Date: 11 December 1982 20:43-EST From: Alan Bawden Subject: The Archive & Administrivia To: CUBE-LOVERS at MIT-MC Those of you who look through the archives of old Cube-Lovers mail will notice that I have split off a new section of the archive. The mail now lives in: MC:ALAN;CUBE MAIL0 ;oldest mail in foward order MC:ALAN;CUBE MAIL1 ;next oldest mail in foward order MC:ALAN;CUBE MAIL2 ;more of same MC:ALAN;CUBE MAIL3 ;yet more of same MC:ALAN;CUBE MAIL4 ;still more of same MC:ALAN;CUBE MAIL ;recent mail in reverse order Those of you who are wondering if Cube-Lovers might be temporarily off the air when TCP is installed on January 1 can rest assured that there is an ITS TPC implementation in progress, and come January 1 the ITS machines will probably be no worse off than the rest of the net. My biggest worry is not that ITS won't be able to send mail (since I can always move the list to another machine if that proves necessary), but that so many of you all will be having difficulty recieving mail that I will be up to my neck in error messages from various mailers around the net. That is one of my reasons for splitting off a section of the archive at this time. I'm willing to bet that a fair number of you will be requesting the back mail that you have missed during the storm at the beginning of the year, and this way I have a small file that I can easily send off to you. Finally let me remind you all to please address your requests to Cube-Lovers-Request at MIT-MC. If that address, or Cube-Lovers at MIT-MC ever seems to be giving you trouble, you might also try Cube-Lovers at MIT-Multics and Cube-Lovers-Request at MIT-Multics. -Alan  Date: 21 December 1982 23:50-EST From: Alan Bawden Subject: Correction to previous message. To: CUBE-LOVERS @ MIT-MC It being so difficult to set up a Multics mailing list I'm retracting the promise that those Multics addresses will ever work and will instead just give out my personal Multics address as an emergency contact.  Date: Wednesday, 22 December 1982, 15:33-EST From: Allan C. Wechsler Subject: Hinton's cubes. To: Cube-Lovers at MC In September I received a query from Phil Servito, who has since joined this mailing list, about a game or puzzle called "Hinton's cubes". Does anybody have any information about this toy besides the following? Charles Howard Hinton was an American mathematician who believed that anybody could learn to do four-dimensional visualization if properly trained. He invented a set of colored blocks which were supposed do this training. Philip -- you said Martin Gardner knows more. Is this because the cubes were once discussed in a Sci. Am. Mathematical Games column? If so, the Sci. Am. cumulative index may help. --- Allan  Date: 29 Dec 1982 0954-PST Subject: article of interest. From: Dave Dyer To: cube-lovers@MIT-MC The Christmas issue of NEW SCIENTIST has an article by Singmaster about the basics and metaphysics of the cube. No great revelations, but a literate discussion of cubeology. -------  Date: 29 Dec 1982 19:01:04-EST From: meister at mit-ccc To: cube-lovers@mit-mc Subject: 4x4x4 poll i am interested i getting information on the various solving methods used on the 4x4x4, and the times that it is being done in. (average) my own method is to solve it as follows: 1) top layer (no general order to this) 2) the rest of the centers by layers 3) the edges on the center slices 4) bottom corners 5) bottom edges average time: 1 min 55 sec - 2 min 5 sec depending on my blood-alcohol level. please send me your general method and solving times. i'll compile the results and put them on the list. thanks for replying, -phil p.s. - send replies to meister@ccc@mc  Date: 30 December 1982 00:41-EST From: Jonathan David Callas Subject: Hinton's cubes. To: ALAN @ MIT-MC cc: CUBE-LOVERS @ MIT-MC I remember reading about Hinton's cubes in an old Mathematical Games anthology, so a cumulative index would be the thing. Gardner wrote about them and apparantly they worked, but there were stern warnings not to play with them, because (!) they cause insanity in some people(!!). Gardner said something to the tune of that the human brain is not meant to do 4-dimensional geometry, and these things will make your brain bugcheck. Ever since then, I have been fascenated by the thought of a puzzle that would blow your mind literally! Of course, this was in the days before it was fashionable to use chemical means to do this. Anyone else know anything about them? Jon  Date: 3 Jan 1983 0915-PST From: ISAACS at SRI-KL Subject: Hinton's Cubes To: cube-lovers at MIT-MC cc: isaacs at SRI-KL Hinton's Cubes are mentioned in Martin Gardners "Mathematical Carnival", Chapter 4, Hypercubes, a reprint of the November, 1966, Mathematical Games column from Scientific American. He doesn't say much, mainly that "Hinton... devised a system of using colored blocks for making three-space models of sections of a tesseract." Hinton believed that they would help develop an intuitive grasp of 4-space. The fullest account of them mentioned is the book k"A New Era of Thought", by C. Howard Hinton, Swan Sonnenschein, 1888, apparantly reprinted in 1910. Gardner also prints a letter he received from Hiram Barton, of England, including "A shudder ran down my spine when I read your reference to Hinton's cubes. I nearly got hooked on them myself in the nineeen-twenties. Please believe me when I say that they are completely mind-destroying. The only person I ever met who had worked with them seriously was Francis Sedlak, a Czech neo-Hegelian philosopher...who lived in an Oneida-like community near Stroud, in Gloucestershire. "As you must know, the technique consists essentially in the sequential visualizing of the adjoint internal faces of the poly-colored unit cubes making up the large cube. It is not difficult to acquire considerable facility in this, but the process is one of autohypnosis and, after a while, the sequences begin to parade themselves through one's mind of their own accord." It goes on, but doesn't say a great deal more. I don't have my copy of Hintons book "The Fourth Dimension", but maybe it will give some more description. If anybody does find some marketed versions of the cubes, put it on the list. --- Stan -------  Date: 4 January 1983 03:54-EST From: Alan Bawden Subject: Humor To: CUBE-LOVERS @ MIT-MC From "Special Deliverance" a recent novel by Clifford Simak: "He also mentioned a cube," said Sandra. "I wonder what it could be. Never before have I heard anything described simply as a cube." Well I certainly have! (And no, the book actually has nothing to do with Hungarian Cubes. It just struck me as strange that someone could write that line at a time when an object that frequently WAS described simply as "the cube" was enjoying some popularity. Perhaps Sandra's ignorance is excusable in light of the fact that she comes from an alternate universe...)  Date: 5 Jan 1983 1802-PST From: ISAACS at SRI-KL Subject: Other news To: CUBE-LOVERS at MIT-MC The new RUBIK'S magazine is out. Not as interesting (to me) as the first. Several articles on the Snake, on cube contests, and some on other games. A computer flow-chart for the game Awari, and an analysis of Chinese Rings are included. The most interesting cube articles are one on how disordered a cube can get; and one on pretty patterns in other cube universes (ie, when an edge is flipped, what symmetric patterns are there). If you haven't seen it, the book "INside Rubik's Cube and Beyond" by Christoph Bandelow is very good. It's translated from German, and contains a fairly technical discussion of the Mathematical Model, solving in the supergroup, other magic polyhedra, a detailed flowchart of a cube-solving program (which also checks the input for legality), and a comprehensive Maneuver Index, which contains, among other things, a complete catalog of 3-cycles. It includes an index, a bibliography, and a good section of colored pictures in the center (including one of a 5x5x5!!!). Recomended. I've also heard a rumor that the solution to "Masquerade" has been written (and maybe published). Has anyone heard anything of this? -- Stan -------  Date: 5 Jan 1983 1806-PST From: ISAACS at SRI-KL Subject: more on Hinton's Cubes To: cube-lovers at MIT-MC I got my "Speculations on the Fourth Dimension, Selected Writings of Charles H. Hinton", Dover, 1980, back. It doesn't reprint Hinton's description of his cubes, but Rudolf v.B.Rucker, in the introduction describes something more of them: "The second part of "A New Era of Thought" consists of a description of how to visualize a tesseract by looking at various 3-D cross sections of it. On is to construct a set of 12 cubes, coloring the faces, edges and corners all manner of different colors. (Eighty-one different colors are used, and some rather unfamiliar ones are resorted to. ...)Eight of these cubes make up the boundaries of the hypercube, and the four others are cross sections taken between pairs of opposite cubes. The way in which all the cubes fit together is really explained rather well, if one has the will to endure not only 81 colors, but the 81 Latin names which Hinton assigns to the parts of the tesseract. "In addition to the set of 12 large cubes, there was also to be a set of 81 small monochrome cubes, each representing a part of the tesseract. By moving theese little cubes about one could better comprehend the fact that rotation through the fourth dimension corresponds to mirror image reflection in 3-D space." In Hinton's book "The Fourth Dimension", published n 1904, he has a streamlined version of the tesseract section models. "There were actually three parts to the complete set of tesseract models. There was a set of 27 "slabs," actually cardboard squares; a set of 81 one-inch monochrome cubes, each a different color; and a set of 12 multicolored "catalogue cubes," which were depicted in a color plate bound into "The Fourth Dimension". When the book came out, one could buy a set consisting of the 27 slabs and the 81 little cubes for 16 shillings, or a set consisting of the 12 catalogue cubes for 21 shillings." If anyone on this list can find a copy of "The Fourth Dimension", I, for one, would like to buy it. I wonder if any of the models are around anywhere? --- Stan -------  Date: 7-Jan-83 10:19:31 PST (Friday) From: Pettit at PARC-MAXC Subject: 4-D Graphics Program In-reply-to: ALAN's message of 5 January 1983 23:40-EST To: Alan Bawden cc: Cube-Lovers at MIT-MC Scott Kim, the author of Inversions, wrote a program about 4 or 5 years ago which ran on Tenex, and allowed one to rotate 4-D skeletal objects and view the 2-D projections (on an Imlac display). I don't know what language it was written in. It was apparently quite good at allowing one to achieve an intuition for 4-D space. I'll send him a message and ask if his program is available anywhere for playing with. --Teri Pettit at Xerox OSD  Date: Monday, 24 January 1983 13:57-EST From: Leonard N. Foner To: Cube-Lovers @ MIT-MC Subject: Shortest sequence of moves? cc: Tk.Foner @ MIT-OZ Reply-to: Foner at MIT-MC I remember hearing about a program, probably running on the color LISP machine, that could take an arbitrary cube and try to see what the shortest move sequence to its slution is. Does anyone remember where this program is and how to use it? Specifically, I have been given the following cube and asked to find the shortest solution: GBG BGW GGG RRR WYW ORO YYY ORO GWY OOO WYB RRR WWW ORO YWY BBB GBY BGB If anyone can either find the program I think I remember, or can determine what they think is the shortest solution, I'd be \very/ grateful! Speed is of the essence. Thanx much, everyone!  Date: 24 Jan 1983 1122-PST From: ISAACS at SRI-KL Subject: Masquerade To: cube-lovers at MIT-MC A small paperback version of Masquerade has come out in England, which contains a history, who solved it, and what the solution is. I don't know if it is available in the U.S. yet, but I have seen a copy, and got the following info: The full title & subtitle is "The Treasure Hunt of the Century / Masquerade / Kit Williams Tells the Answer to the Riddle", ISBN 0 224 02937 1; Jonathan Cape (pub), Thirty Bedford Square, London. I will check around to see if it is available --- Just did; At lease one local book store (Printers Inc, in Palo Alto, Ca.) says it's on order, and they expect in in a day or two. If anyone wants a rough idea of how to find the solution, I can put it on the net. -- Stan -------  Date: 24 January 1983 16:51 EST From: David C. Plummer Subject: Shortest sequence of moves? To: FONER @ MIT-MC cc: CUBE-LOVERS @ MIT-MC Date: Monday, 24 January 1983 13:57-EST From: Leonard N. Foner Reply-to: Foner at MIT-MC I remember hearing about a program, probably running on the color LISP machine, that could take an arbitrary cube and try to see what the shortest move sequence to its slution is. Does anyone remember where this program is and how to use it? Sorry, no such program. Don't we wish though!! If we understood the mathematics of the cube well enough to write such a program, we probably wouldn't have this mailing list anymore.  