube-lovers at MIT-AI In-Reply-To: Msg of 04/27/81 11:23 from Isaacs *nothing* could make my congressman think!  Date: 27 Apr 1981 1322-EDT From: IC.RAG at MIT-EECS Subject: Re: Re: cubes, barrels, and stuff To: VaughanW.REFLECS at HI-MULTICS, Isaacs at SRI-KL cc: cube-lovers at MIT-AI In-Reply-To: Your message of 27-Apr-81 1315-EDT *NOTHING* could make me think of my congressman! -------  Date: 27 Apr 1981 1139-PDT From: Gary R. Martins Subject: Re: Re: cubes, barrels, and stuff To: VaughanW.REFLECS at HI-MULTICS cc: Isaacs at SRI-KL, cube-lovers at MIT-AI, gary at RAND-AI In-Reply-To: Your message of 27-Apr-81 1215-PDT Try *M* *O* *N* *E* *Y* !! Worked wonders for the FBI ! G -------  Date: 28 Apr 1981 0233-PDT From: Peter D. Henry Subject: mailing list add request To: cube-lovers at MIT-MC please add me to the mailing list... thanks Peter D. Henry PDH@sail  Date: 28 April 1981 08:34-EST From: David C. Plummer Subject: mailing list add request To: PDH at SU-AI cc: CUBE-LOVERS at MIT-MC Done.  Date: 29 April 1981 1334-EDT (Wednesday) From: Guy.Steele at CMU-10A To: cube-lovers at MIT-MC Subject: New member for mailing list Message-Id: <29Apr81 133430 GS70@CMU-10A> Please add Paul.Haley @ CMUA to the cube-lovers mailing list?  Date: 6 May 1981 2030-EDT From: ROBG at MIT-DMS (Rob F. Griffiths) To: cube-lovers at MIT-MC Message-id: <[MIT-DMS].196841> While visiting the local gaming shop today, I heard a rumour about a pending lawsuit between (I think) The Original maker of Rubik's cube and Ideal.. Anyone know anything about this? -Rob.  Date: 8 May 1981 1036-PDT From: Isaacs at SRI-KL Subject: Non-twisting corner moves To: cube-lovers at MIT-MC Does anyone know good move sequences for exchanging a pair or corner cubies on a face without twisting? (Of course, a pair of edges will have to exchange also.) Or of cycling 3 corners without twisting? I'm looking for the "shortest" sequence, and the "easiest to remember" sequence. Most of the moves I've seen are long and complicated. --- Stan Isaacs -------  Date: 8 May 1981 14:30-EDT From: David C. Plummer Subject: Non-twisting corner moves To: Isaacs at SRI-KL cc: CUBE-LOVERS at MIT-MC How about L' [(R' DD R) U (R' DD R) U'] L for moving the top three corners around perserving the top color. Or, (R' D' R) U' (R' D R) U For moving three front pieces around, preserving a couple colors. 12 and 8 moves is pretty short...  Date: 9 May 1981 08:47-EDT From: Lars S. Hornfeldt Sender: LSH0 at MIT-MC To: CUBE-LOVERS at MIT-MC, isaacs at SRI-KL Kimmo Eriksson is a 14 year old computer fan who lives in Stockholm. As mentioned, he solved a series of 10 cubes in 47-61 s, average 52, using 70-120 moves, average 95 (half-turns and slices counted as one). * He always starts with the WHOLE yellow layer (regardless of ini- tial state, probably because the regularity allows faster reflexes) * Then the middle layer (betw. yellow and white) * Then all top-corners into place, then into correct orientation. * Finally turn and move the top-edges (requires 0-3 macro-moves). He keeps strictly to this scheme, but uses a large set of macros, that are different longer varities of the following basic five: For middle: RUR'U'F'U'F Move corner: RU'L'UR'U'LU Turn corner: RUR'URU2R'U2 Move edge: MU2M'UMU2M'UMU2M' (M moves the Mid-line of the Bottom Move and turn edge:MUM'U2MUM' up Front, ie = LR' ) For timing, he starts a stopwatch, grabs the cube, solves it - while watching (easy) the watch during the last macro in order to read off the time exactly as the last macro is completed. After re-mixing the cube, the procedure is repeated (10 times). -lsh  Date: 10 May 1981 1258-EDT From: Jerry Agin Subject: Counting moves To: Cube-Lovers at MIT-MC Frequently when I play with the cube, I try to solve it in as few moves as possible. I find this to be more intellectually challenging than going for speed. Does any one else do this? I'd be interested in comparing notes. Presently it takes me between 70 and 100 quarter- twists, provided I don't make gross errors. (My guess is that if I were counting slices and half-twists as one, the number would be between 50 and 70.) -------  Date: 15 May 1981 0456-PDT Sender: OLE at DARCOM-KA Subject: New pseudo-cube From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-lovers at MIT-MC Cc: Oyvind Message-ID: <[DARCOM-KA]15-May-81 04:56:31.OLE> Pardon me if this object has been described before, but I don't remember seeing it. While browsing for cubes in a local store here in Newcastle the other day, I came accross a new "cube" which I shall try to describe and I invite you all to think of a good name for it. First of all let me point out that this new toy is really just a Rubik's Cube with some modification with respect to coloring and construction. Imagine taking your normal cube and making 4 vertical slices along the corner diagonals. Your top and bottom faces would now look like: --- / \ I I \ / --- (sorry about poor ratio, but I hope you get the idea) Now recolor the new faces and voila, your new toy is complete. The construction has the following consequences: 1. The object is no longer symmetrical, U and D faces are different from L R F and B. 2. The "corners" have only got TWO colors, but act as corners of the Rubik's Cube, the mechanics is identical. 