Date: 28 Jan 1983 3:33-EST From: Dan Hoey Subject: The shortest sequence of moves. To: Foner at MIT-MC Cc: Cube-Lovers at MIT-MC In-Reply-To: Leonard N. Foner's message of Monday, 24 January 1983 1357-EST Leonard, The process (R U^2 B^2 L')^2 will restore your cube in twelve quarter-twists when executed with the Green face Up and the White face Front, and twelve is the minimum sufficient number of quarter-twists. Dave Plummer's discouraging word is usually right--we know of no algorithm to let us find optimal processes for most positions. This is because the only known algorithms involve exhaustive breadth-first search, and there are far too many positions of the cube to make this practical in either time or space. But when the optimal process is sufficiently short, some headway can be made. Having some megabytes and CPU-hours at my disposal, I was able to list (A) all positions reachable in five qtw from your cube, and (B) all positions reachable in five qtw from SOLVED. Finding that sets (A) and (B) are disjoint, I conclude that there is no ten qtw process for the pattern, so the twelve qtw process is optimal. I discovered the optimal process by hand. Of course, I could have just run the program one more qtw and it would give me the process, along with any other twelve-qtw processes that may exist. The problem with that approach is that I don't have that many megabytes and CPU-hours. My program, by the way, is written in C and runs under Unix. It trades time and storage efficiency for programmer laziness, making extensive use of the Unix sort utility. Dave Plummer has written a much optimized program, in assembler language for the PDP-20, that uses very clever hacks (some of them my own). As I recall, he and I estimated that with about 150 megabytes and a day or two elapsed time on an unloaded machine it could take the search three more quarter-twists. Does anyone need to settle a bar bet on an eighteen qtw process? Dan  Date: 28 Jan 1983 1838-PST From: ISAACS at SRI-KL Subject: Circle Puzzles To: cube-lovers at MIT-MC cc: isaacs at SRI-KL I just got some very nice examples of circle (group-theory) puzzles - similar to the Lorente Grill puzzles shown in Hofstadters' Sci.Am. article in July, 1982 (p. 26). They are made by Douglas Engel, under the name of General Symmetrics, Inc, 2935 W. Chenango, Englewood, Co, 80110. He is selling some as experimental items, and one version he hopes to have marketed "soon". They all consist of two intersection circles, each of which rotate; I have 3 kinds: the "21 piece", similar to 2/3 of Lorente's Trebol puzzle; the "35 piece", similar to the one in the upper right corner of the article "Another Grill puzzle by Lorente", and a "19 piece", similar to the 21 piece, except the circles intersect in the center. Each of these are quite different from each other; the 35 piece has 3 different pieces which rotate in 4 different cycles (compare to the Cubes' 3 kinds of pieces - edge, corner, and center). In the 35 piece, a straightforward "in in out out" (R'LRL') type move will move the centers in a 3-cycle, the "square" pieces in a pair of 2-cycles, and the "triangles" in 2 separate 5-cycles. (Doug is selling these at $10.00 for the 19 or 21 piece, $20 for the 35 piece.) Doug Engel has also written a paper called "Some Problems Suggested By Circle Puzzles", mostly asking questions about various kinds of symmetry in this type, whether there are an infinite number of them, etc. I wonder what the best way to talk about them and analyze them from a group theory perspective is. I'll try to put in more information and theories as I work on them; I'd like to hear any thoughts or suggestions. Right now, I want to go home for dinner. -- Stan -------  Date: Saturday, 29 January 1983 23:33-EST From: Leonard N. Foner To: Dan Hoey Cc: Cube-Lovers @ MIT-MC, Tk.Foner @ MIT-OZ Reply-to: Foner at MIT-MC Subject: The shortest sequence of moves. In-reply-to: The message of 28 Jan 1983 03:33-EST from Dan Hoey Thanx for the solution... we determined independently that 12 qtw seemed to be minimal, but couldn't be absolutely sure. Since you say with a couple of days of CPU time you could deal with 18 qtw processes, doesn't this mean that \any/ cube process can be checked for minimality? I can't recall whether the maximum number of moves from solved is 17 or 19... if the former, 18 qtw may be unnecessary. If the latter, then almost every cube process can be checked directly, save the absolute longest ones (which are then provably 19 anyway, since that's all they can be). Is my reasoning faulty here? In any case, exactly how much more work in involved for each qtw in solving the cube? In other words, how fast does it grow (what's the power on the exponent, if describable this way)? Thanx much folks.  Date: 30 January 1983 00:40 EST From: Alan Bawden Subject: The shortest sequence of moves. To: FONER @ MIT-MC cc: CUBE-LOVERS @ MIT-MC, Tk.Foner @ MIT-OZ, HOEY @ CMU-CS-A In-reply-to: The message of 29 Jan 1983 23:33-EST from Leonard N. Foner The maximum number of moves (qtw) from solved is certainly not 17 or 19. All we know about that number is that it is greater than or equal to 21. It could be even larger. There is a fair amount of sentiment that it is probably around 28, but that's just intuition.  Date: 30 Jan 1983 3:15-EST From: Dan Hoey Subject: Finding optimal processes To: Cube-Lovers at mit-mc The best known bounds put the maximum number of qtw at between 21 and 104. At most about 1 percent of the cube's positions are within 18 qtw of solved. At each qtw, the number of positions increases by a factor of almost sqrt(12+5*sqrt(6))+sqrt(6)+2 ~ 9.374. [I say ``almost'' because this rate of increase does not take into account the identities of length 12 qtw or greater. Eventually, long identies reduce this factor to zero. But between six and seven qtw, the factor is about 9.356]. If we want to show a 2n qtw process optimal, we need only scan the table of n-1 qtw processes, so the increase is more like a factor of 3 per qtw. Finally, I would like to call attention to the fact that showing an 18 qtw process to be optimal requires about 150 megabytes and two days on a PDP-20 for each such proof. Finding someone to lend you a big machine and a big disk for two days is not something you can take for granted. And if you want to do it on your Apple, be prepared to sit around swapping 2400 floppies in and out for the next decade or so, as that two day figure is mostly disk latency rather than CPU time.  Date: Monday, 31 January 1983 21:48-EST From: Leonard N. Foner To: Alan Bawden Cc: CUBE-LOVERS @ MIT-MC, HOEY @ CMU-CS-A, Tk.Foner @ MIT-OZ Reply-to: Foner at MIT-MC Subject: The shortest sequence of moves. In-reply-to: The message of 30 Jan 1983 00:40 EST from Alan Bawden Ugh. I see the problem is far more intractable than I had first assumed. I suppose it'll be quite some time indeed before we see any sort of complete optimality solution if the maximum number of moves from solved is anything larger than the low twenties. I'll hunt through the list archives for what's been accomplished on proving what that number is... if anyone feels like giving me a brief refresher instead, I'd appreciate it, though the whole list might not like to see it all again. Thanx again everyone.  Date: 1 February 1983 13:20 EST From: David C. Plummer Subject: The shortest sequence of moves. To: FONER @ MIT-MC cc: CUBE-LOVERS @ MIT-MC, ALAN @ MIT-OZ, Tk.Foner @ MIT-OZ, HOEY @ CMU-CS-A What you really want is some deep mathematical insight. You know, the kind Gauss and Fermat had...  Date: Tue 15 Feb 83 15:04:02-PST From: ISAACS@SRI-KL.ARPA Subject: "6-T" pretty pattern To: cube-lovers@MIT-MC.ARPA A friend of a friend says he can do the 6-T pattern in 16 moves (he may call 1/2 turns a move, I'm not sure). Does anybody know if this is a minimum? I'll try to get the sequence. -- Stan -------  Date: Wednesday, 16 February 1983, 09:24-EST From: Bernard S. Greenberg Subject: 5 x 5 x 5 To: cube-lovers at MIT-MC I have seen and touched a 5 x 5 x 5 Rubik cube device. It was allegedly a prototype by Meffert, the people who created the Megaminx. It was about the same size as a 4 x 4 x 4, with slightly smaller cubies. It had never been devirginized, nor did its owner want it to be. The face coverings are of the same material as the coverings of the Megaminx. No clue as to how it operates, although looking at the black surface exposed by partial twists, there seemed to be new techniques involved, of protuberances coming down from cubies sliding behind other cubies, at least. Although stated to be a prototype, it seemed completely developed, and I would expect to see it available soon.  Date: 24 Feb 1983 0618-PST From: JAY Subject: NxNxN: Finite N To: cube-lovers@MIT-MC I have a program, not finished, which allows manipulation of a NxNxN (N^3) cube. Presently it WILL manipulate any NxNxN cube (i tried up to 23) but it has a VERRY BAD user interface. I plan to improve the interface in the near future (this morning?) I hope to be able to parse the Cube-Talk mentoined in the first message in the archive alan;cube mail1 (or 2?), or a super-set of it (like being able to save cube's on disk, save move seq's (macros), defining macros with parameters, and make logfiles of especialy hairy session). As for output... For small cubes (up to 8^3) i can use the normal air-plane pattern of display, for medium cubes (8^3 <= size <= 12^3) i can put it all on the normal terminal, but for larger cubes (24^3 >= cube >= 12^3) all i can do is display a face, or maybe two. [Implementation] I am interested in a, WORKING, N^3 simulator, not speed or space. As a result my representation of the cube is loosing on both counts. (yes it realy is a NxNxN array of cubes (a record of six integers!)) It is written in CLU for TOPS-20 os. However my rep. brought up an interesting super-groop, immagin a cube realy made of N^3 cubies. Each of these cubies would be a 'Miniature' N^3 in color scheme. Now this new device (a 'compleatly colored' cube) is solved only when ALL cubies are oriented correctly (ie. all have red up and blue forward), and positioned correctly. In the 3^3 we would not only have to solve the centres, but also the imaginary inside cubie (is it ever un-solved?) [Questions] What do you (readers) think is a good display scheme for a N^3, remember it should be useable on a 24x80 h19? Is a N^3 simulator even interesting? What sort of speed improvement could be gained from a comp-cube? with or without macros? Is the 'compleatly colored' cube interesting? For what sizes is it similar to the 'partialy colored' (normal) cube (1^3 and 2^3 for sure...)? Could the solution of compleat-5^3 be a solution of the outer shell, and the inner 3^3? Is a simulation of a 'compleatly colored' cube interesting? How would one view the manny inside faces? What other reps for the cube are there? (other than the obvious two; an array of color tabs, or a 3d array of cubies..) Which reps optimize storage, time, or simplicity to compute a twist, or even ease of compairison? Just occured to me that each cubie could be rep'd as a number twixt 1 and 24 (as a cube has only 24 orientations). This optimization would reduce storage by a factor of 6 (not too bad!) enough..... enough..... j' -------  Date: 24 February 1983 1815-EST (Thursday) From: Dan Hoey at CMU-CS-A To: Cube-Lovers at mit-mc Subject: Whole-Cubing Message-Id: <24Feb83 181536 DH51@CMU-CS-A> When we describe cube positions, we typically fix the position of the face centers. This avoids counting different positions of the same pattern more than once. But suppose we wished to distinguish between different positions of a pattern? How should we form this group G* of spatially oriented patterns? A simple way of generating G* is to adjoin generators for C, the motion group of the cube, to generators for G, the usual Rubik group. Generators for C were discussed in early Cube-Lovers mail as I, J, and K, three orthogonal quarter-twists of the whole cube in space, although there was some disagreement about which was which. We generate G with B, F, U, D, L, and R as usual. Unfortunately, the two kinds of generators are not conjugates, which was a nice thing about our generators for G. Also the generators do not interact very strongly. If we have an identity on {IJKBFUDLR}, the substring on {IJK} must be an identity in C, and there is a simple way of transforming the substring on {BFUDLR} to be an identity in G. In @i, Berlekamp, Conway, and Guy described another way of generating G*, by appending the slice moves, which they name by mnemonic greek letters; this labeling scheme was also reported in Hofstadter's column. Here we have some stronger interaction between the two kinds of generators, but they are still two different kinds: they are not conjugate. This has never really bothered the English, though; they gratuitously include half-twists as generators as well. I thought up a scheme involving what I will call depth-2 moves named B2, F2, U2, D2, L2, and R2. Readers of my reports on the 4^3 and 5^3 cubes will find these familiar. Essentially, a depth-2 move is performed by turning a face together with its adjacent center slice. Thus F2 involves holding the B face immobile and turning the rest of the cube a quarter-turn clockwise, as seen from the front. Clearly the depth-2 moves are M-conjugate. Unfortunately, they do not generate all of G*, nor even all of G. This can be seen easily by noticing that each depth-2 move is an even permutation of the edge cubies, while G includes odd permutations of edges. We can look at this a different way by observing that a depth-2 move is the same as a vanilla depth-1 move of the opposite face together with a whole-cube quarter-twist. Thus we can perform any depth-1 process using depth-2 moves, and observe the spatial orientation of the cube at the end. If we performed an odd number of depth-2 moves, then there were an odd number of whole-cube moves, so the cube cannot be in its home orientation at the end. So just what group do the depth-2 moves generate? It turns out that they generate precisely half of G*, the half that contains even edge permutations. Suppose we want to produce such a position in G*. First operate in G, simulating depth-1 moves as described above, to produce a position that is an even number of whole-cube moves from the desired one. Then use M-conjugates of the process F2 L2 D2' B2' D2' F2 L2 B2 L2 U2' F2' D2' R2' B2, which performs a whole-cube third-twist. [This was derived from identity I14-4]. Since all the even whole-cube moves are generated from third-twists, this is all you need. What can we do about generating all of G* with a set of conjugate generators? Sad to say, that is impossible. For supposing we had such generators, their cycle structures would have to be the same; in particular, they would have to have the same permutation parities. Applying an odd number of such generators would yield those parities, and applying an even number would yield even parity on every kind of cubie. But G* includes four different parity classes: Face centers Edges Corners even even even even odd odd odd even odd odd odd even so at most half of G* can be generated with a set of conjugate generators.  Date: 24 May 1983 2218-EDT From: James.Saxe@CMU-CS-A (C410JS30) To: Cube-Lovers@MIT-MC Subject: Cubists do it . . . Message-Id: <24May83.221845.JS30@CMU-CS-A> . . . in over three billion positions.  Date: 25 May 1983 1001-PDT Sender: OLE at SRI-CSL Subject: Anybody there? From: Ole at SRI-CSL (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Cc: Alan at MIT-MC Message-ID: <[SRI-CSL]25-May-83 10:01:13.OLE> Several months have gone by without a word from this list. Have we lost our interest (or have I been removed from the list)!?? What about the 5x5x5 that somebody reported having actually touched, anyone seen any on sale? Regards, |-+-+-| |O+L+E| |O+L+E| |O+L+E| |-+-+-|  Date: 25 May 1983 20:17 EDT From: David C. Plummer Subject: Anybody there? To: Ole @ SRI-CSL cc: CUBE-LOVERS @ MIT-MC, ALAN @ MIT-MC Date: 25 May 1983 1001-PDT From: Ole at SRI-CSL (Ole J. Jacobsen) Several months have gone by without a word from this list. Have we lost our interest (or have I been removed from the list)!?? I don't know that much of interest has happened. Maybe I'll find a little time some year and try a couple things. What about the 5x5x5 that somebody reported having actually touched, anyone seen any on sale? It will never be sold in mass. MANY stores are stuck with 4x4x4s that nobody will buy.  Date: Thu 26 May 83 09:27:30-PDT From: ISAACS@SRI-KL.ARPA Subject: anybody there; 5x5x5; circular To: cube-lovers@MIT-MC.ARPA There doesn't seem to be much happening with the cube, but the (at least) 3 newletters are still being published. Ideals "Rubik's Cube Newsletter" no. 3 came, marked winter, 1983. Mostly about contests, also an article on computer cubing and the calendar cube. Rubik's magazine from Hungary is also on issue 3, with article on contests, other games and toys, and a little bit of mathematics. Singmasters "Cubic Circular" is on issue 5/6 (phisically, the forth issur), and is the best of the lot mathematically. He describes and talks some about rotating rings, about Tsukuda's square (and how to solve it), generalized Hungarian rings, the Incredi-ball (equivalent to the corners of a Magic Dodecahedron without centers?? anyway, about 2*10**25 positions), rotating rings of cubes and tetrahedra (STarburst, Kaleido Puzzle, etc), some notes on the 4x4x4 (his suggested order: 2 opposite face centers, say L and R; Pair up and place the F,C,B edge pairs on the L & R faces; D corners; Pair up FD & store them, then pair BD edges, then put these edge pairs in place; check parity of U corners - if odd, apply U; 3-cycle U edges to get UR & UL edge pairs; EXAMINE, carefully, the parity of the 4 edge pieces at UF&UB - if odd, r2D2l'D2r2, which 4 cycles these edges; get all edges together and place; do U corners; do centers with 3-cycles like [[r,b],U]. phew. If people are interested, he gives slightly more information on his method, also "Useful moves", and some other strategies taken from the literature. He talks about "Evisceration", that is, the principle of Duality, that if you replace capitals with small letters and vice versa, in any move sequence, you turn the cube inside out - corners become invisable centers, edges become face centers, etc. Singmaster uses this to transfer knowledge of 3x3x3 to 4x4x4. He guesses the minimum for God's Algorithm on the 4x4x4 to be about 44 "hand moves" or 66 quarter turns. (question, from DS: how should we count moves? "r" is really 2 hand moves - Rr followed by R' - while Rr and even Rrl' could be counted as a single move.) The circular talkes about Pyraminx, and gives God's Algorithm as between 8 and 17 moves; there was some work on that done in the Journal of Recreational Mathematics (a minimum solution was presented), but I can't find the issue right off hand. Also a bunch of Pyraminx patterns are given. There is an article on the prehistory of the cube,and a little on the legal problems. Anyway, this parenthetical paragraph is way out of hand - if anybody wants more, or more organized, on the Circular, let me know.) The 5x5x5: Yes, I saw one and played with it (a very little bit) at Jerry Slocums' puzzle party. I also got an add from Meffert for one, and have sent him my money, but it hasn't come yet. Neither has my Skewb, for which I sent money half a year ago or so. What else is new? Well, nothing much. Sales in the stores are slow, and the number of puzzles available has lessened. Doug Engels rotating circles are ready for production, and his manufacturer is mainly waiting for the "right time" to release them. (These are 2 overlapping rotating circles, with 3 piece overlap and 21 pieces all together.) He is trying to decide between 2 colorings; if he choses the more difficult one, it is a fairly hard puzzle, very cube like, but harder to get the already solved pieces "out of the way". Sorry this turned out so long and rambling - I was just going to write a short note, and got carried away. Happy puzzling. -- Stan -------  Date: 26 May 1983 23:44 EDT From: Alan Bawden Subject: [DCP: 5x5 cubes] To: CUBE-LOVERS @ MIT-MC Date: Thursday, 26 May 1983 13:20-EDT From: DCP at SCRC-TENEX To: Bernard S. Greenberg cc: ALAN, cmb at SCRC-TENEX Re: 5x5 cubes Date: Thursday, 26 May 1983, 10:02-EDT From: Bernard S. Greenberg From: David C. Plummer What about the 5x5x5 that somebody reported having actually touched, anyone seen any on sale? It will never be sold in mass. What, pray tell, are the details and source of this information? As if there were anything I could do with it other than stare at it... A person at Games People Play believes Ideal will never produce a 5x5x5 commercially. He believes the only people that seriously bouhght the 4x4x4 were those that solved the 3x3x3 without a solution book. He finally said he is stuck with many 4x4x4 that he fears he won't be able to get rid of. The economics of making a 5x5x5 just don't sound very good.  Date: Tue 31 May 83 09:52:44-PDT From: ISAACS@SRI-KL.ARPA Subject: economics To: cube-lovers@MIT-MC.ARPA Meffert claimed in an early letter, that he could produce his strange products (Skewb, 5x5x5, and many others) if he had some not-to-big number of guaranteed sales, on the order, if I remember, of 1000 or so. He has a "club", that is, we paid about $10 to him up front. Then, when he is "ready" with something, he sends an order form; they cost about $20 or so. So far, he has done this with the Skewb, about 5 months ago, and the 5x5x5 recently. However, we have not actually received either of these yet. I talked to about 3 other people who ordered them and have not received them. So we don't actually know whether this system will work. We know he has produced real, working prototypes, though, since we have seen them. If they ever do come, I'll put up a message here. (Assuming I still have access to this list.) -- Stan -------  Date: 1 June 1983 1939-EDT (Wednesday) From: Dan Hoey@CMU-CS-A To: cube-lovers@MIT-MC Subject: Eccentric Slabism, Qubic, and S&LM Message-Id: <01Jun83.193917.DH51@CMU-CS-A> I don't know whether Isaacs or Singmaster know just what a bombshell was contained in the Cubic Circular. I am somewhat frightened at the possibilities. Section 1 discusses the history of metrics for N^3 puzzles and proposes a new one. Section 2 describes new symmetries of the generators of the 4^3 puzzle. Section 3 outlines a theory of symmetry and local maxima for the 4^3 puzzle. Section 4 indicates directions for further work. 1. Cutism, Slabism, and Eccentric Slabism ========================================= Let's start with the question that was posed by Stan Isaacs in his message of 26 May: what is a quarter-turn on the 4^3 or larger cube? On the N^3 puzzle, each set of N-1 parallel ``cuts'' divides the cube into N ``slabs''. There seem to be two straightforward metrics. The Slabist defines a move to be a turn of one slab with respect to the rest of the cube. The Cutist defines a move to be a turn of a connected part of the cube with respect to another connected part, the two parts being separated by a cut. [In the terminology I used on 2 September 1982, the Slabist counts ``slice moves'' while the Cutist counts ``twist moves.''] I have heretofore espoused Cutist theory. For one thing, it agrees with our current theories on the 3^3 in disallowing a turn of a center slice as a single move. This seems to be a good idea, since the current quarter-turn theory has the advantage of conjugate generators, which it would lack if we allowed center-slice moves. [This is presumably not a problem for Singmaster, who allows the squared moves, which are not conjugate.] Another reason for Cutism is that it makes it easier to equate positions that arise from a whole-cube move of the N^3. A third reason is that it makes the parity hack (see my message of 1 June 1982) easier. The last two reasons are for convenience only; the arguments can still be made in a Slabist formulation. But as I admitted, I solve the cube as a Slabist. Slabs are probably convenient because they minimize the degree of each generator. I casually dismissed this tawdry practicality until I was struck by Evisceration. In the course of my examination of Evisceration I have experienced an epiphany which converted me to Eccentric Slabism. I now define a move to be a turn of any slab except one whose interior contains the center of the cube. In other words, any slab except a center slice. At first glance, Eccentric Slabism looks like a hack, since there is an excluded slab only in the case of a puzzle of odd size. I believe that the truth is more complicated, but the explanation is partly beyond the scope of this note and partly beyond my knowledge. If you really want an answer I suggest you study tic-tac-toe. 2. Evisceration, Inflection, and Exflection =========================================== The (Eccentric) Slabist moves of the 4^3 puzzle form the 24-element set Q4={B,B',b,b',...,r'}, where upper-case refers to turning a side (an outslab move) and lower-case refers to turning the adjacent internal slab (an inslab move). We consider these moves as generators of G4, the ``Theoretical Invisible Group'' [Invisible Revenge, 9 August 1982] in which the inslabs turn the eight stomach cubies like a 2^3 puzzle. Thus two positions in G4 are equal if and only if all sixty-four pieces of the cube are in their home position and orientation. [Actually, this is not quite the Theoretical Invisible Group, since we do not equate positions that differ by a whole-cube move. I feel confident that the identification can be performed, but I will speak of the unidentified group here.] Consider the following permutations on Q4: Rotations: I=(FRBL)(F'R'B'L')(frbl)(f'r'b'l'), J=(FUBD)(F'U'B'D')(fubd)(f'u'b'd'), Reflection: R=(FB')(F'B)(RL')(R'L)(UD')(U'D)(fb')(f'b)(rl')(r'l)(ud')(u'd), Evisceration: V=(Ff)(F'f')(Bb)(B'b')(Rr)(R'r')(Ll)(L'l')(Uu)(U'u')(Dd)(D'd'), Inflection: N=(fb')(f'b)(rl')(r'l)(ud')(u'd), Exflection: X=(FB')(F'B)(RL')(R'L)(UD')(U'D). Permutations I, J, and R are familiar generators of M, the group of rotations and reflections of the cube. Singmaster introduced Evisceration, which consists of swapping each outslab with the adjacent inslab. I extend the list with Inflection and Exflection. Inflection consists in swapping each inslab with the inverse of its parallel inslab; Exflection swaps each outslab with the inverse of its parallel outslab. It is well known that M is a group of automorphisms on G4. Singmaster observed that Evisceration is also an automorphism. I observe that Inflection and Exflection are automorphisms, too. Thus M4, the 192-element group generated by is a group of automorphisms on G4. [Actually, since R=NX and X=VNV, M4=. The group M4 is also the automorphism group of the game of Qubic, or 4^3 tic-tac-toe.] I began to doubt Cutism when I noticed that M4 sometimes maps cut moves to pairs of cut moves. I went home last night wondering why this might be so. I nearly got to sleep before I realized the big news: M4 is Q4-transitive! Eccentric slabs are conjugate! 3. Symmetry and Local Maxima ============================ This section relies especially heavily on ``Symmetry and Local Maxima'' [14 December 1980; available as file "MC:ALAN;CUBE S&LM" on MIT-MC]. Following the argument in S&LM, consider the symmetry group of the Pons Asinorum (with the face-centers each half-twisted, as is customary). We already know Pons is M-symmetric; by examination, the symmetry group of Pons also contains Evisceration and Inflection. Thus Pons is M4-symmetric. The result is that Pons is a local maximum in G4. This is the first local maximum to be found in a close relative of Rubik's Revenge. It is not hard to show that we can dispense with fixing the Pons in space, and it is only slightly harder to carry out in general. Unfortunately, I see no way of showing that Pons is a local maximum if we ignore the stomach cubies without ignoring the corners. 4. Open problems ================ This is a pretty random collection of directions for further work. Some of these were posed in the text. The ones I think likely to be impossible are labeled (*). Conjecture: The automorphism group of the Eccentric Slabs of the N^3 puzzle is the same as the automorphism group of N^3 tic-tac-toe. I don't believe this has been rigorously done for any N>1. Stronger conjecture: The automorphism groups of the N^D puzzle and N^D tic-tac-toe are the same. (Hint: There are at least two definitions of the N^D puzzle. I think both work.) Extension: The relation between the automorphism groups is too amazing to be accidental. What is really going on here? Search: There is published literature on tic-tac-toe automorphisms; in particular the group of automorphisms of N^D tic-tac-toe is well known. I seem to recall seeing some horribly theoretical work, approaching the problem from the standpoint of algebraic geometry or some such. At that time I settled for scanning the results. Now I have questions that need a general treatment. If the world's leading expert on Qubic has his bibliography on line, I think there's a reference I'd appreciate. Actually, I'll take references from anybody and send the compilation to any requestors. Query: Why must slabism be eccentric? Query: Can Cutism be saved? Are cut moves conjugate in some sense? Easy extension: Equate positions that differ by whole-cube moves. Hard extensions (*): Equate positions that differ only internally. Equate positions that differ only in the permutation of like-colored face cubies. Problem: Prove that the Pons requires 12 quarter-turns in the 4^3 puzzle. Ditto for 12 qtw in the N^3 puzzle(*?). Prove or disprove for 4D qtw in the N^D puzzle (*). Problem: Find the Q4-transitive subgroups of M4, then list all the Q4-symmetric local maxima in the 4^3. Problem: Describe all symmetric local maxima of the N^3(*), or place useful conditions on them. Problem: Demonstrate an infinite class of local maxima (Ponses?). Final query: Did someone ask if Cubism was dead? Dan Hoey HOEY@CMUA.ARPA  Date: Wed 22 Jun 83 09:54:45-PDT From: ISAACS@SRI-KL.ARPA Subject: 5x5x5 and "bricks" To: cube-lovers@MIT-MC.ARPA I got my 5x5x5 today, from Meffert in Hong Kong. Whether they will appear in regular (puzzle) stores, I don't know, but I would assume they will be available from him. His address is Puzzles Club P.O.Box 31008 Causeway Bay Hong Kong. I haven't had a chance to do much with it yet; it seems well and strongly made. I have not taken it apart because I don't know how to do it safely. The corners seem to have the usual flanges. The center edges seem to have a very deep tab, perhaps to go completely through the second layer. The non-center middle pieces seem to be some kind of "caps" over the edge extensions. That's all I can tell from "peeking" in the cracks. One of the suggestions in Singmaster's "Cubic Circular" was to take a regular 3x3x3 and tape over some cracks to limit the movement. The idea is to take each pair of cubies and tape them together into one "brick", 2x1x1. If you do it right, it makes a very interesting variation. For taping, I used labels from another cube; that way the colors can be make to look right. The pattern I used is as follows: Choose a corner. It will be the only single cubie (except for a center). Make the other 2 cubies along each edge projecting from the chosen corner, into one brick (by putting tape on both cracks separating the cubies). Now, in each of the 3 faces bordering the chosen corner are 4 untaped facies in a square. Tape to make 2 adjacent bricks (don't forget the other side of the edge bricks). Note that when you have done