3. Four new "edges" which I will call wedges have appeared in the middle layer. These have only ONE color, but as you will discover when using your edge moves: the orient- ation matters. Edges and wedges may be interchanged. I will now describe the coloring of my particular cube, note that there are 10 different colors. The U face is blue and the D face is white. Then starting at the 6'o clock edge column (i.e 1/3 of the F face) we have: GOLD(e),ORANGE(w), RED(e),PURPLE(w),YELLOW(e),PINK(w),GREEN(e),LIGHT BLUE(w). (Where e=edge and w=wedge colums respectively). I chose this particular orientation because it makes red=left and green=right which is nice. Note however that this "cube" may be reassembled in various legal patterns since the edge column/wedge column neighbouring properties are not forced. This further complicates solving since there in no way of knowing which sequence the "corners" and edges go in layer1 unless you have a map. Once this is known, solving is straight forward, but as said the wedges will confuse you. Qestion: Is there some way of deter- mining the parity etc, such that the object may be solved without a map of layer one? I invite comments from Jim and Dan. If it is the case that these pseudo-cubes (how about Rubik's Drum) are not available in the US, I can send one or two samples. The drums are made in Taiwan and are not as well finished or as smooth turning as Ideals cubes. Random twisting produces very strange shapes and the Cruxi Plummeri et Cristmani are simply out of this world. OK, thats it. Hope this made sense, but this thing is more difficult to describe that its predecessor,- so forgive me if I haven't succeeded. Cheers OLE  Date: 15 May 1981 19:04 edt From: Greenberg.Symbolics at MIT-Multics Subject: Re: New pseudo-cube To: Ole at DARCOM-KA (Ole J. Jacobsen) cc: Cube-lovers at MIT-MC, Oyvind at DARCOM-KA In-Reply-To: Message of 15 May 1981 07:56 edt from Ole J. Jacobsen I have seen this thing under the name "space shuttle" - in a cylinder like rubiks cubes used to come in. One way to determine the color layou is to make an assumption (as I did) and proceed to solve-- if you lose, you get the "impossible" single-pair-swap. Then swap two edges of your assumption and try again. Incidentally, the plural of crux is cruces, not cruxi.  Date: 18 May 1981 1046-EDT (Monday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Drum info Message-Id: <18May81 104634 DH51@CMU-10A> Most of this note is pretty straightforward application of the known cube properties, but if you want to know about the drum.... The drum shows everything you see on a regular cube except the orientation of the four truncated edges, or wedges. Because the (invisible) edge parity is preserved, each visible position of the drum corresponds to 2^4/2 cube positions. Thus there are 5.406x10^18 drum positions. To count the number of solutions, note that as in the normal cube, the face centers force each edge to its home position and orientation. In addition, each corner has a facelet that says whether it is top or bottom and fixes the corner's orientation. This means that solved positions are obtained from each other by permuting top corners, bottom corners, and wedges. But the three cubies on a diagonal face must match, and so the three permutations are the same. Only even permutations are achievable in this way (since the cube of an odd permutation is odd) and there are 4!/2=12 of these. One easy process that goes from one solved position to another is FF RR FF BB RR BB. I asked Ole Jacobsen what he meant when he said of the wedges that "as you will discover when using your edge moves: the orientation matters." It turns out this is because he solves by layers: top-middle-bottom, and doesn't know which way to orient the edges in the middle so that the edges on the bottom will have the right parity. There are several ways out of that problem; one is to turn the drum sideways and solve left-middle-right. The problem of solving without knowing the order of the wedges is trickier. Solving sideways is one method: do the left side any way; on the right side there two possibilities, one of which will work. (This is Bernie Greenberg's suggestion, modified so you don't need to memorize the whole map.) One interesting thing to do with a drum is to turn it into baseball. Using colored tape and disassembly, change the colors and positions so that the wedges appear in the UF, DF, BL, and BR positions when the colors match. On a baseball, there are only two solved positions.  Date: 18 May 1981 17:59-EDT From: Richard Pavelle Subject: the magic barrel := scraping the bottom thereof To: CUBE-LOVERS at MIT-MC Now from Ideal comes "MAGIC PUZZLE" ta ta..ta ta... It is a new barrel puz (not worth the extra typing) and sells for about $5.00 in Boston. It is not deserving of a discussion but so you may recognize and avoid it let me say the following: Barrel shaped, it has 6 slots perpendicular to 3 frames which rotate about the axis. Five slots contain 3 colored panels each and the 6th has only two. The puz comes solved with all 6 slots the same color. One then tries to disarrange it (more difficult than actually solving it). The greatest challenge might be to run over it with a car and humpty dumpty it.  Date: 10 Jun 1981 2327-PDT From: Alan R. Katz Subject: edge flip anyone... To: cube-lovers at MIT-MC cc: katz at USC-ISIF Does anyone know of a way to flip two edges without changing anything else, ie interchange the FR and FB edges, keeping their same orientation. If not, how about one that will interchange FR and FB edges without disturbing the top layer, but can mess up the bottom?? This may have been given a while back, but I couldn't find it looking through the old CUBE-LOVERS mail. Also, GAMES magazine says they have 100 of the old cubes left for $10.95 plus $2.50 for postage and handling. These are the Ideal ones WITHOUT the IDEAL stamped on the white face. Order from: Games Shop 515 Madison Ave. New York, NY 10022 (first come, first serve) Alan -------  Date: 11 June 1981 0323-EDT (Thursday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Re: edge "flip" anyone... Message-Id: <11Jun81 032344 DH51@CMU-10A> Does anyone know of a way to flip two edges without changing anything else, ie interchange the FR and FB edges, keeping their same orientation. The term "flip" generally means "change orientation" (of an edge). I take it you mean "exchange" (or, acceptably, "interchange"). This is provably not possible, because an exchange is an odd permutation. I will send you the two messages which deal with what can be done with a cube. (Anyone else who wants a copy, request of Hoey@CMU-10A rather than the list.) If you don't know what an even (or odd) permutation is, or how it acts, any elementary algebra text should help. If you try and fail, drop me a line and I'll try to help. If not, how about one that will interchange FR and FB edges without disturbing the top layer, but can mess up the bottom?? There is no FB edge. I'd guess you mean UR and UB, for which RU'R' UUFFBB DDFFBB RUR' suffices.  Date: 12 Jun 1981 1022-PDT From: ISAACS at SRI-KL Subject: Re: Re: edge "flip" anyone... To: Dan Hoey at CMU-10A cc: cube-lovers at MIT-MC In-Reply-To: Your message of 11-Jun-81 0023-PDT It is simpler in actually manipulating the cube to avoid moves that use the back; in particular, the FFBB slice move is a difficult one to actually do. So I would recommend BUB' and BU'B' at the ends of a RRLL slice. In addition, if you define X = RL' (or R'L) followed by an entire cube rotation (ie, move the slice and allow the orientation to change - turn the ham, not the bread ) then the sequence becomes: BUB' (UUXX)**2 BU'B'. ---- Stan -------  Date: 12 Jun 1981 1522-PDT From: Dolata at SUMEX-AIM Subject: Set of operators To: cube-lovers at MIT-MC I am new on cube-lovers mailing list, so please excuse this message if it is a repeat of one in the past... Does anyone know where I can find a good dictionary of operators? HAs one been compiled and is it available online? Thanks for your help Dan Dolata (dolata@sumex-aim) -------  Date: 14 Jun 1981 1404-PDT (Sunday) From: Mike at UCLA-SECURITY (Michael Urban) Subject: Howtodoit books To: cube-lovers at mit-ai There are presently at least TWO books on how to solve Rubik's Cube (TM). One is from Bantam Books, and the other is from a more obscure publisher. They run about $2 each. I haven't given them more than a cursory look as yet. The nonBantam one seems to use BEFUDLR notation, though. -------  Date: 15 Jun 1981 0954-PDT From: ISAACS at SRI-KL Subject: re. howtodoit books To: cube-lovers at MIT-AI I haven't seen the Bantom book, but I will keep an eye open. Sigmasters "Notes on Rubik's Magic Cube" is now out and available (at least in the Bay Area), published by Enslow Publishers (Bloy St and Ramsey Ave, Box 777, Hillside, N.J. 07205) , lists for $5.95. Seems to be EXACTLY the "Fifth Edition Preliminary Version", as in a photo-copy, although it may have some minor corrections and changes. There is also a solution booklet by J.G.Nourse, published by Storc, marketed by Paul N. Weinberg, Mountain View, Ca. And still another solution book privately printed by Kirsten Meiers. The solutions (happily) proliferate; each book has variations, most use non-BFUDLR notation - in many cases, just diagrams. Norse intends his solution to be "interruptable" - that is, if you drop the cube in the middle, or get a telephone call (or your computer beeps), you shouldn't have to start over. Meiers is meant to be easy to do. Singmasters ( I think), is meant to be easy to remember - mostly variations on a few basic algorithms. It would be nice to analyse all these solutions and arrange them for speed, ease of action, minimal twists, minimal time, better for right-handers etc.etc. --- Stan -------  Date: 18 Jun 1981 2101-PDT From: Alan R. Katz Subject: My previous message To: cube-lovers at MIT-AI cc: katz at USC-ISIF I didn't quite get what I wanted. I would like to interchange the RB and RD edges, the following does it: RD'R'D RD'D'R'D R D'D'R'D RD'R' This will mess up the bottom, but leaves the top and middle alone. I was wondering if there is a way to do it in less steps. Also, there is a mini-cube on a keychain out, its called the mini- Wonderful puzzler. The normal size Wonderful puzzler is a Rupic's cube of inferior quality; I don't know who makes them, but they are not from Ideal. Alan -------  DENG at MIT-AI (Dave English, University of Newcastle, UK)@MIT-AI (Sent by DENG@MIT-AI) 06/25/81 05:34:38 Re: Royal Wedding To: cube-lovers at MIT-MC CC: DENG at MIT-AI Someone here in Newcastle upon Tyne, England is selling cubes to celebrate the forthcoming marriage of His Royal Highness, Charles Mountbatten, Prince of Wales to Lady Diana Spencer. Said cubes have a portrait of Prince Charles on one face & Lady Di on the opposite. The four faces have identical Union Jacks. Consequently the orientations of only two centres are significant. Each cube comes complete with a leaflet describing a solution. It gives away nearly all my fastest spells Presumably the solution has been included in order to avoid spoiling THE DAY for the poor unfortunates who scramble the cubes - and therefor the portraits. The case is marked "Made in England", & so I claim this as the first observation of c. Britannicus. The material used is white plastic, but not the brittle stuff of c. Albinus. Movement is quite free, but not as swish as c. Americanus. Yours, Dave E. English.  Date: 4 July 1981 06:17-EDT From: Richard Pavelle Subject: 4 layered cube To: CUBE-LOVERS at MIT-MC I heard that a 4x4x4 is being marketed in Boston. The person who told me claimed to have seen and tried one but did not know who was selling them. Can anyone add anything to this?  Date: 5 July 1981 2217-edt From: Ronald B. Harvey Subject: re: 4 layered cube To: CUBE-LOVERS@MIT-MC If anybody does find out where these are being sold, please try to get an address for a distributor so that those of us not lucky enough to live in Boston can have a local store stock them. Thanks  Date: 12 July 1981 1343-EDT (Sunday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Happy birthday In-Reply-To: ALAN@MIT-MC's message of 12 Jul 80 13:42-EDT Message-Id: <12Jul81 134343 DH51@CMU-10A> \ /\ / / \ \ \I\ I/ ### ### ### ### ### --------------------------###------- | -- -- ### -- | ---------------------###------------- | | -- -- ### -- -- | | ------------------------------------- | | | -- -- -- -- | -- | | ------------------------------------- | -- | | | | | | | -- | | H a p p y B i r t h d a y | | | -- | | | | | | | -- | | | | | -- | | |-----------+-----------+-----------| | -- | | | C U B E | | | | -- | | | | | | | -- | | | | - | | -- | |L O V E R S| | -- | | |-----------+-----------+-----------| -- | | | | | -- | | O