the first face, you will have 3 parallel bricks; the other 2 faces should have their parallel bricks with the "same handedness" - If you look down from the chosen corner, the 3 faces should have rotational symmetry. (I haven't tried an irregular pattern yet; it might have different properties than this one). Anyway, all that is left is a 2x2x2 sub-cube, diagonally across from the original chosen corner. Picture that as 4 1x1x2 bricks piled with one pair crossways from the other. One brick, of course, includes the very center of the cube (invisible), and will leave the only other untaped facie - namely one center. Which one is the center doesn't matter, but I think there is a clockwise/counter-clockwise (relative to the outer handedness) choice here. When all that is done, you will have a very interesting new puzzle. Any twist must include the chosen corner! Home position is easy to recognize - it is when the chosen corner is back in its original place, and the bricks around it are in their positions (not necessarily with the correct colors together). I have not solved it yet, but it seems that the sub-cube is both hard to mix up, and easy to fix once you have. It looks as if all the interesting moves can be done with just the 3 faces surrounding the chosen corner. Have fun.. -- Stan -------  Date: 23 Jun 83 12:20:43-EDT (Thu) From: Stuart Stirling Return-Path: Subject: Singmaster's "Cubic Circular" To: cube-lovers@Mc Cc: silver.Emory@Rand-Relay Via: Emory; 23 Jun 83 23:31-PDT Where is this available? Stuart Stirling {sb1,sb6,msdc,gatech}!emory!silver Math and Computer Science silver.emory@rand-relay Emory University Atlanta, GA 30322 404/329-5637  Date: 27 June 1983 19:04 EDT From: Alan Bawden Subject: Bizarre Bricks To: CUBE-LOVERS @ MIT-MC A bizarre variant of the "bricks" described by Isaacs last week is currently sitting on the front desk of the SIPB office at MIT. Imagine that you took three cubies along any edge of a cube and welded them together (perhaps by taping over the cracks as Singmaster suggests). Now imagine that you perform the same mutilation on a second cube. Now imagine that you arrange to \share/ the same 1 x 1 x 3 cubie between the two cubes. The resulting arrangement looks like this from above: XXX XXX XXXXX XXX XXX A moment's thought should convince you that sharing the 1 x 1 x 3 cubie has done \nothing/ to increase the difficulty of the puzzle. Amazing what lengths puzzle manufacturers will go to to try and squeeze a few more bucks out of the Cube Craze.  Date: Tue 28 Jun 83 09:04:29-PDT From: ISAACS@SRI-KL.ARPA Subject: Cubic Circular address To: silver.emory@RAND-RELAY.ARPA cc: cube-lovers@MIT-MC.ARPA The address for the "Cubic Circular" is indeed that of David Singmaster: David Singmaster 87 Rodenhurst Road London, SW4 8AF ENGLAND. Hope this helps. -- Stan -------  Date: Thu 30 Jun 83 10:07:06-PDT From: ISAACS@SRI-KL.ARPA Subject: double cube To: alan@MIT-MC.ARPA cc: cube-lovers@MIT-MC.ARPA I've heard of the double cube - do you have any idea where to buy one? Another strange cube I've seen, is a normal 3x3x3 with extra cubies glued on three faces so that it looks like a 4x4x4; of course, it twists in a very lopsided manner. The main problem in making one (which I have not solved) is how to grind or cut the flanges off cubies before gluing. Without equipment, can anyone think of a good, practical way to do this? -- Stan -------  Date: 30 June 1983 20:47 EDT From: Alan Bawden Subject: [Barry Margolin: double vision] To: CUBE-LOVERS @ MIT-MC Date: 30 June 1983 16:57 edt From: Barry Margolin at MIT-MULTICS To: Alan Bawden Re: double vision Well, I got it at the Harrod's department store in London (they say you can buy ANYTHING at Harrod's). I was in Games People Play last week and didn't see it, so maybe it hasn't gotten to this country yet. Sorry I can't be of more help. barmar  Date: 21 July 1983 1450-PDT (Thursday) From: cline at AEROSPACE (Ken Cline) Subject: put me on your mailing list! To: cube-lovers at mit-mc Please include me (CLINE@AEROSPACE) in your Rubik's Cube mailing-list. My address will change in September to SCGVAXD!4CCVAX!KCLINE@CIT-VAX, but I will let you know when this occurs. Thank you, Ken Cline  Date: Monday, 21 November 1983 10:08-EST To: cube-lovers at mc From: BSG Subject: Meffert cube availability Someone asked me from afar where to get the hairy Meffert cube-like things. It was a long time ago that I ordered a five-cube from somewhere in Los Angeles. Would somebody care to repeat the name and address of that store, or some other stores? Thank you.  Date: 21 November 1983 20:30 EST From: Alan Bawden Subject: Rubik's cube cornered To: CUBE-LOVERS @ MIT-MC >From mit-eddie!genrad!decvax!dartvax!andyb Tue Oct 25 00:03:36 1983 Relay-Version: version B 2.10.1 6/24/83; site mit-vax.UUCP Path: mit-vax!mit-eddie!genrad!decvax!dartvax!andyb From: andyb@dartvax.UUCP Newsgroups: net.puzzle Subject: Rubik's cube cornered Message-ID: <307@dartvax.UUCP> Date: Tue, 25-Oct-83 00:03:36 EDT Article-I.D.: dartvax.307 Posted: Tue Oct 25 00:03:36 1983 Date-Received: Tue, 25-Oct-83 08:53:25 EDT Lines: 7 It is possible to solve the corners of the Rubik's Cube in no more than 11 turns. Over 4/5 of all possible patterns can be solved in 9 turns or fewer, but over half take a full 9 turns. Ten turns are enough to solve more than 99.9% of all scrambled states. (Only 2,644 states require 11 turns). This from L. John Kelley, solver of the Magic Pyramid.  Date: 23 November 1983 17:01 EST From: Alan Bawden Subject: [genrad!decvax!dartvax!andyb: Rubik's cube cornered] To: CUBE-LOVERS @ MIT-MC Date: 23 Nov 1983 06:56:09-EST From: genrad!decvax!dartvax!andyb at mit-eddie, Received: by decvax.UUCP (4.12/4.2) id AA26306; Wed, 23 Nov 83 03:10:45 est Date: Wed, 23 Nov 83 03:10:45 est From: decvax!dartvax!andyb Message-Id: <8311230810.AA26306@decvax.UUCP> To: decvax!genrad!mit-eddie!ALAN@MIT-MC, genrad!decvax!dartvax!andyb@MIT-EDDIE Subject: Rubik's cube cornered Cc: ALAN@MIT-MC Alan: Feel free to rebroadcast any of this discussion to the cube-lovers list. (Could you enroll me on the list, also?) Both solutions were generated by "brute force", using a fairly short PL1 program running on a VAX. The solutions are tables which express the information "if the pyramid/cube's state is X, the best move is Y" for all possible states. In that sense, they are optimal solutions, but if you wanted to know the best solution for any particular state, you would have to carry the entire table around with you. Perhaps we could publish the list :-) The Magic Pyramid is the one sold under the name "Pyraminx" -- it's the tetrahedron that rotates in only 4 planes. The cube solution is for corners only, with no reference to centers or edges. You could think of it as a complete solution to the "Pocket Cube", which is a 2x2x2 cube. Glad to help. Andy Behrens (& L. Kelley) decvax!dartvax!andyb  Date: Sat 26 Nov 83 22:35:59-EST From: Tai Subject: cube solvers To: cube-lovers@MIT-MC.ARPA cc: Jin@COLUMBIA-20.ARPA hi, are there any programs out there that solve the cube? i would like to get a copy of the source if possible. also, are there any general methods for solving the cube? thanks...tai -------  Date: 29 February 1984 23:08-EST From: Alan Bawden Subject: [CL.BOYER: A Programming Language for Group Theory (Dept. of Math)] To: CUBE-LOVERS @ MIT-MC Some of the group theory hackers amoung us might be interested in this abstract. Of course the talk has already happened, but perhaps one of you knows more about this? Date: Sun 26 Feb 84 17:06:23-CST From: Bob Boyer To: AIList Re: A Programming Language for Group Theory (Dept. of Math) [Forwarded from the UTexas-20 bboard by Laws@SRI-AI.] DEPARTMENT OF MATHEMATICS COLLOQUIUM A Programming Language for Group Theory John Cannon University of Sydney and Rutgers University Monday, February 27, 4pm The past 25 years has seen the emergence of a small but vigorous branch of group theory which is concerned with the discovery and implementation of algorithms for computing structural information about both finite and infinite groups. These techniques have now reached the stage where they are finding increasing use both in group theory research and in its applications. In order to make these techniques more generally available, I have undertaken the development of what in effect is an expert system for group theory. Major components of the system include a high-level user language (having a Pascal-like syntax) and an extensive library of group theory algorithms. The system breaks new ground in that it permits efficient computation with a range of different types of algebraic structures, sets, sequences, and mappings. Although the system has only recently been released, already it has been applied to problems in topology, algebraic number theory, geometry, graphs theory, mathematical crystalography, solid state physics, numerical analysis and computational complexity as well as to problems in group theory itself.  Received: From ti-csl.csnet by csnet-relay; 25 Jun 84 20:44 EDT Date: Mon, 25 Jun 84 14:30 CST From: Gil Gamesh To: cube-lovers@mit-mc.arpa Subject: Recent Activity? Is this mailing list still alive? Does anyone know of any recent activity in the cube world? Are there still any Cube Newsletters that anyone knows of? Has anyone ever seen any of the many puzzles I read about in places such as 'Scientific American' and 'Omni', ex. the Skewb, the Double Pyramid, the 5x5x5 cube, etc.? Dan Nichols Dnichols.ti-csl@csnet-relay  Date: Tue 26 Jun 84 11:00:18-PDT From: ISAACS@SRI-KL.ARPA Subject: cubes dead? To: cube-lovers@MIT-MC.ARPA I don't see much about cubes these days. And, as a puzzle collector, I'm still looking. Singmaster and Ideal have both discontinued their newsletters; however, the Hungarian "Rubics Magazine" still is being published. It has broadened somewhat to talk more about puzzles in general. I also found last Saturday, that there is a childrens cartoon about Rubics Cube - in fact the cube is the hero. It is truly awful. As Alan says, most of the objects mentionsed have been seen. But they are hard to get nowadays. Meffert, in Hong Kong, is silent. I ordered a dozen or so 5x5x5's more than 6 months ago, and have yet to hear anything. Or get my money back. And he never sent me any skewbs. Well, it was an exciting couple of years in the puzzle world; now, I guess, things are more normal. Stores no longer have puzzle sections, and I have to write to specialty places to find new puzzles. For me, wood puzzles are back in. I've been digging up old puzzle patents, and hope to find someone to make some of the puzzles described for me. -- Stan Isaacs (isaacs@hplabs) -------  Received: from SCRC-CONCORD by SCRC-STONY-BROOK via CHAOS with CHAOS-MAIL id 54973; Tue 26-Jun-84 10:22:30-EDT Date: Tue, 26 Jun 84 10:20 EDT From: "Bernard S. Greenberg" Subject: Recent Activity? To: ALAN@MIT-MC.ARPA cc: cube-lovers@MIT-MC.ARPA In-Reply-To: The message of 25 Jun 84 23:23-EDT from Alan Bawden Message-ID: <840626102050.3.BSG@CONCORD.SCRC.Symbolics> Date: 25 June 1984 23:23-EDT From: Alan Bawden Date: Mon, 25 Jun 84 14:30 CST From: Gil Gamesh Subject: Recent Activity? Is this mailing list still alive? Does anyone know of any recent activity in the cube world? More significantly, has anyone heard of any progress in the important theoretical problems (God's number/algorithm)?  Date: 28 June 1984 00:17-EDT From: Alan Bawden Subject: The Cube meets Massive Parallelism To: CUBE-LOVERS @ MIT-MC cc: CM-I @ MIT-MC In-reply-to: Msg of Tue 26 Jun 84 10:20 EDT from Bernard S. Greenberg Since I spend most of my time these days thinking about designing and programming massively parallel computers, it occured to me to think about applying such machines to exploring the Cube's Group. Here are some preliminary thoughts. When we say "massively parallel" we are talking at least a quarter million simple processors. This is enough processors to give all of the positions 5 or fewer quarter twists away from home their own processor. A million processors would be enough to get up to 6 Qs, but lets not push our luck. Given a machine like the MIT Connection Machine we could set up a database in which every processor representing a configuration knew the addresses ot the 12 other processors representing its 12 closest neighbors, in almost no time at all. (Processors 5 Qs away from home would have null pointers for their unrepresented 6 Q neighbors.) A conservative statement would be that operations like generating a list of all identities of length 10 or less (which has previously taken us hours to accomplish) could be done so fast that the machine could generate output faster than you could read it. Since this is all so absurdly easy, there must be ways to go beyond this to generate significant new results using this (promised) new kind of hardware. Perhaps Dave Christman, who is both a cube hacker, and a designer of algorithms for massively parallel machines, could be persuaded to devote some spare cycles to figuring out ways to brute-force the Cube using such a machine.  Date: Fri 13 Jul 84 15:24:23-PDT From: Micheal Hewett Subject: Please add me to your mailing list To: cube-lovers@MIT-MC.ARPA Thanks. Mike Hewett (HEWETT@SU-SCORE.ARPA) -------  Date: Tue 24 Jul 84 15:30:18-PDT From: Haym Hirsh Subject: God's number To: cube-lovers@MIT-MC.ARPA Can anyone verify a rumor - that some Princeton student, as an undergraduate thesis, solved the problem of how far from start one can get (i.e., the longest sequence God's algorithm would give for any position)? The number I heard was 26. Haym -------  Date: 1 Aug 1984 11:57-EDT From: Dan Hoey Subject: Pocket cube program To: cube-lovers at mit-mc The UNIX-SOURCES@BRL mailing list recently forwarded this note from Usenet. Date: Tue, 31 Jul 84 13:47:22 EDT From: news@BRL-TGR.ARPA Subject: /usr/spool/news/net/sources/397 Path: brl-tgr!seismo!hao!hplabs!zehntel!ihnp4!ihnet!eklhad From: eklhad@ihnet.UUCP (K. A. Dahlke) Subject: solve the 2x2x2 Rubix cube in a minimum number of moves Date: Mon, 30-Jul-84 10:49:48 EDT Article-I.D.: ihnet.142 Organization: AT&T Bell Labs, Naperville, IL After solving the Rubix cube 4 years ago, I turned my attention to more interesting (and more difficult) questions. How can one find the minimum path solution for an arbitrary position? How far away is the farthest positions? Is there one position diametrically opposed to start, or does it fan out into billions? Recently, I have started playing again, and have made some progress. Here is a computer program (C/unix) which solves the 2x2x2 cube in a minimum number of moves. The 2 cube is not as common as the 3 cube, but it is commercially available. If you only have a 3 cube (standard), just ignore the sides and centers, and use the corners. This effectively simulates a 2 cube. Thanks to ATT-BL for the use of their computing facilities. Later versions may come, if i am ambitious. Unfortunately, my program cannot be expanded to handle the 3 cube. Nobody has that much memory/CPU time. I will have to come up with something better. Feel free to contact me with any ideas about this subject. ---------------------------------------------------------------- The note is followed by about 1000 lines of c code that I can make available if you want it. Unfortunately, the program seems to believe that there are 870 * 729 = 634230 positions of the 2^3, while assiduous cube lovers realize the number is actually (7! / 2) * 3^6 = 2520 * 729 = 1837080 positions. The number 870 = 29 * 30 is strange. I guess it is an approximation of 6! + 5! + 4! + 3! + 2! = 872, since the code for encoding a position as an integer contains a table of those factorials. Dan  Date: 7 Aug 1984 19:24-EDT From: Dan Hoey Subject: The pocket cube: correction, calculation, and conjectures To: cube-lovers at MIT-MC Well, maybe this list is dead after all, if I can tell you there are (7!