n e Y e a r | -- | | | | -- | 342 messages -- 73000 words | -------------------------------------  Date: 13 Jul 1981 11:33 PDT From: McKeeman at PARC-MAXC Subject: Missing Link To: Dan Hoey at CMU-10A cc: Cube-Lovers at MIT-MC, Ramshaw Have you seen the Missing Link? ["By the people who brought you Rubik's Cube"; oh well, if they say so] It is a cylindrical 15-puzzle, and about as hard to solve. Here are some notes on it. The actual puzzle is a square tower of height four. When solved, the puzzle has the appearance of linked chains. It has 15 square pieces leaving one hole through which you can peek to see the innards of the puzzle. The hole can move up or down by sliding the piece next to it down or up. The top and bottom slices rotate about the vertical axis which permits the hole to move to different columns. A rough idea of the pattern is: |~|n| can rotate edge-on view |n|X| |X|X| |u|u| can rotate I use two transformations, r and R to do things with it. If you look at an edge of the puzzle with the hole showing, there is a little cycle and a big cycle as noted below. Both leave the back unchanged. transformation from: to: name r |::|1| |::|2| |7|2| => |1|7| |6|3| |6|3| |5|4| |5|4| R |::|1| |1|2| |7|2| => |::|3| |6|3| |7|4| |5|4| |6|5| The move set is: U move the hole up D move the hole down T twist the top clockwise 90 degrees B twist the bottom clockwise 90 degrees for convenience, /T = TTT = T inverse, etc. r = D TU/T D /TUT RR = /TDT DD /BUB U TU/T DD BD/B U Actually, I can "feel" my way to a solution easier than I can figure my way to one. It feels very much like the 15 puzzle.  Date: 13 July 1981 20:36-EDT From: Alan Bawden Subject: Forwarded message To: CUBE-LOVERS at MIT-MC cc: LSR at MIT-XX Date: 13 Jul 1981 1441-EDT From: Larry Rosenstein Subject: Contest Thought you might be interested: This past weekend I heard a radio ad (WBZ-AM 1030) advertising a Rubik's Cube contest being held this weekend at Jordan Marsh in Burlington, Mass. The contest is being sponsored by Ideal Corp. and apparently will be world-wide; at least, the national winner is supposed to receive a trip to Monte Carlo for the world competition. At the time, I did not pay too much attention to the ad, so I am not positive about the details (specifically, the time and place). You might try calling Jordan Marsh in Burlington for more details. Larry -------  Date: 14 Jul 1981 1129-PDT From: ISAACS at SRI-KL Subject: HOWTODOIT books again To: cube-lovers at MIT-MC An update on How To Do It books on the Magic Cube: There now seem to be 3 "officially" published books on the Cube, besides an unknown number of privately printed books, pamphlets, and single-page solutions. The three are: 1) "Notes on Rubik's Magic Cube", by David Singmaster, Enslow Publishers, Bloy Street and Ramsey Avenue, Box 777, Hillside, New Jersey, 07205($5.95). This is, of course, THE book on the magic cube. It has the history, anecdotes, the math theory, and a solution. It is almost exactly the same as the fifth printing of his privately printed pamphlets. His solution is top, turn over, middle, top. The final face is done: orient edges, position edges, position corners, orient corners. 2) "The Simple Solution to Rubik's Cube(TM)", by James G. Nourse, Bantam Books, 1981($1.95). Illustrated by Dusan Krajan. This is a somewhat revised edition of Nourse earlier publication "Solution to Rubik's Magic cube", Storc Publications. It contains a solution, plus a little other information on the cube and some "other games to play" (speed, competition, pretty patterns). The main thing new to me is the idea of pretty-patterning the alphabet, and then spelling out 3 or 4 letter names around the cube. Some are pretty stretched, but it's a nice idea. His solution goes top-middle-bottom, and bottom is: place corners, orient corners, place edges, orient edges. His moves tend to be longer than "normal", but with the purpose of being able to recover fairly easily from a mistake or from a dropped cube. He tries to move no more than one cube from a previously solved position at a time, and to make it always possible to back up only one step (rather than to the beginning). Each section has error correction (in case you go wrong), and short cuts (to speed things up once you get used to it). His notation uses Top, Bottom, Front, Posterior, Left, Right. He tries to use Posterior as little as possible (which I think is very good - but for the same reason, its easier to turn the whole cube over after the first couple of steps to be able to work on the top.) The book is roughly paper-back size (though of course, much thinner), and on crummy paper. But the layout and pictures are good. I noticed a couple of errors - on page 45, you might have to error correct back to the end of step 2, since an upper corner might be misplaced. On p. 46, Short Cut 2 is a replacement for step 4C, not 4D, and the sequence should end with B+ or B-. 3) "Mastering Rubik's Cube(TM), The solution to the 20th century's most amazing puzzle", by Don Taylor, an Owl Book, Holt, Rinehart and Winston, 1980, $1.95. This is a straightforward solution book, with a few additional games and pretty patterns, but not much. He does it top, turn over, middle, top; and the final top is place corners, place edges, twirl corners, and flip edges. He uses BFUDLR, place(to move), and position(to orient). The moves are similar to Singmasters, but with some reductions. This note is getting long. Does anybody know of any other published books on the Cube? I notice that almost all the published solutions I've seen are top-middle-bottom (or that turned over in the middle). Except that Singmasters original solution was first all the corners, then all the edges. Does anybody do edges before corners? I nowadays use a variant which is easier for me: First I do all the top except one corner. Then I use that corner to make it easier to get in the middle layer (except for one). Then I finish off the top, and put in the last middle. I turn the cube over, and orient first the edges, then the corners. With the top face now showing the right colors, I find it easier to see where I am. I position the corners (with a move that doesn't twist), and finally position the edges. What other variants are around? -- Stan -------  Date: 18 July 1981 17:15-EDT From: David C. Plummer Subject: Speed cubing contest To: CUBE-LOVERS at MIT-MC Second hand info: (first hand in the Boston Phoenix (day unknown)) SPEED CUBING COMPETITION Date: Saturday, July 25, 1981 Place: Jordan Marsh, Burlington, Massachusetts Times: 1100-1500 3 minute qualifiers 1530-1800 Registration and Playoffs 1800-.... Finals First prize is N hundred dollars, where N is about 5. (Sorry for the lack of info, we got this over the phone in a hurry.)  Date: 18 Jul 1981 14:28 PDT From: Pasco at PARC-MAXC Subject: Please remove my name To: CUBE-LOVERS at MIT-MC Sorry to bother all of you, but it's not obvious to whom I should address this request: Please remove my name (Pasco at PARC-MAXC) from CUBE-LOVERS. Thank you. Rich Pasco  Date: 18 July 1981 17:39-EDT From: Alan Bawden Subject: Conserve computrons! To: CUBE-LOVERS at MIT-MC cc: Pasco at PARC-MAXC cube-lovers-request@mit-mc is the mailing list to send adminitrivia to. Since mailing to cube-lovers causes machines all over the ARPA net to have to deal with the mail, you should all take a moment to remember that fact, so that when YOU want to be removed, or want someone new added, you will know who to mail to.  Date: 20 July 1981 23:14-EDT From: Alan Bawden Subject: problem To: CUBE-HACKERS at MIT-MC Here's a problem I don't think I have seen asked before: Suppose you left your cube out in the rain and one of its six axles froze so that you were unable to turn that face. How badly is your cube damaged? Can you still reach all the positions you could before? or are you now limited to some proper subgroup? (how big is it?) I can see that all of the configurations of corner cubies are still easily attainable. But without a cube at home to fool with, I can't figure out if I can reach all the edge positions, and I can't figure out how the edge and corner groups interact given this restriction. Another way of asking the problem would be: How large is the subgroup generated by just five of the six quarter twists.  Date: 20 July 1981 2332-EDT (Monday) From: Dan Hoey at CMU-10A To: Cube-hackers at MIT-MC Subject: Re: One stuck axle In-Reply-To: Alan Bawden's message of 20 Jul 81 22:14-EST Message-Id: <20Jul81 233257 DH51@CMU-10A> Singmaster gives the answer. Let A = RL'FFBBRL', then AUA = D. Thus it is possible to dispense with D moves entirely. Jim Saxe also has a solution, but I don't know if it's the same one.  Date: 20 July 1981 23:35-EDT From: David C. Plummer Subject: problem To: ALAN at MIT-MC cc: CUBE-HACKERS at MIT-MC Date: 20 July 1981 23:14-EDT From: Alan Bawden Here's a problem I don't think I have seen asked before: .... Another way of asking the problem would be: How large is the subgroup generated by just five of the six quarter twists. According to the Fusrt-Hopcroft-Luks algorithm, they are the same size. I would appreciate Hoey and Saxe to confirm this (I think I have all the bugs out of my code). Sorry for such a brute force method, but it only took a few minutes.  Date: 21 Jul 1981 0908-PDT From: ISAACS at SRI-KL Subject: More problem To: CUBE-HACKERS at MIT-MC Well, then. How about four of the six quarter twists? Three? -- Stan -------  Date: 21 July 1981 2350-EDT (Tuesday) From: Dan Hoey at CMU-10A To: Cube-Hackers at MIT-MC Subject: The ten stuck-axle subgroups In-Reply-To: ISAACS@SRI-KL's message of 21 Jul 81 11:08-EST Message-Id: <21Jul81 235052 DH51@CMU-10A> 1. No faces stuck. The familiar cube group. 2. D face stuck. As previously noted, all positions can be reached. In addition, all Supergroup positions that fix the orientation of the D face center are achievable. 3. B and D faces stuck. All Supergroup positions that fix the BD edge and the B and D face centers are achievable. 4. U and D faces stuck. Edges cannot be flipped. If we define edge orientation by marking the F and B facelets of the F and B edges, and the U and D facelets of the others [cf Jim Saxe's message of 3 September 1980], then all Supergroup positions that fix the orientation of all edges and the U and D face centers are achievable. 5. L, B, and D faces stuck. All Supergroup positions that fix the BLD corner, the LB, BD, and DL edges, and the L, B, and D face centers are achievable. 6. U, B, and D faces stuck. Again, edges cannot be flipped. All Supergroup positions that fix the orientation of all edges, the position of the UB and BD edges, and the orientation of the U, B, and D face centers are achievable. 7. U, L, B, and D faces stuck. Singmaster has a very nice description of this group [indexed as Group, Two Generators]. The group of achievable permutations of the six movable corners is isomorphic to the group of all permutations on five letters. All Supergroup positions that permute the corners in an achievable permutation, fix edge orientation, and fix the unmovable two corners, five edges, and four face centers are achievable. 8. U, L, D, and R faces stuck. Sixteen positions 9. U, L, D, B, and R faces stuck. Four positions. 10. All faces stuck. One position.  Date: 24 July 1981 2220-EDT (Friday) From: Dan Hoey at CMU-10A To: Cube-Hackers at MIT-MC Subject: A new problem Message-Id: <24Jul81 222049 DH51@CMU-10A> Suppose you buy a new cube and the arrangement of the colors is different from your old cube. Naturally, you want the new one to be like the old, so you decide to switch the colortabs around. A. What is the smallest number of faces you have to recolor? B. What is the smallest number of colortabs you have to move? Note the hidden variable: the permutation of the new cube with respect to the old one. This variable has thirty values, including the identity. There are two kinds of answers I am interested in. 1. A minimax value -- a recoloring algorithm and a proof of its optimality. 