/2)(3^6) positions of the pocket cube, and have it stand for a week. The correct number is of course (7!)(3^6) = 3674160, since the generators are odd. But not being one to eat crow with a straight face, I have hacked the good hack, so I can give you the exact number P(N) of pocket cube positions exactly N quarter-turns from solved. (This was done in September 1981 for the half-twist metric; see the archives.) I have also computed the number L(N) of local maxima at each distance. These numbers are given below. N P(N) L(N) 0 1 0 1 6 0 2 27 0 3 120 0 4 534 0 5 2256 0 6 8969 0 7 33058 16 8 114149 53 9 360508 260 10 930588 1460 11 1350852 34088 12 782536 402260 13 90280 88636 14 276 276 An approach for dealing with these numbers (suggested to me by Dale Peterson) is to form the Poincare polynomial p(x) = SUM P(i) x^i i in hopes that it can be factored nicely. Unfortunately, this doesn't work out--with the exception of the obvious factor (x+1), p(x) is irreducible. I have also tried to decompose p(x) using the power (1+x)^2 series for ------------, which agrees with the first five terms of p(x) 3 - 2(1+x)^2 due to the lack of non-trivial identities. I haven't found any good ways of expressing p(x), but there may be something there. The point of all of this is that it could conceivably lead to a conjecture--or even a derivation--of God's number for the 3^3 puzzle. I might pass along another fuzzy recollection from a year and a half ago, in hopes that it is more informative than incorrect. Dale mentioned another classical method for dealing with group diameters. It seems there is a class of groups, called reflection groups, for which tight diameter bounds can be derived. A reflection group is a group of matrices with eigenvalues of plus and minus one. Some properties generalize to pseudo-reflection groups, where the eigenvalues all have complex magnitude one. We managed to construct isomorphisms between the 3^3 edge group and a reflection group, and between the corner group and a pseudo-reflection group. As I recall, he was fairly certain that the full cube group did not qualify, but that was beyond my depth. So if you think cubes are dead, remember it's not because the results are all in. Dan  Date: 20 Aug 1984 4:34-EDT From: Dan Hoey Subject: The pocket cube and corners of the full cube To: cube-lovers at mc Cc: umcp-cs!seismo!ihnp4!ihnet!eklhad at NRL-AIC Karl Dahlke, the author of the pocket cube program I mentioned on 1 August, sent me a note about the appearance of the unusual constant 870 in his program. It turns out that the program is correct, and the constant arises in an interesting way. Recall that the pocket cube has 729 orientations and 5040 permutations of the pieces. Dahlke had noticed that the ``reflections and rotations'' of a position need not be stored, since they are the same distance from start. By reflections and rotations, he means the S-conjugates, where S is the six-element symmetry group of the pocket cube with one corner fixed. It turns out that the pocket cube has 2 permutations with a six-element symmetry group, 16 permutations with a three-element symmetry group, 138 permutations with a two-element symmetry group, and 4884 permutations with a one-element symmetry group. Thus the number of permutations that are distinct up to S-conjugacy is 2 + 16/2 + 138/3 + 4884/6 = 870. This discussion of symmetry recalls a question I have meant to propose to Cube-Lovers for some time: How many positions are there in Rubik's Cube? We know from Ideal that the number is somewhat over three billion. Most cube lovers will tell you a number of about 43 quintillion. But I really don't see why we should count twelve distinct positions at one quarter-twist from solved--all twelve are essentially the same position. So the question, suitably rephrased, is of the number of positions that are distinct up to conjugacy in M, the 48-element symmetry group of the cube. I think this is an interesting question, but I don't see any particularly easy way of answering it. My best guess is that it involves a case-by-case analysis of the 98 subgroups of M, or at least the 33 conjugacy classes of those subgroups. In ``Symmetry and Local Maxima'', Jim Saxe and I examined five of the classes, which we called M, C, AM, H, and T. Even finding the numbers for the pocket cube is a little tricky. If we limit ourselves to symmetry in S, I believe the pocket cube has 2 positions with a six-element symmetry group, 160 positions with a three-element symmetry group, 3882 positions with a two-element symmetry group, and 3670116 positions with a one-element symmetry group, for 613062 positions distinct up to S-conjugacy. But the numbers for M-conjugacy are still elusive; I am not even sure how to deal with factoring out whole-cube moves in the analysis. I hope to find time to write a program for it. I expanded my pocket cube program to deal with the corner group of Rubik's cube. This group is 24 times as large as the group of the pocket cube, having 3^7 * 8! = 88179840 elements. The number of elements P(N) and local maxima L(N) at each (quarter-twist) distance N from solved are given below. N P(N) L(N) 0 1 0 1 12 0 2 114 0 3 924 0 4 6539 0 5 39528 0 6 199926 114 7 806136 600 8 2761740 17916 9 8656152 10200 10 22334112 35040 11 32420448 818112 12 18780864 9654240 13 2166720 2127264 14 6624 6624 The alert reader will notice that rows 10 through 14 contain values exactly 24 times as large as those for the pocket cube. This is not surprising, given that the groups are identical except for the position of the entire assembly in space, and each generator of the corner cube is identical to the inverse of the corresponding generator for the opposite face except for the whole-cube position. Thus when solving a corner-cube position at 10 qtw or more from solved, it can be solved as a pocket cube, making the choice between opposite faces in such a way that the whole-cube position comes out right with no extra moves. Dan  Date: 22 Aug 1984 18:06-EDT From: Dan Hoey Subject: An outer automorphism of the cube group To: cube-lovers at mit-mc If you have a Rubik's cube where all the edges flip on each quarter-turn, you can solve it by using bifocals when it's odd. I noticed this while drawing a Hasse diagram of the subgroups of M. It turns out that M has a similar automorphism, where the odd elements are reflected through the center of the cube. If anyone wants the Hasse diagram, I can send it--it takes about 30 minutes to draw in the lines, for which directions are included. If you know whether there are other outer automorphisms of M, please let me know. Dan  Date: Fri, 7 Sep 84 9:16:44 EDT From: Michael Frishkopf Subject: list request To: CUBE-LOVERS@mit-mc.arpa Cc: mfrishkopf@BBNCCY.ARPA Dear sir/ms: Please place me on your arpa mailing list: mfrishkopf@bbnccy Thank you, Michael Frishkopf  Received: by gyre.ARPA (4.12/4.7) id AA09303; Mon, 1 Oct 84 21:10:44 edt Date: Mon, 1 Oct 84 21:10:44 edt From: Rehmi Post Message-Id: <8410020110.AA09303@gyre.ARPA> To: cube-lovers@mc Subject: Archives Does anyone have copies (partial or otherwise) of the cube-lovers mailing list? I need it as a reference generator/thought provoker, and if I could ftp it all from someone, it'd be nice... - rehmi at maryland  Date: 5 October 1984 17:07-EDT From: Alan Bawden Subject: Moleculon wins a round! To: CUBE-LOVERS @ MIT-MC From today's Boston Globe: DOVER, Del. --- A massachusetts firm that claimed it had the first patent on the Rubik's cube puzzle has won a patent infringement suit against CBS Inc. and its subsidiary Ideal Toy Corp., which manufactures the puzzle. US District Court Judge Walter Stapleton issued the ruling Tuesday, validating the patent held by Moleculon Research Corp. of Cambridge, Mass. The ruling said CBS infringed on the patent with its various Rubik's cube products. Moleculon's attorney, Robert Perry, said if no appeal is filed, the case would continue to a second trial to determine monetary damages. Moleculon is seeking $60 million in damages and an undetermined amount in profits from Rubik's cube and related sales.  Received: from CheninBlanc.ms by ArpaGateway.ms ; 08 OCT 84 08:57:05 PDT Date: 8 Oct 84 08:58:06 PDT (Monday) From: Lynn.es@XEROX.ARPA Subject: Re: Moleculon wins a round! In-reply-to: ALAN's message of 5 Oct 84 18:01 EDT To: Cube-Lovers@MIT-MC.ARPA cc: Lynn.es@XEROX.ARPA A similar article in the LA Times went on to say that the fellow at Moleculon had patented a 2x2 cube held together with magnets, but mention was made in the patent of larger cubes and mechanical linkages. How specific the mentions were is not clear. He was also quoted to the effect that they do not dispute that Rubik invented the present cube, but it infringes on their patent. Moleculon apparently offered to sell rights to their toy to Ideal before Rubik and were turned down. /Don Lynn  Received: from SCRC-DUPAGE by SCRC-STONY-BROOK via CHAOS with CHAOS-MAIL id 185428; Tue 26-Feb-85 11:09:54-EST Date: Tue, 26 Feb 85 11:00 EST From: Richard Pavelle Subject: cube-freak To: cube-lovers@MIT-MC.ARPA cc: rp@SCRC-QUABBIN.ARPA Message-ID: <850226110043.5.RP@DUPAGE.SCRC.Symbolics.COM> I received a letter from Georges Helm 76, Bomicht 4936 Bascharage Luxembourg who seems to have outdone most of us. He has collected 14000 pages of material on the 3x3 and its variations and has composed a bibliography on them. He would like to communicate with anyone who is still interested and is willing to swap material.  Received: from Gamay.ms by ArpaGateway.ms ; 01 MAR 85 08:55:07 PST Date: 1 Mar 85 08:54:47 PST (Friday) From: Hoffman.es@XEROX.ARPA Subject: 5-by-5-by-5 To: Cube-Lovers@MIT-MC.ARPA cc: Hoffman.es@XEROX.ARPA Anyone have a written solution algorithm for the 5-by-5-by-5 cube? I got mine a year and a half or so ago from Meffert in Hong Kong by mail order. After a couple of weeks of just trying out "pretty patterns" on it, I finally got up the nerve to mess it up. Well, of course, it's never been altogether solved since. (I was never a great cube wizard.) It sits on display as the base of my cube tower, with those few nagging cubies out of place. I haven't touched it in many months. --Rodney Hoffman  Received: from SCRC-NEPONSET by SCRC-VALLECITO via CHAOS with CHAOS-MAIL id 371; Fri 1-Mar-85 17:16:32-EST Date: Fri, 1 Mar 85 17:16 EST From: David C. Plummer in disguise Subject: 5-by-5-by-5 To: Hoffman.es@XEROX.ARPA, Cube-Lovers@MIT-MC.ARPA In-Reply-To: The message of 1 Mar 85 11:54-EST from Hoffman.es@XEROX.ARPA Message-ID: <850301171626.2.NFEP@NEPONSET.SCRC.Symbolics.COM> Date: 1 Mar 85 08:54:47 PST (Friday) From: Hoffman.es@XEROX.ARPA Anyone have a written solution algorithm for the 5-by-5-by-5 cube? I got mine a year and a half or so ago from Meffert in Hong Kong by mail order. After a couple of weeks of just trying out "pretty patterns" on it, I finally got up the nerve to mess it up. Well, of course, it's never been altogether solved since. (I was never a great cube wizard.) It sits on display as the base of my cube tower, with those few nagging cubies out of place. I haven't touched it in many months. It took me about half an hour to solve the 4x4x4 by generalizing techniques for the 3x3x3. I think many of us convinced ourselves that the 5x5x5 is just a similar extension. The generalization is one of those "Aha!" insights. Unfortunately for me, I've never seen a 5x5x5 to prove myself.  Received: from WISCVM.ARPA by MIT-MC.ARPA; 11 MAR 85 23:00:01 EST Received: from (ENG130)BOSTONU.BITNET by WISCVM.ARPA on 03/11/85 at 21:58:19 CST Date: 11 Mar 85 22:51 EST From: Subject: Please remove me from this list To: cube-lovers@mit-mc ----- Please take me off this mailing list. Thanks alot. -----  Received: from csnet-relay by MIT-MC.ARPA; 3 APR 85 17:23:55 EST Received: from ti-csl by csnet-relay.csnet id aa14999; 3 Apr 85 17:00 EST Date: 3 Apr 1985 1028-CST From: Dan Subject: articles help To: cube-lovers@mit-mc.ARPA Received: from csl60 by ti-csl; Wed, 3 Apr 85 13:41 CST I would very much like to obtain a copy of the following articles. If anyone can send me a copy or give me more information about where and how to obtain them, I will really appreciate it. Thanks. Eidswick, Jack - How to solve the nxnxn cube. Mathematics and Statistics Dept - Univ of Nebraska 1982 Kamack, HJ & T R Keane - The Rubik Tesseract - Unpublished manuscript 1982 Kim, Scott E - The Impossible Skew Quadrilateral: A Four-Dimensional Optical Illusion - Proceedings of 1978 AAAS Symposium on Hypergraphics: Visualizing Complex Relationships in Arts & Sciences Marx, George, Eva Gajzago, & Peter Gnadig - The Universe of Rubik's Cube European Journal of Physics 3 (1982) pp 34-43 Daniel Nichols 1721 E. Frankford Rd Apt. 1514 Carrollton, TX 75007 ARPA: dnichols%ti-csl@csnet-relay CSNET: dnichols@ti-csl UUCP: {ut-sally,smu,texsun,rice}!waltz!dnichols VOICE: 214-492-3275 -------   Received: from csnet-relay by MIT-MC.ARPA; 24 APR 85 18:42:16 EST Received: from ti-csl by csnet-relay.csnet id ab16901; 24 Apr 85 14:45 EST Date: 24 Apr 1985 1051-CST From: Dan Subject: information requests To: cube-lovers@mit-mc.ARPA Received: from csl60 by ti-csl; Wed, 24 Apr 85 12:49 CST 1. Does anyone know of anywhere I can obtain any of the following puzzles: skewb pyraminx crystal pyraminx ball pyraminx ultimate double pyramid 2. I would very much like to obtain copies of Singmaster's Cubic Circular. I have the first 4 issues, but saw a reference to there being 8 volumes. I would be glad to pay for copying and postage. Can anyone help? 3. Does anyone know the status of Meffert's Puzzle Club? 4. Do Singmaster or Doug Hofstadter have net addresses? Thanks for any help. Daniel Nichols 1721 E. Frankford Rd Apt. 1514 Carrollton, TX 75007 ARPA: dnichols%ti-csl@csnet-relay CSNET: dnichols@ti-csl UUCP: {ut-sally,smu,texsun,rice}!waltz!dnichols VOICE: 214-492-3275 -------  Received: from CISL-SERVICE-MULTICS.ARPA by MIT-MC.ARPA; 30 APR 85 09:51:49 EDT Received: FROM HIS-PHOENIX-MULTICS.ARPA BY CISL-SERVICE-MULTICS.ARPA WITH dial; 30 APR 1985 09:47:08 EDT Acknowledge-To: "Craig L. Senft" Date: Tue, 30 Apr 85 06:44 MST From: "Craig L. Senft" Subject: Re: information requests To: Alan Bawden cc: Cube-Lovers@MIT-MC.ARPA In-Reply-To: Message of 30 Apr 85 01:09 MST from "Alan Bawden" Message-ID: <850430134407.956530@HIS-PHOENIX-MULTICS.ARPA> Please remove me from Cube-Lovers mailing list. Thank you, ==cls==  Date: Tue, 30 Apr 85 17:01:09 EDT From: Alan Bawden Subject: information requests To: Senft.Multics%PCO-Multics@MIT-MULTICS cc: CUBE-LOVERS@MIT-MC In-reply-to: Msg of Tue 30 Apr 85 06:44 MST from Craig L. Senft Message-ID: <[MIT-MC].477537.850430.ALAN> Date: Tue, 30 Apr 85 06:44 MST From: Craig L. Senft Please remove me from Cube-Lovers mailing list. Thank you, Removed.  Received: from Xerox.ARPA by MIT-MC.ARPA 31 May 85 18:33:35 EST Received: from CheninBlanc.ms by ArpaGateway.ms ; 31 MAY 85 15:31:50 PDT Date: 31 May 85 15:31:46 PDT (Friday) From: Hoffman.ES@Xerox.ARPA Subject: Re: information requests In-reply-to: <[MIT-MC].476352.850430.ALAN> To: Alan Bawden cc: Cube-Lovers@MIT-MC.ARPA Message-ID: <850531-153150-1581@Xerox> Dan, Did you learn anything about Skewb availability or the status of Meffert's club and company? In January, 1983, I sent Meffert a money order for a skewb and followed up twice with letters in Fall '83 and January '84. I never heard a thing. However, in August '83 (after Stan Isaacs said it worked for him), I sent Meffert money for the 5-by-5-by-5, and it arrived very promptly! I would LOVE a Skewb. Have you even seen one? I do have a pyraminx ball and a small double pyramid. I don't recall what stores I found those in a couple of years ago, but they were local toy or department stores. Doug Hofstadter usually is connected to the net, but I haven't corresponded with him in a year and a half, and I know he's moved since then. Isn't he visiting UMass just now? If you ever get hold of Scott Kim, he usually knows how to reach DRH. --Rodney  Received: from Xerox.ARPA by MIT-MC.ARPA 31 May 85 18:36:52 EST Received: from CheninBlanc.ms by ArpaGateway.ms ; 31 MAY 85 15:34:09 PDT Date: 31 May 85 15:34:05 PDT (Friday) From: Hoffman.ES@Xerox.ARPA Subject: Re: information requests In-reply-to: Your message of 24 Apr 1985 1051-CST To: Dan cc: Cube-Lovers@MIT-MC.ARPA Message-ID: <850531-153409-1583@Xerox> Dan, Did you learn anything about Skewb availability or the status of Meffert's club and company? In January, 1983, I sent Meffert a money order for a skewb and followed up twice with letters in Fall '83 and January '84. I never heard a thing. However, in August '83 (after Stan Isaacs said it worked for him), I sent Meffert money for the 5-by-5-by-5, and it arrived very promptly! I would LOVE a Skewb. Have you even seen one? I do have a pyraminx ball and a small double pyramid. I don't recall what stores I found those in a couple of years ago, but they were local toy or department stores. Doug Hofstadter usually is connected to the net, but I haven't corresponded with him in a year and a half, and I know he's moved since then. Isn't he visiting UMass just now? If you ever get hold of Scott Kim, he usually knows how to reach DRH. --Rodney  Date: Mon, 15 Jul 85 16:02:41 EDT From: Alan Bawden Subject: [alice!amo: Cube-lovers] To: CUBE-LOVERS@MIT-MC.ARPA Message-ID: <[MIT-MC.ARPA].576851.850715.ALAN> Date: Mon, 15 Jul 85 05:54:21 pdt From: alice!amo at Berkeley To: alan Re: Cube-lovers You might like to post the following message to the Cube-lovers list: One of the best sources for various "twisting puzzles" (and purportedly the only remaining supplier of the 5x5x5 cubes, which he is selling for $ 15 plus postage) is Dr. Christoph Bandelow Haarholzer Strasse 13 4630 Bochum-Stiepel West Germany phone: 0234-797794 I have just obtained two 5x5x5 cubes from him.  Received: from mitre-bedford by MIT-MC.ARPA 30 Jul 85 12:50:06 EDT Date: Tuesday, 30 Jul 1985 12:47-EDT From: pes@Mitre-Bedford To: ALAN@MIT-MC.ARPA Cc: Cube-Lovers@MIT-MC Subject: Re: The elusive 5x5x5 In-reply-to: Your message of Monday, 22 Jul 1985 07:05-EDT. <[MIT-MC.ARPA].583932.850722.ALAN> How should I go about buying a 5 cubed cube? Can I just mail 20 or so American dollars to Dr. Bandelow in W. Germany? Would a personal check be acceptable? Thanks for any info. "Just because you're a cube lover doesn't mean you're square" -Paul Silvey pes@mitre-bedford.arpa The MITRE Corporation  Received: from Xerox.ARPA by MIT-MC.ARPA 18 Sep 85 18:42:54 EDT Received: from CheninBlanc.ms by ArpaGateway.ms ; 18 SEP 85 15:38:16 PDT Date: 18 Sep 85 15:35:09 PDT (Wednesday) From: Hoffman.es@Xerox.ARPA Subject: Puzzlers Club resurrected? To: Cube-Lovers@MIT-MC.ARPA cc: Hoffman.es@Xerox.ARPA Message-ID: <850918-153816-1795@Xerox> This came Tuesday (spelling and grammar as you see it). See my comments at the end. Dear Sirs, It has come to my attention that apparently quite a lot of the puzzles we sent to Puzzlers Club members were not received by them. Upon careful investigation of this matter, it now appears that they were confiscated by the Customs who must have thought at the time that they were Taiwan copies. As we did send the goods and paid the Airmail postage and then had so many claims to send the goods a second and third time, the Puzzlers Club suffered a very big loss and consequently had to closed down. Due to the suppressed Puzzle market, we are not producing new Puzzles at this time. However, we will be releasing later this year the "I.Q. DIE" which is the Skewb with Die markings, and a Game called 'King/Ace'. It is the Pyraminx without the 4 apexes, decorated in the 4 Card Suits, the Game is similar to Back-Jack. The "I.Q. DIE" and 'King/Ace' including instructions are now available to Ex-Puzzlers Club members and collectors at US$25 each including registered Airmail postage. Also a few 5 x 5 cubes and Regular Skewbs, Timber finished Pyraminx and Immposi-ball are still available as per price list below, including registered Airmail postage. A copy of the registered receipt will be sent by separate airmail. Stocks, however are limited, as there will be no additional production for several years at least. Please let me know if you wish some of the above items. I do apologize if you were one of the unfortunate ones who were affected by the customs action. However if you would like to send us your new order with payment, I personally guarantee that you will get the goods this time. (The customs have been notified that we are the legal copyright holder.) Once again my apologizes for any inconvenience caused. With warm regards, Mr Uwe Meffert 1985 PRICE-LIST Descriptions Item No. Unit Price 5 x 5 x 5 cube CE8426 US$25/Air Skewb CE8431 US$15/Air Pyraminx CE8423-24 US$20/Air Timber finished Pyraminx US$20/Air Impossi-Ball CE8429 US$29/Air 3D Nought's + Crosses CE8434 US$ 9/Air *All plus US$1 for registered mail. Please rush me with Item No...............Amount:............... PRICEWELL (FAR EAST) LIMITED Excellente Commercial Bldg., (Fifteenth Floor) 456 Jaffe Road Hong Kong. Postal Address: P.O. Box 31008 Causeway Bay, Hong Kong. Tel.: 5-8939944 Cable: "REFOUNDEL" Telex: 75600 REF HX ------------------------------------------------------ It was accompanied by one color page (sloppily torn out of a brochure) containing pictures of the puzzles. Well, I'm one of those who got burned, though I never was a member of the Puzzlers Club. In January 1983, I sent Meffert a Cashier's Check for a Skewb. I followed up by letter late that Spring. In Summer '83, I paid for and received very promptly a 5x5x5 cube from him. In January 1984, I wrote a second letter about the missing Skewb. I never got the Skewb nor any replies to my letters until this arrived! So now it's all the (U.S.) government's fault. Hmph. (Maybe I should attend those occasional Customs sales I've heard about!) I'm just a Skewb-hungry sucker, for I'm going to try again. But I'm not very reassured by his "personal guarantee". What good will that do me? The man doesn't even answer his mail. Oh, well. Order at your own risk. --Rodney Hoffman  Date: Wed, 18 Sep 85 21:38:00 EDT From: Alan Bawden Subject: Administrivia To: CUBE-LOVERS@MIT-MC.ARPA Message-ID: <[MIT-MC.ARPA].650223.850918.ALAN> Several years ago, when Cubes were a big rage, the great volume of mail sent to this mailing list forced us into making it into a digest. For most of the last few years the list has been fairly low-volume, nevertheless I have refrained from converting the list back to direct-distribution because some of the messages sent directly to Cube-Lovers were really add-me requests. Now I find that I am experiencing a new phenomena: There is a growing tendency for people to address replies to messages I forward to Cube-Lovers to -me-, rather than to the original sender. This is a bit of a pain for me, so as of right now, I am converting Cube-Lovers back to direct-distribution. Let me take this opportunity to remind you of two bits of administrative trivia: 1. Please continue to send all administrative requests (like if you want to get off of the list) to Cube-Lovers-Request@MIT-MC. 2. Old cube-lovers mail is archived in the following places: MC:ALAN;CUBE MAIL0 ;oldest mail in forward order MC:ALAN;CUBE MAIL1 ;next oldest mail in forward order MC:ALAN;CUBE MAIL2 ;more of same MC:ALAN;CUBE MAIL3 ;still more of same MC:ALAN;CUBE MAIL ;recent mail in reverse order Files can be FTP'd from MIT-MC without an account.  Date: Wed, 18 Sep 85 22:18:28 EDT From: Alan Bawden Subject: Puzzlers Club resurrected? & Administrivia To: CUBE-LOVERS@MIT-MC.ARPA Message-ID: <[MIT-MC.ARPA].650261.850918.ALAN> Date: 18 Sep 85 15:35:09 PDT (Wednesday) From: Hoffman.es@Xerox.ARPA Subject: Puzzlers Club resurrected? To: Cube-Lovers@MIT-MC.ARPA cc: Hoffman.es@Xerox.ARPA This came Tuesday (spelling and grammar as you see it). See my comments at the end. Dear Sirs, It has come to my attention that apparently quite a lot of the puzzles we sent to Puzzlers Club members were not received by them. Upon careful investigation of this matter, it now appears that they were confiscated by the Customs who must have thought at the time that they were Taiwan copies. As we did send the goods and paid the Airmail postage and then had so many claims to send the goods a second and third time, the Puzzlers Club suffered a very big loss and consequently had to closed down. Due to the suppressed Puzzle market, we are not producing new Puzzles at this time. However, we will be releasing later this year the "I.Q. DIE" which is the Skewb with Die markings, and a Game called 'King/Ace'. It is the Pyraminx without the 4 apexes, decorated in the 4 Card Suits, the Game is similar to Back-Jack. The "I.Q. DIE" and 'King/Ace' including instructions are now available to Ex-Puzzlers Club members and collectors at US$25 each including registered Airmail postage. Also a few 5 x 5 cubes and Regular Skewbs, Timber finished Pyraminx and Immposi-ball are still available as per price list below, including registered Airmail postage. A copy of the registered receipt will be sent by separate airmail. Stocks, however are limited, as there will be no additional production for several years at least. Please let me know if you wish some of the above items. I do apologize if you were one of the unfortunate ones who were affected by the customs action. However if you would like to send us your new order with payment, I personally guarantee that you will get the goods this time. (The customs have been notified that we are the legal copyright holder.) Once again my apologizes for any inconvenience caused. With warm regards, Mr Uwe Meffert 1985 PRICE-LIST Descriptions Item No. Unit Price 5 x 5 x 5 cube CE8426 US$25/Air Skewb CE8431 US$15/Air Pyraminx CE8423-24 US$20/Air Timber finished Pyraminx US$20/Air Impossi-Ball CE8429 US$29/Air 3D Nought's + Crosses CE8434 US$ 9/Air *All plus US$1 for registered mail. Please rush me with Item No...............Amount:............... PRICEWELL (FAR EAST) LIMITED Excellente Commercial Bldg., (Fifteenth Floor) 456 Jaffe Road Hong Kong. Postal Address: P.O. Box 31008 Causeway Bay, Hong Kong. Tel.: 5-8939944 Cable: "REFOUNDEL" Telex: 75600 REF HX ------------------------------------------------------ It was accompanied by one color page (sloppily torn out of a brochure) containing pictures of the puzzles. Well, I'm one of those who got burned, though I never was a member of the Puzzlers Club. In January 1983, I sent Meffert a Cashier's Check for a Skewb. I followed up by letter late that Spring. In Summer '83, I paid for and received very promptly a 5x5x5 cube from him. In January 1984, I wrote a second letter about the missing Skewb. I never got the Skewb nor any replies to my letters until this arrived! So now it's all the (U.S.) government's fault. Hmph. (Maybe I should attend those occasional Customs sales I've heard about!) I'm just a Skewb-hungry sucker, for I'm going to try again. But I'm not very reassured by his "personal guarantee". What good will that do me? The man doesn't even answer his mail. Oh, well. Order at your own risk. --Rodney Hoffman ------- Date: Wed, 18 Sep 85 21:38:00 EDT From: Alan Bawden Subject: Administrivia To: CUBE-LOVERS@MIT-MC.ARPA Several years ago, when Cubes were a big rage, the great volume of mail sent to this mailing list forced us into making it into a digest. For most of the last few years the list has been fairly low-volume, nevertheless I have refrained from converting the list back to direct-distribution because some of the messages sent directly to Cube-Lovers were really add-me requests. Now I find that I am experiencing a new phenomena: There is a growing tendency for people to address replies to messages I forward to Cube-Lovers to -me-, rather than to the original sender. This is a bit of a pain for me, so as of right now, I am converting Cube-Lovers back to direct-distribution. Let me take this opportunity to remind you of two bits of administrative trivia: 1. Please continue to send all administrative requests (like if you want to get off of the list) to Cube-Lovers-Request@MIT-MC. 2. Old cube-lovers mail is archived in the following places: MC:ALAN;CUBE MAIL0 ;oldest mail in forward order MC:ALAN;CUBE MAIL1 ;next oldest mail in forward order MC:ALAN;CUBE MAIL2 ;more of same MC:ALAN;CUBE MAIL3 ;still more of same MC:ALAN;CUBE MAIL ;recent mail in reverse order Files can be FTP'd from MIT-MC without an account. -------  Date: Wed, 25 Sep 85 10:14:28 EDT From: Alan Bawden Subject: Well, it kept me entertained for an evening. To: CUBE-LOVERS@MIT-MC.ARPA Message-ID: <[MIT-MC.ARPA].658952.850925.ALAN> Dewdney's column in this month's Scientific American presents a puzzle which he claims to be comparable in difficulty to a Rubik's cube. (Like he claims there are people who can do the cube but haven't done this thing.) Interested Cube-Hackers might find it diverting to give it a try. It didn't take me long to devise a sufficient set of tools for solving it using only pencil-and-paper. Personally I think it was significantly easier than a cube, but perhaps it is harder than the majority of other permutation puzzles I have gotten my hands on in the last few years. (But perhaps not. I did this without having a physical model of it in my hands, so perhaps that has caused me to overlook something.) You know, I don't recall ever hearing anybody speculate about just what makes a permutation puzzle interesting and/or difficult. I guess the group has to be large and have a large diameter, and there should be a scarcity of short identities...  