2. A probability distribution of optimal recolorings. Any takers?  Date: 27 July 1981 10:29-EDT From: Dennis L. Doughty Subject: Cube Championship To: CUBE-LOVERS at MIT-MC cc: DUFTY at MIT-MC Does anyone know what the winning time was in the recent regional cube contest? (Last saturday at Jordan Marsh)? --Dennis  Date: 27 Jul 1981 1111-EDT From: PDL at MIT-DMS (P. David Lebling) To: DUFTY at MIT-MC Cc: CUBE-LOVERS at MIT-MC In-reply-to: Message of 27 Jul 81 at 1029 EDT by DUFTY@MIT-MC Subject: Regional Cubing Championship Message-id: <[MIT-DMS].205137> According to the Boston Globe, the fastest times were; 48.31 sec. - Jonathan Cheyer, 10 51.16 sec. - Jeffery Varafano, 14 51.59 sec. - Peter Pezaris, 11 (these are for the "junior" division; under 17). The fastest "seniors" were; 69.64 sec. - Herbert H. Thorp, 17 69.83 sec. - Charles Hawes 77.26 sec. - Rick Miranda Jordan Marsh says they sell about 2000 cubes per week. As the Jordan Marsh V.P. who was standing next to me said, "You can't buy this kind of publicity!" The competition was organized reasonably well, consisting of three rounds: 1) The qualifying round consisting of being able to solve a cube in under three minutes. No official timing other than "under three minutes" was done in this round. About 20 people were tested per qualifying round, and from 20-30% qualified. The cubes were allegedly "broken in" in advance, and all had the same color orientation. They were re-randomized between rounds. 2) Those who qualified in the first round were given two tries to solve a random cube in under two minutes. 3) Three "patterned" cubes were solved (presumably everyone got the same patterns). I didn't see this round so I don't know the details of it. My impression of the qualifying rounds was that those who qualified differed from those who didn't largely in speed. They didn't seem to use any macros I haven't seen, they just did them extremely fast and rarely paused more than fractions of a second to decide what to do next. The fact that the three top finishing juniors all had better times than the three top-finishing seniors indicates that competitive cubing is a young person's game.  Date: 29 JUL 1981 2103-EDT From: RMC at MIT-MC (R. Martin Chavez) To: CUBE-LOVERS at MIT-MC  Date: 29 JUL 1981 2107-EDT From: RMC at MIT-MC (R. Martin Chavez) To: CUBE-LOVERS at MIT-MC If it's at all possible, I would very much like to be added to the Cube-lovers mailing list. Though I will not go so far as to call myself a "cube-meister", I have spent a great deal of time developing my own personalized algorithm (without the book.) I can solve the cube in about five minutes (don't laugh), yet I still need to discover an operator for swapping a pair of edges and a pair of corners. Thanks....  Date: 30 July 1981 16:53-EDT From: Carl W. Hoffman Subject: Another Japanese cube To: CUBE-LOVERS at MIT-MC Sitting in the SIPB office is a super-group cube. At least I think it's a super-group cube. Four sets of facies show the symbols on playing cards, and the other two sets show five pointed stars of different types. So the center facie carrying the diamond can have one of two orientations. Is it possible to rotate one center facie 180 degrees? (The cube is currently in the solved state, thanks to Jim Saxe.)  Date: 30 Jul 1981 16:59 PDT From: Woods at PARC-MAXC Subject: Misc Cube replies In-reply-to: recent messages To: RMC at MIT-MC (R. Martin Chavez), CWH at MIT-MC (Carl W. Hoffman) cc: CUBE-LOVERS at MIT-MC To RMC: The proper way to get yourself added to any of the large lists at MIT is to send a message to -REQUEST, e.g., CUBE-LOVERS-REQUEST. Likewise for getting yourself removed from the list. Also, a macro for swapping two corners and two edges (all on the same face, and without flipping any of them), is (UUFLF')*5. There are probably shorter ones, but I like this one because it requires remembering only a very short sequence, making it easier to do when stoned/drunk/tired/etc. To CWH: I don't believe you can rotate a single center facie 180 degrees without changing anything else, which means you can make a lot of confusion for someone by inverting the diamond and coming up with a cube that appears to have only one center facie flipped. -- Don.  Date: 31 July 1981 00:04-EDT From: Steve B. Waltman To: CUBE-LOVERS at MIT-MC It is possible to rotate one center face any multiple of 90 degrees. In fact, many apparently 'null operations' rotate one or more center faces.... Steve  Date: 30 Jul 1981 23:01 PDT From: Woods at PARC-MAXC Subject: Re: Supergroup To: Steve B. Waltman cc: CUBE-LOVERS at MIT-MC It was my understanding that the size (number of possible positions) of the supergroup was only (4^6)/2 times as large as the normal group, i.e., that there was another level of parity involved. This would imply that you can rotate a center facie 180 degrees, but not 90 (at least, not without changing something else). I'd be most interested in seeing one of your "many apparently 'null operations'" that rotates one center face 90 degrees. -- Don.  Date: 31 July 1981 0626-EDT (Friday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Re: Supergroup In-Reply-To: Woods@PARC-MAXC's message of 31 Jul 81 01:01-EST Message-Id: <31Jul81 062607 DH51@CMU-10A> Don Woods's answer is correct. When the edges and corners are in their solved positions, the total face center twist is zero modulo 180 degrees. Thus the number of face centers that are twisted 90 degrees must be even. A proof of this appeared in Cube-Lovers last year. A process to twist the U face center 180 degrees without changing anything else is (U LR UU L'R')^2. This process comes from Singmaster's book. I sent Cube-Lovers a general procedure for solving the supergroup in January. If you want a copy of either of these messages, send a note to Hoey@CMU-10A.  Date: 2 August 1981 03:43-EDT From: Alan Bawden Subject: Administrivia and an assault on God's number. To: CUBE-LOVERS at MIT-MC First the administrivia. Starting with this message mail to Cube-Lovers is no longer automatically redistributed to everyone on the list. This was done because the "please add me to this list" type of message is now almost as frequent as messages discussing topics of general interest. Now the "editors" (Dave Plummer and myself) will have a chance to catch these boo-boos. Hopefull this change will be completely invisable to the rest of you except that the mail headers will contain our names as the senders, and the turn-around time will be a little slower. If this really offends anyone then we can put it back the way it was, but lets try it this way for a while. You should continue to send your messages to Cube-Lovers at MIT-MC. Now, on to God's number! As you may recall, a somewhat complicated counting argument sets a lower bound on God's number (the worst case counting of the best possible algorithm) at 21 Quarter twists. Dan Hoey's message of January 9 1981 (the second message in a series about the "Supergroup") contains an excellent summary of that argument, so I won't repeat any of it here. That argument takes into account certain trivial identities such as FFFF=I and FB=BF in order to reduce the amount by which the counting overestimates the number of configurations a certain number of twists away from "solved". The same argument ignoring the identities only leads to a lower bound of 19. It is thus natural to expect that taking even more identities into account would lead to an even higher lower bound. Well, the next smallest identities are those of the form FR'F'RUF'U'FRU'R'U=I. It is known that there are none smaller that aren't a consequence of those "trivial" identities mentioned above (See Hoey's message of 22 March 1981), although there might be others of the same length. What happens if we take these additional no-ops into account? The conceptual problems in applying these new identities to the counting have had me stumped for quite some time now, but last week I finally figured a way that would cover at least some (maybe even all, I haven't worked on a proof of that yet) of the consequences. Well, all right, I'm keeping you in suspence, what did I learn? Nothing. The lower bound still stands at 21 (similarly the Supergroup lower bound still stands at 25). Even after taking FR'F'RUF'U'FRU'R'U=I and his many friends into consideration it seems that the numbers that fall out of the counting scheme (and it is amazingly complicated!) are only slightly smaller than those we already knew. The relevant numbers are: Size of the cube group: 43252003274489856000 Under old counting: positions 20 Q's or less away from start: 39812499178877773072 positions 21 Q's or less away from start: 373188814849923987472 Under new counting: positions 20 Q's or less away from start: 39726356237445007600 positions 21 Q's or less away from start: 372326146413193718032 As you can see the numbers are depressingly close. This seems to shut the door on any further improvements of this kind to this argument. It is hard to imagine that the effects of any other identities (remember they have to be at least 12 Q's long) could be signifigantly greater than the effect here. (Of course, if we knew ALL of them... but then we would understand the group completely!) It is of course possible that some deeper property, deeper than just the knowledge of one identity, could improve this style of counting argument. It is of course also possible that I have screwed up somewhere. I sould really let some of the rest of you into the details of this thing. As you can guess, I am not very excited at the idea of having to explain the details of the argument to you all. The proof is complicated and kludgey, and I at least am convinced that it leads nowhere. People who are interested in the gory details can contact me and we can work something out.  ZILCH@MIT-MC 08/02/81 06:06:43 Re: Identities (galore)! To: CUBE-LOVERS at MIT-MC CC: ALAN at MIT-MC This message may be overdo, but I figuered it ought to be sent now since ALAN has done some work on this subject and may be able to use my results. Recently I have found many Identity transformations and this message is basically a catalog of them. I12-1 FR'F'RUF'U'FRU'R'U I12-2 L'D2F'D'FLD2BDB' I12-3 FR'F'RUF'UL'U'LFU' I14-1 D'L'DRD'LR'DF'D'RDR'F I14-2 D'L'D'F'DFLDF'R'D'RDF I14-3 LD2RL'F2L'F2LR'D2 I14-4 F'R'D'RDFUF'D'R'DRFU' I14-5 (LRD2L'R'D')^2 NOT GOOD IN SUPERGROUP DOES I14-6 LR'FRL'U'DFB'R'F'BUD' " " I16-1 (LRD2L'R'D2)^2 I16-2 F'R'D'RDFU2F'D'R'DRFU2 I16-3 F'R'D'RDFD2B'D'L'DLBD2 I16-4 F'R'D'RDFUDR'D'B'DBRU'D' I16-5 F'R'D'RDFU'DR'D'B'DBRUD' I16-6 UF2U'DR2D'U'R2UD'F2D I16-7 LR'D2RL'F2RL'F2LR'D2 I16-8 FDLD'F'LDL'F'L'D'LFD'L'D I16-9 LD'L'D'F'DFUF'D'FDLDL'U' I16-10 FL'F'LF'D'FUF'DFL'FLF'U' I16-11 F'D2R'D'RD'FLD2BDB'DL' I16-12 (F2B2R2L2)^2 I16-13 LR'FRL'U2D2L'RB'LR'U2D2 NOT IN SUPERGROUP I16-14 LR'F2RL'U'DFB'R2F'BUD' " " I18-1 F'BD2F'D'FD2B'LBDB'L2FL I18-2 LRD'L'R'D'LRDL'R'DLRDL'R'D' I18-3 D'RD'R'DBDB'DBD2B'D'RD2R' I18-4 LR'F2RL'U2D2L'RB2LR'U2D2 I18-5 LDR'L'D'LRDL'R'D'RLDR'L'D'R I18-6 (RBL'R'B'L)^3 I18-7 LDR'L'D'LRDL'R'DRLD'R'L'DR NOT IN SUPERGROUP I18-8 (F'D'FD'RD2R'D)^2 " " These are not all of the identities that I have found but are generators of them. How a generator generates other identities: 1. Inversion (2) 2. Rotation (24) 3. Reflection (2) 4. Shifting (N) where N is the length of the identity The numbers in () are the number of different ways that can be gotten for each of these methods. Combining gives 96N. For example an identity of length 12 generates a possible 96*12=1152. However this number is usually not reached becauseof inherent symmetry. If you take the inverse of I12-1 --> U'RUR'F'UFU'R'FRF' Then its reflection -->UL'U'LFU'F'ULF'L'F Then a rotation U->F,L->R,F->U -->FR'F'RUF'U'FRU'FR'U You get back what you started with. When shifting is included in this process there are a total of 6 different ways this can be done giving 1152/6=192 different identities generated by I12-1. Shifting: Basically you chop the transform in its interior and append the first part to the second part. For instance. I12-1 FR'F'R / UF'U'FRU'R'U Becomes UF'U'FRU'R'U FR'F'R Note that this is just a rotation away from the origional. From these some equivilences may be deduced: DL'F'D2R'D'R = L'D2F' = BD'B'D2L'F'D DF'R'DRD'FD' = FL'F'L =D'LD'BDB'L'D FR'F'RUF' = U'RUR'F'U = UF'L'ULU' Unfortunatly these equivilences only generate the 3 identities of length 12 , using the idea that midpoint of an identity must be the unique maximum along the path of the transform.  