Date: Thu, 10 Oct 85 20:47:33 EDT From: Alan Bawden Subject: Has anybody heard of this? To: CUBE-LOVERS@MIT-MC.ARPA Message-ID: <[MIT-MC.ARPA].675661.851010.ALAN> Date: Thu 10 Oct 85 16:21:40-PDT From: Haym Hirsh ... 3. (forward to cube-lovers if necessary) I hear there is a textbook on group theory using Rubik's Cube for all its examples (or some such gimmick). Have you heard anything about it? ... Not I. Anyone else? I'd be interested in seeing this myself.  Received: from Xerox.ARPA by MIT-MC.ARPA 15 Oct 85 01:27:00 EDT Received: from CheninBlanc.ms by ArpaGateway.ms ; 14 OCT 85 22:24:05 PDT Date: 14 Oct 85 22:23:48 PDT (Monday) From: Hoffman.es@Xerox.ARPA Subject: Skewbs To: Cube-Lovers@MIT-MC.ARPA cc: Hoffman.es@Xerox.ARPA Message-ID: <851014-222405-4931@Xerox> Hey, it worked! On Sept. 18, I sent an order for a skewb with a cashier's check for US$16 to Meffert in Hong Kong, in response to his apology/order form. I included a cover letter mentioning that I was one of those who had never received an earlier order (that is, my skewb order of January 1983). On Wednesday, Oct. 9, I received the promised airmail letter containing the registered mail receipt showing the package was mailed Oct. 2, and on Saturday, Oct. 12, I received a package containing TWO skewbs! The thing seems rather easy, I think, but it's great fun because it's so weird -- so odd looking in mid-turn, so awkward to manipulate quickly, so difficult to keep your bearings (no face centers that stay put to navigate by). I haven't fully explored it, but I've found a few handy "macros" and identities that seem to get me out of most trouble. Anyway, if you have wanted a skewb ever since first seeing the drawings of them in Hofstadter's "Scientific American" column, go ahead and try Meffert. In case you've lost my note of Sept 18, here's the relevant portion again: 1985 PRICE-LIST Descriptions Item No. Unit Price 5 x 5 x 5 cube CE8426 US$25/Air Skewb CE8431 US$15/Air Pyraminx CE8423-24 US$20/Air Timber finished Pyraminx US$20/Air Impossi-Ball CE8429 US$29/Air 3D Nought's + Crosses CE8434 US$ 9/Air *All plus US$1 for registered mail. Please rush me with Item No...............Amount:............... PRICEWELL (FAR EAST) LIMITED P.O. Box 31008 Causeway Bay Hong Kong --Rodney Hoffman  Received: from CSNET-RELAY.ARPA by MIT-MC.ARPA 17 Oct 85 12:49:05 EDT Received: from ti-csl by csnet-relay.csnet id ab07525; 17 Oct 85 12:46 EDT Date: 17 Oct 1985 0923-CDT From: Dan Subject: group theory book To: cube-lovers@mit-mc.arpa Received: from csl60 by ti-csl; Thu, 17 Oct 85 09:40 CST 10-Oct-85 20:47:33-EDT,00000540;000000000001 Date: Thu, 10 Oct 85 20:47:33 EDT From: Alan Bawden Subject: Has anybody heard of this? To: CUBE-LOVERS%mit-mc.arpa@CSNET-RELAY Date: Thu 10 Oct 85 16:21:40-PDT From: Haym Hirsh ... 3. (forward to cube-lovers if necessary) I hear there is a textbook on group theory using Rubik's Cube for all its examples (or some such gimmick). Have you heard anything about it? ... Not I. Anyone else? I'd be interested in seeing this myself. There was a book published a few years ago which had one chapter dealing with the cube. Groups and Geometry - Peter Neumann, Gabrielle A. Stoy, & E. C. Thompson The Mathematical Institute - Oxford University The last chapter is "The Group Theory of the Hungarian Magic Cube". Dan Nichols -------  Received: from CSNET-RELAY.ARPA by MIT-MC.ARPA 17 Oct 85 12:49:05 EDT Received: from ti-csl by csnet-relay.csnet id ab07525; 17 Oct 85 12:46 EDT Date: 17 Oct 1985 0923-CDT From: Dan Subject: group theory book To: cube-lovers@mit-mc.arpa Received: from csl60 by ti-csl; Thu, 17 Oct 85 09:40 CST 10-Oct-85 20:47:33-EDT,00000540;000000000001 Date: Thu, 10 Oct 85 20:47:33 EDT From: Alan Bawden Subject: Has anybody heard of this? To: CUBE-LOVERS%mit-mc.arpa@CSNET-RELAY Date: Thu 10 Oct 85 16:21:40-PDT From: Haym Hirsh ... 3. (forward to cube-lovers if necessary) I hear there is a textbook on group theory using Rubik's Cube for all its examples (or some such gimmick). Have you heard anything about it? ... Not I. Anyone else? I'd be interested in seeing this myself. There was a book published a few years ago which had one chapter dealing with the cube. Groups and Geometry - Peter Neumann, Gabrielle A. Stoy, & E. C. Thompson The Mathematical Institute - Oxford University The last chapter is "The Group Theory of the Hungarian Magic Cube". Dan Nichols -------  Received: from Xerox.ARPA by MIT-MC.ARPA 25 Oct 85 12:14:59 EDT Received: from CheninBlanc.ms by ArpaGateway.ms ; 25 OCT 85 09:14:40 PDT Date: 25 Oct 85 09:14:33 PDT (Friday) From: Hoffman.es@Xerox.ARPA Subject: Rubik's Revenge: The Group Theoretical Solution To: Cube-Lovers@MIT-MC.ARPA cc: Hoffman.es@Xerox.ARPA Message-ID: <851025-091440-1219@Xerox> I only yesterday came across this from a couple of months ago: "Rubik's Revenge: The Group Theoretical Solution" by M.E. Larsen, in the 'American Mathematics Monthly', Vol. 92, No. 6, 1985, pages 381-390.  Received: from ddn2 by MIT-MC.ARPA 5 Nov 85 11:36:09 EST Date: 5 Nov 85 11:04 EST From: mccliman @ DDN2.ARPA Subject: mailing list To: cube-lovers @ mit-mc.arpa CC: mccliman @ DDN2.ARPA I recently discovered your mailing list involving the CUBE, and would very much like to be added to it. Thanks. 'mcclimans at ddn2'  Received: from ngp.UTEXAS.EDU by MIT-MC.ARPA 14 Nov 85 18:06:20 EST Date: Thu, 14 Nov 85 14:27:38 cst From: jknox@ngp.UTEXAS.EDU (John W. Knox) Posted-Date: Thu, 14 Nov 85 14:27:38 cst Message-Id: <8511142027.AA02634@ngp.UTEXAS.EDU> Received: by ngp.UTEXAS.EDU (4.22/4.22) id AA02634; Thu, 14 Nov 85 14:27:38 cst To: cube-lovers@mit-mc.ARPA, mccliman@DDN2.ARPA Subject: Re: mailing list Cc: mccliman@DDN2.ARPAn q  Received: from CSNET-RELAY.ARPA by MIT-MC.ARPA 15 Nov 85 02:48:31 EST Received: from ti-csl by csnet-relay.csnet id a006258; 15 Nov 85 2:31 EST Date: 14 Nov 1985 1008-CST From: Dan Subject: new article To: cube-lovers@mit-mc.arpa Received: from csl60 by ti-csl; Thu, 14 Nov 85 10:46 CST A new article on the cube has appeared. "Rubik's Groups" by Edward Turner and Karen Gold in The American Mathematical Monthly v92n9 November 1985 -------  Received: from Xerox.ARPA by MIT-MC.ARPA 11 Dec 85 14:44:07 EST Received: from CheninBlanc.ms by ArpaGateway.ms ; 11 DEC 85 10:35:51 PST Date: 11 Dec 85 10:35:16 PST (Wednesday) From: Hoffman.es@Xerox.ARPA Subject: Gift packs To: Cube-Lovers@MIT-MC.ARPA cc: Hoffman.es@Xerox.ARPA Message-ID: <851211-103551-1873@Xerox> I never really asked to be, but I seem to be a member of Meffert's Puzzler's Club. The latest mailing offers several individual items plus a Christmas special: 9 skewbs plus 9 5x5x5 cubes for US$99 registered airmail or $US82 registered surface mail. It says the individual prices are $25 for the 5x5x5 and $15 for the Skewb, so this is said to be a $360 value. If you're like me, you're the only cube-lover among those you exchange gifts with. But if you happen to have several cube-lovers on your list, maybe this can solve your shopping worries. The offer is "limited to December 15, 1985" All orders should have "bankdraft or money order in US Dollar Currency". The address is: PRICEWELL (FAR EAST) LIMITED P.O. Box 31008 Causeway Bay Hong Kong --Rodney Hoffman  Received: from LLL-MFE.ARPA by MC.LCS.MIT.EDU 6 Jan 86 21:11:01 EST Date: Mon, 6 Jan 86 18:02 pst From: "del tufo joseph%e.mfenet"@LLL-MFE.ARPA Subject: "deltufo%d"@lll-mfe.arpa To: cube-lovers@mit-mc.arpa requesting any information  Received: from Louie.UDEL.EDU by MC.LCS.MIT.EDU 27 Jan 86 12:33:53 EST Received: from ccvax1 by Louie.UDEL.EDU id a005979; 27 Jan 86 12:32 EST Received: by vax1.acs.udel.edu (4.12/4.7) id AA09563; Mon, 27 Jan 86 11:42:48 est Date: Mon, 27 Jan 86 11:42:48 est From: TOMCHANY Message-Id: <8601271642.AA09563@vax1.acs.udel.edu> To: cube-lovers@mc.lcs.mit.EDU Please add me to your mailing list concerning THE CUBE and other puzzles. Thank you, M. Tomchany  Received: from Xerox.COM by MC.LCS.MIT.EDU 21 Mar 86 21:07:36 EST Received: from CheninBlanc.ms by ArpaGateway.ms ; 21 MAR 86 17:36:22 PST Date: 21 Mar 86 17:35:55 PST (Friday) From: Hoffman.es@Xerox.COM Subject: Eidswick article To: Cube-Lovers@MC.LCS.MIT.EDU cc: Hoffman.es@Xerox.COM Message-ID: <860321-173622-1700@Xerox> In case no one's mentioned it yet: Eidswick, J. A., "Cubelike Puzzles -- What Are They and How Do You Solve Them?", 'American Mathematical Monthly', Vol. 93, #3, March 1986, pp. 157-176. From the introductory section: "This article is an attempt to put some algebraic order into the business of solving cubelike puzzles. Ideally, an algebraic theory would unfold that would, in an elementary way, yield highly efficient algorithms for all such puzzles. The work here is a step in that direction." Included in the applications section are the tetrahedron, octahedron, dodecahedron, icosahedron, and the general n X n X n cube. --Rodney Hoffman  Received: from SCRC-STONY-BROOK.ARPA by MC.LCS.MIT.EDU 26 Mar 86 19:25:52 EST Received: from SORA.SCRC.Symbolics.COM by SCRC-STONY-BROOK.ARPA via CHAOS with CHAOS-MAIL id 445199; Mon 24-Mar-86 10:30:16-EST Date: Mon, 24 Mar 86 10:30 EST From: Bernard S. Greenberg Subject: Eidswick article To: Cube-Lovers@MC.LCS.MIT.EDU In-Reply-To: <860321-173622-1700@Xerox> Message-ID: <860324103055.2.BSG@SORA.SCRC.Symbolics.COM> Now that I saw a letter on this list.... Try to find a store that sells an ordinary Rubik's cube. The largest Cambridge game shop hasn't had them for eternity....  Date: Tue, 15 Apr 86 04:40:29 EST From: Alan Bawden Subject: Cube-Lovers moves! To: CUBE-LOVERS@AI.AI.MIT.EDU Message-ID: <[AI.AI.MIT.EDU].27214.860415.ALAN> The Cube-Lovers mailing list has moved from MIT-MC to MIT-AI. All this really means is that the archives are in a new place. Sending mail to Cube-Lovers@MIT-MC and Cube-Lover-Request@MIT-MC will still work, although it will be infinitesimaly faster if you mail to Cube-Lovers@MIT-AI and Cube-Lover-Request@MIT-AI instead. (Please do not take this move as an opportunity to start spreading false rumors about the status of MIT-MC. I am moving Cube-Lovers from MC to AI because I am -personally- moving to AI, and it is thus convenient for the mailing list to move with me. There will be a machine named MIT-MC for years and years to come.) For the record: Old cube-lovers mail is archived on MIT-AI in the following places: AI:ALAN;CUBE MAIL0 ;oldest mail in forward order AI:ALAN;CUBE MAIL1 ;next oldest mail in forward order AI:ALAN;CUBE MAIL2 ;more of same AI:ALAN;CUBE MAIL3 ;still more of same AI:ALAN;CUBE MAIL4 ;yet more AI:ALAN;CUBE MAIL ;recent mail in reverse order Files can be FTP'd from MIT-AI without an account. Unfortunately the archives are way too large to be conveniently mailed by either computer or physical mail.  Received: from MC.LCS.MIT.EDU by AI.AI.MIT.EDU via Chaosnet; 2 AUG 86 16:12:56 EDT Received: from ICSE.UCI.EDU by MC.LCS.MIT.EDU 2 Aug 86 16:14:39 EDT Received: from localhost by ICSE.UCI.EDU id a014758; 2 Aug 86 11:05 PDT To: cube-lovers@mc.lcs.mit.edu Subject: 5x5x5 Date: Sat, 02 Aug 86 11:05:08 -0800 From: Mark Wadsworth On July 15 I sent a check for $20 to Christoph Bandelow in Germany, and I got my cube sometime last week. He also sent me a letter in which he says, I shall be at the International Congress of Mathematicians at Berkeley this August 2 - 10 with most of my "cubes". After the congress and some travelling time, that is in September, I will send you my then updated catalogue - but I hope I can sell the complete rest in Berkeley. If you happen to be around Berkeley this week, it might be worth checking out. If he sends a catalog with anything in it, I will post info.  Received: from Xerox.COM by AI.AI.MIT.EDU 4 Aug 86 13:44:42 EDT Received: from CheninBlanc.ms by ArpaGateway.ms ; 04 AUG 86 10:40:38 PDT Date: 4 Aug 86 10:40:31 PDT (Monday) From: Hoffman.es@Xerox.COM Subject: Rubik Redux To: Cube-Lovers@AI.AI.MIT.EDU cc: Hoffman.es@Xerox.COM Message-ID: <860804-104038-1367@Xerox> From the New York Times, Sunday, August 3, Business section, p. 6: WILL 'SON OF CUBE' BRING ANOTHER BOUT OF RUBIKMANIA? By Martin Gottlieb ... Erno Rubik... is about to mark his return with a puzzle that, if anything, is more difficult to sove than the once-ubiquitous cube.... Professor Rubik... said his new puzzle, called Rubik's Magic, may have even more configurations. But, in his view, who cares? "You can solve the puzzle by discovering the possibilities, but it is magic, I think," Professor Rubik said, absently fondling a prototype of his new invention in the local offices of Matchbox International Ltd., the key arm of a Bermuda-based holding company, that will make and sell it. ... Within the toy industry ... there is a fair amount of anticipation about Rubik and his Magic. "We hope we can create a craze," said David C.W. Yeh, chairman of Universal Matchbox Group, Matchbox International's parent company. Matchbox ... has already taken on 2,000 production workers in China to make the puzzle, and plans to market hundreds of thousands of copies around the world this fall, for about $10.... The [Hungarian] Government two years ago approved [Rubik's] plans for a private business, Rubik Studio, which employs 20 and develops designs for what Mr. Rubik hopes will be a wide range of items, from buildings to work flow charts and puzzles. Rubik's Magic is its first commercial venture. ... When Professor Rubik traveled to the Nuremburg Toy Fair this year with his latest puzzle, by all accounts, he caused a commotion. ... Key to [Matchbox's successful proposal to Rubik] were a three- to five-year game plan and concepts for developing more advanced versions of Rubik's Magic to be introduced in later years. Rubik's Magic is marked by the same sort of handsome design as its predecessor. Palm-sized, it is made of eight squares of impact-resistant transparent plastic that in their original position form two equal rows. Spread across the squares are depictions of three unconnected rainbow-colored rings printed on a black background. The object of the puzzle is to intertwine the three rings by rejiggering the squares, which are linked by an ingenious hinge patented by Professor Rubik, that flexes on the four sides of each square. Unlike the cube, Mr. Rubik's new puzzle can be nameuvered into a plethora of different shapes. [Accompanying picture shows this, but little more. Rubik is seen leaning on a table which has four Magic toys arrayed in different configurations.] The multi-colored loops break into fanciful swirls and any number of variations would probably look pretty good on coffee tables. "On the way to the solution," Mr. Rubik said, "you can have wonderful discoveries because you have beautiful shapes. The cube was very intellectual. This item could be more fun and more pleasure -- it is beautiful and changeable." ... "You sort of expected a lot from him after the cube and I think he delivered," said Rick Anguilla, editor in chief of Toy & Hobby World, the leading industry journal, who believes the new puzzle is marked by the same quality as The Cube. Mr. Anguilla is one of the few industry analysts to get a glimpse of the new puzzle and he has a hunch it could be this year's Teddy Ruxpin. .... Matchbox has devised a plan for producing and promoting the puzzle aimed in no small part at heading off the knockoff artists who polluted the Cube market with counterfeits. It has applied for patents the world over, and plans to kick off a simultaneous international sales campaign in October. Professor Rubik's signature is on the puzzles, and both professor and puzzle will be promoted with an exceptionally high sales budget... ... For Professor Rubik, filming commercials that might well place him second only to Soviet chief Mikhail S. Gorbachev as the most recognized Eastern bloc citizen in America, is another adventure.  