ZILCH@MIT-MC 08/02/81 06:36:52 Re: 2edge-2corner swappers To: CUBE-LOVERS at MIT-MC I know of 4 different transforms that swap 2 edges and 2 corners. LD'BRDR'B2LBL2D (FD,RD),((DLF,DFR) R'DRD2R'L'DRD'LD2 (FD,LD),(FDL,FRD) FD2F'B'DFD'BD2F'D (FD,LD),(LFD,BDR) FLFL'UBLB'U2FUF (FD,DL),(DLF,DFR) I found the first 3 of these before I saw Singmaster and modified one of Singmaster's into the fourth one which he didn't even have recorded.  Date: 2 August 1981 19:41-EDT From: Alan Bawden To: CUBE-LOVERS at MIT-MC This is a test message that should NOT get forwarded!  Date: 2 August 1981 19:42-EDT From: Alan Bawden Subject: again To: CUBE-LOVERS at MIT-MC This an another test message to see if COMSAT really sucks rocks as big as I think!  ALAN@MIT-AI 08/02/81 19:44:44 Re: and finally. To: CUBE-LOVERS at MIT-AI Here is the last test in the series.  Date: 2 August 1981 23:51-EDT From: Steve B. Waltman Subject: Supergroup To: Woods at PARC-MAXC cc: CUBE-LOVERS at MIT-MC Oops.. Woods is right and I was out-to-lunch. Sorry... Steve Waltman  Date: 3 Aug 1981 0934-EDT From: Jerry Agin Subject: One more identity To: zilch at MIT-MC cc: Cube-Lovers at MIT-MC U'F'UBU'FUB'URU'L'UR'U'L -------  Date: 3 Aug 1981 1250-PDT From: ISAACS at SRI-KL Subject: another new cube To: cube-lovers at MIT-MC Last week I saw still another type of new cube that a friend had picked up on a chinese boat. Each face was a magic square - that is, each facelet contained a number from 1 to 9, and, when solved, each line of 3 facelets added to 15. This particular cube was also colored, so wasn't very interesting (you could solve it by colors and the magic-ness was automatic), but it should be a very difficult problem with the numbers colorless. Apropos to this, my son says he saw two other new types (but by the time he went back to get them for me, they were gone). They were both related to the 10-sided one that essentially cut off 4 parallel edges; one cut off all EXCEPT 4 parallel edges (how?), and the other cut off corners. Has anybody else seen thesed? Can you describe them more accurately? By the way, there are lots of "do it yourself" variations on the cube. For instance, a good present (though it takes a lot of work to make, and I still don't know what the best glue to use is) is a picture cube. Find 6 family photos (or other appropriate pictures) of the right size and glue away. Or (and this is based on earlier discussion of a braille cube), make a tactile cube - put a different material on each face. I've tried cloth (which doesn't work too well because of the glue and the fraying), misc stuff (washers, broken toothpicks, etc), and next want to try with different sandpaper grades (the ideal tactile cube would be ONLY solvable by touch, and have the faces look the same to the eye). Another, more difficult to make suggestion, would be to make each face a different thickness, or perhaps a different contour. Any ideas on how to actually construct these? Any ideas on a really good glue for cube faces? --- Stan Isaacs -------  Date: 3 Aug 1981 1304-PDT From: ISAACS at SRI-KL Subject: Another Cube Book To: cube-lovers at MIT-MC Just got still another book on solving the cube: "Solve That Crazy Mixed-Up Cube", by Don Frederick, Frederick Enterprises, P.O. Box 1016, Oceano, Ca, 93445. This one has a sense of humor, and likes to make up new names. He talks about "slabs" for a layer, "cranking the keepers" to save some already positioned cubies while you do something else, a "magic move", etc. For instance, his top slab moves are: the STICK-UP, the HANG-UP, the PUT-DOWN, the PICK-UP, the DOUBLE-DIP, DUMP TRUCK, and the SLING-SHOT GOTCHA! He goes top-middle-bottom, bottom done: twist edges, position edges, position corners, twist corners. He includes hints, memory aids, short-cuts, patterns, a two person game called "Widow's Revenge", etc. Even cartoons ('if all the unsolved cube puzzles were put in one pile, they'd stay that way.'). All in all, a very nice addition to the litterature. $3.99. --- Stan Isaacs -------  Date: 3 August 1981 16:47-EDT From: Allan C. Wechsler Subject: God's Number To: CUBE-LOVERS at MIT-AI We know that some positions are "global maxima". We don't know how many such positions there are. We would dearly love to know how far away they are. Suppose that God's Number is N. (For some obscure theological reason I have the irrational belief that N=28, but we'll leave such hunches out of the discussion.) Let's say there are K global maxima at that distance from solved. What if we could show that there are at least 12K states at distance N-1? This is a little bit reasonable. All it means is that global maxima are all separated from each other by more than 2q. If that were true, mightn't we be able to increment our lower bound on N? Can anybody prove that it's true? I would also like to hear from ZILCH where he gets those identities, and whether any of those impressive lists are exhaustive. Alan, do the other two order 12 nulls enable you to stretch your results? I am busy wading through Wielandt's "Finite Permutation Groups" and will report if I learn anything applicable. --- Wechsler  Date: 3 August 1981 18:07-EDT From: Alan Bawden To: CUBE-LOVERS at MIT-MC The answer to everybody's questions about whether the additional identities help my proof is: no. My idea depended heavily on the peculiar properties of that one identity. Dan Hoey has compiled a list of ALL the 12Q identities by brute forcing down to 6 twists, we suspect that ZILCH may have found them all, but it will take a while to sort throught them to be sure. We will let you know the results soon.