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 14 Aug 86 15:44:23 EDT Date: 14 Aug 86 15:34:00 EST From: "CLSTR1::BECK" Subject: CUBE EXCHANGE To: "cube-lovers" Reply-To: "CLSTR1::BECK" I have been a cube person for 5 or 6 years now and I am upset by the almost total disappearance of the cube and its asociated memorabilia and paraphernalia. Therefore, I propose to establish a "CUBE EXCHANGE" to buy, sell and trade cube items in suport of the greater development of cubing. To this end it will be necessary to catalog what was available, what is avaiable and from whom it can be obtained. Your assistance in this endeavor will be much appreciated, including pointers through the archives. So far I have located a source for 3x3x3 cubes (solid colors, card suits, flowers, numbers, domino dots, fruits, sports balls, and ball). Your assistance in who has what for sale would also be much appreciated. peter beck BECK@ARDEC-LCSS.ARPA ------  Received: from PROPHET.BBN.COM by AI.AI.MIT.EDU 14 Aug 86 16:36:18 EDT Date: Thu, 14 Aug 86 16:28:18 EDT From: Bernie Cosell To: cube-lovers@ai.ai.mit.edu cc: cube-lovers Subject: Re: CUBE EXCHANGE ``The Games People Play'' in Cambridge, MA (617-492-0711) had, and still has, lots of cubes. Carol (who owns the place) continually complains about the basement-full of 4x4x4 cubes that she's ended up stuck with... /bernie  Received: from harvard.HARVARD.EDU by AI.AI.MIT.EDU 19 Aug 86 10:45:41 EDT Received: by harvard.HARVARD.EDU; Tue, 19 Aug 86 10:44:51 EDT Date: Tue, 19 Aug 86 10:44:51 EDT Received: by h-sc4.HARVARD.EDU; Tue, 19 Aug 86 10:44:36 edt From: mazzarel%h-sc4@harvard.HARVARD.EDU To: cube-lovers@mit-ai.arpa Subject: Paraphernalia I bought a book about four years ago in Boston that had procedures for generating all sorts of patterns on the 3x3 cube. I think it was about five to ten 8-1/2 x 11" pieces of paper folded in half and stapled to form a book. The cover was blue paper. Unfortunately, I lost it moving about half a year after that. By that time I guess interest in the cube was waning; I have not been able to find another copy. If anyone else has a copy or just knows where to find them, could you please post it to this newsgroup? I thought it was quite a find, and think no cube lover should be without one. --Paul (pm@harvard.HARVARD.EDU)  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 25 Aug 86 10:34:12 EDT Date: 25 Aug 86 10:15:00 EST From: "CLSTR1::BECK" Subject: singmaster's book To: "cube-lovers" Reply-To: "CLSTR1::BECK" RE: Paraphernalia, 19 aug 86 The book you are talking about is Sinmaster's "Notes on Rubik's Cube". It was available as you describe in the 1981 time frame. I believe it is currently avaialble in hardcover with an expanded bibliography from "Enslow Publishers" . They are located in northern NJ. If you can't get their address send me a message and I will look it up for you, if you just want the patterns and not the book I can send them to you since I have a copy of the book. They also have a second book by Singmaster that was oriented towards teaching math concepts with the cube. pete ------  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 25 Aug 86 13:59:23 EDT Date: 25 Aug 86 13:33:00 EST From: "CLSTR1::BECK" Subject: CUBE AVAILABILITY To: "cube-lovers" Reply-To: "CLSTR1::BECK" CUBE AVAILABILITY: 1. REVENGE - on 21 aug 86 I bought a revenge for $1.99 at the KAY BEE toy store in the Gallery mall, phila, pa. NOTE: kay bee is a national chain, so check your area if interested. 2. GLOBE PUZZLE - This is a 3x3x3 sphere with the following difference, the edge pieces are broken into two pieces. Thus the edges and centers are interchangeable. (does this make it a planar puzzle, for edges and centers) I haven't received mine yet. Is there any discussion of this variant in the archives. AVAILABLE from: the nature company pob 2310 berkley, ca 94702 800/227-1114 globe puzzle, cat # 1449, $12.50 ------  Received: from CSNET-RELAY.ARPA by AI.AI.MIT.EDU 26 Aug 86 00:54:03 EDT Received: from dartmouth by csnet-relay.csnet id ah03666; 25 Aug 86 22:00 EDT Received: by dartmouth.EDU (4.13D/2.3D) id AA19439; Mon, 25 Aug 86 21:13:13 edt Date: Mon, 25 Aug 86 21:13:13 edt To: cube-lovers@AI.AI.MIT.EDU From: Andy Behrens Subject: Re: Paraphernalia References: <8608191515.AA23076@EDDIE> Organization: Dartmouth College, Hanover, NH The blue-covered book that Paul (pm@harvard.HARVARD.EDU) refers to is probably David Singmaster's pamphlet Notes_on_Rubik's_'Magic_Cube'. It is 60 pages long, and includes sections on the basic mathematical problem, some of the simple subgroups, pretty patterns, algorithms to restore the cube in the shortest number of moves. It lists many processes for flipping and rotating edges and corners. If the book is still in print, it is available from Enslow Publishers Bloy St. and Ramsey Ave. Hillside, New Jersey 07205 (USA) -- ISBN 0-89490-043-9 or David Singmaster & Co. 66 Mount View Road London N4 4JR (UK) -- ISBN 0-907395-00-7 Andy Behrens {astrovax,decvax,ihnp4,linus,harvard}!dartvax!andyb.UUCP andyb@dartmouth.EDU andyb%dartmouth@csnet-relay.ARPA  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 26 Aug 86 08:05:35 EDT Date: 26 Aug 86 07:55:00 EST From: "CLSTR1::BECK" Subject: HOFSTADTER To: "cube-lovers" Reply-To: "CLSTR1::BECK" HOFSTADTER I haven't noticed any reference to his 3rd book which is a reprint of his scientific american column with postscripts. I.E. there are 8pgs of new info; trivia, distribution of patterns from start, reverse scramblings, measuring scrambleness. METAMAGICAL THEMAS by DOUG HOFSTADTER BASIC BOOKS 1985 COPYRIGHT ISBN 0-465-04540-5 PRICE ABOUT $20 PETE ------  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 26 Aug 86 08:05:54 EDT Date: 26 Aug 86 07:57:00 EST From: "CLSTR1::BECK" Subject: MEFERT To: "cube-lovers" Reply-To: "CLSTR1::BECK" MEFFERT I have written mefert in march and april 86 and received no response. Does anybody out there know if he is still in business. By the way I ordered and received his XMAS special of 9 5x5s and 9 skewbs by boat (about 2 months delivery time) with no problems. PETE ------  Received: from STONY-BROOK.SCRC.Symbolics.COM by AI.AI.MIT.EDU 26 Aug 86 08:21:27 EDT Received: from PEGASUS.SCRC.Symbolics.COM by STONY-BROOK.SCRC.Symbolics.COM via CHAOS with CHAOS-MAIL id 87545; Tue 26-Aug-86 08:19:07 EDT Received: by scrc-pegasus id AA00685; Tue, 26 Aug 86 07:40:08 edt Date: Tue, 26 Aug 86 07:40:08 edt From: Bernard S. Greenberg To: cube-lovers%ai.ai.mit.edu@stony Subject: Paraphernalia Date: Mon, 25 Aug 86 21:13:13 edt To: cube-lovers@AI.AI.MIT.EDU From: Andy Behrens Subject: Re: Paraphernalia References: <8608191515.AA23076@EDDIE> Organization: Dartmouth College, Hanover, NH The blue-covered book that Paul (pm@harvard.HARVARD.EDU) refers to is probably David Singmaster's pamphlet Notes_on_Rubik's_'Magic_Cube'. It is 60 pages long, and includes sections on the basic mathematical problem, some of the simple subgroups, pretty patterns, algorithms to restore the cube in the shortest number of moves. It most certainly made no claim to minimal algorithms. Minimal cube algorithms are still an unsolved problem. It lists many processes for flipping and rotating edges and corners. If the book is still in print, it is available from (**PLEASE DO NOT REPLY TO THE ABOVE ADDRESS, IT WILL HANG YOUR MAILER. I AM SORRY I AM FORCED TO SEND THIS FROM A BROKEN COMPUTER. I CAN'T PUT IN A FROM: or REPLY-TO: Either. I UNDERSTAND IT's MY PROBLEM, NOT YOUR PROBLEM, PLEASE DON'T FLAME AT ME. Reply to BSG@SCRC-STONY-BROOK not about mailers. Sorry.)  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 29 Aug 86 08:02:18 EDT Date: 29 Aug 86 07:54:00 EST From: "CLSTR1::BECK" Subject: globe puzzle To: "cube-lovers" Reply-To: "CLSTR1::BECK" GLOBE PUZZLE - I made a mistake. The "Globe Puzzle" is NOT a cube. It looks like a cube with the edge pieces broken up into its two faces but the corners are fixed and only the three equators are free to move. It has a metal surface painted as a globe and each surface area has a dimple. These dimples do not make it any easier to work the puzzle. It is the same 3" diameter as a cube ballThe puzzle is made in Hungary. Thus the globe puzzle is three intersecting rings and appears to be simpler then the cube. If the corners could move like the cube would that puzzle also be no more difficult then the cube? Any ideas on how to engineer such a puzzle? AVAILABLE from: the nature company pob 2310 berkley, ca 94702 800/227-1114 globe puzzle, cat # 1449, $12.50 peter beck ------  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 10 Sep 86 15:09:04 EDT Date: 10 Sep 86 13:38:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" Reply-To: "CLSTR1::BECK" 1 - I have an incomplete reference to a PUZZLE MUSEUM in the Wash DC area. Anybody have any details. 2 - Is there any good (complete) mailorder catalog for puzzles. 3 - "INTERLOCKING" CUBE, this is a puzzle with 27 cubies held together in a series connection by a rubber band. The joint that connects cubies is not universal. The cubies all have the same six colors, but not in the same orientation. THE OBJECT OF THE PUZZLE IS TO FOLD UP THE CUBIES to make a 3x3x3 cube with solid colored faces. ANYBODY DONE THIS PUZZLE, ANY REFERENCES. / \ |\ / | | \ / / | |\ |\ /\ /| /| | \| \/ \/ |/ | \ |\ |\ /| /| / \| \| \/ |/ |/ \ |\ | /| / \| \|/ |/ \ | / \|/ ------  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 19 Sep 86 11:28:14 EDT Date: 19 Sep 86 10:48:00 EST From: "CLSTR1::BECK" Subject: interlocking cube To: "cube-lovers" Reply-To: "CLSTR1::BECK" RE: msg from Randel Shane requesting clarification of the "INTERLOCKING" CUBE PUZZLE, this is a puzzle with 27 cubies held together in a series connection by a rubber band, like a snake. The cubies faces each have a different color, ie, 6 per cubie. The blue and green colors are along the connecting axis while the red,white,yellow and orange colors are parallel to the connecting axis. The cubbies are not all identical. The joint that connects cubies is not universal. The joint is made by a slot in the blue and green faces. These slots are perpendicular to each other. This is limits the possible orientations. THE OBJECT OF THE PUZZLE IS TO FOLD UP THE CUBIES to make a 3x3x3 cube with six solid colored faces. HAS ANYONE DONE THIS PUZZLE, ANY REFERENCES. PS Randell, I hope this description is somewhat better. If you would like to see it first hand I can send you one for $2. PPS I need a better ARPANET address to send you msgs directly. / \ |\ / | | \ / / | |\ |\ /\ /| /| | \| \/ \/ |/ | \ |\ |\ /| /| / \| \| \/ |/ |/ \ |\ | /| / \| \|/ |/ \ | / \|/ ------  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 6 Oct 86 08:44:03 EDT Date: 6 Oct 86 08:19:00 EST From: "CLSTR1::BECK" Subject: rubiks magic To: "cube-lovers" Reply-To: "CLSTR1::BECK" RUBIK'S MAGIC: 1. In the NYC area the puzzle is available from Macy's for $10. 2. Matchbox Toys "is the distributor" 141 West Commercial Ave Moonachie, NJ 07074 212/696-5400 3. HOTLINE: sat & sun from 9am to 5pm until April 26, 1987. 1800-843-1202, NJ RES 1800-843-1203 4. Repairs from matchbox $3 5. The mechanism is similar to a 2-dimensional Jacobs ladder. 6. Trivial observations: a. The puzzle has a front and back. b. For three intersecting rings the 8 squares have to be arranged as a 2x3x3 and not as a 2x4. 7. It is a good puzzle. / \ |\ / | | \ / / | |\ |\ /\ /| /| | \| \/ \/ |/ | \ |\ |\ /| /| / \| \| \/ |/ |/ \ |\ | /| / \| \|/ |/ \ | / \|/ ------  Received: from ARDEC-LCSS.ARPA.ARPA by AI.AI.MIT.EDU 10 Oct 86 08:40:07 EDT Date: 10 Oct 86 08:32:00 EST From: "CLSTR1::BECK" Subject: rubiks magic To: "cube-lovers" Reply-To: "CLSTR1::BECK" RUBIK'S MAGIC: >>> UPDATE <<< ADMINISTRIVIA 1. In the NYC area the puzzle is available from Macy's for $10. In your area it may available from "TOYS R US". 2. Matchbox Toys "is the manufacturer" 141 West Commercial Ave Moonachie, NJ 07074 212/696-5400 3. HOTLINE: sat & sun from 9am to 5pm until April 26, 1987. 1800-843-1202, NJ RES 1800-843-1203 4. Repairs from matchbox $3 PUZZLE NOTES: 1. It took about 4 days for my crack puzzle team to solve Rubik's Magic. 2. The mechanism (similar to a 2-directional Jacobs ladder) is durable and the puzzle is fun to manipulate. There are many 3-dimensional shapes/configurations that the squares can be arranged as. 3. It is necessary to establish a notation scheme and establish moves to do the puzzle systematically. The puzzle is similar in difficulty to the Skewb or Pyramid. I would say about 10 - 15 moves. 4. Observations: a. The puzzle has a front and back. b. Each piece always has a hinge with two (2) other pieces. This causes the pieces when rotated to rotate as if they were gears, ie, adjacent pieces rotate in opposite directions. c. The 2x4 layout can go either up and down or left and right. d. It is possible to turn the open/loop layout inside out, ie, the non-intersecting rings can be either on the inside or outside. This is the same as changing how the hinge works on the 2x4 layout. This also exchanges the top and bottom of the 2x4 layout. e. For three intersecting rings (the object of the puzzle) the 8 squares have to be arranged as a 2x3x3 and not as a 2x4. / \ |\ / | | \ / / | |\ |\ /\ /| /| | \| \/ \/ |/ | \ |\ |\ /| /| / \| \| \/ |/ |/ \ |\ | /| / \| \|/ |/ \ | / \|/ ------  Received: from MX.LCS.MIT.EDU by AI.AI.MIT.EDU via Chaosnet; 10 OCT 86 13:42:48 EDT Date: Fri, 10 Oct 86 13:41:11 EDT From: Richard Pavelle Subject: Rubiks Magic To: CUBE-LOVERS%MX.LCS.MIT.EDU@MC.LCS.MIT.EDU Message-ID: <[MX.LCS.MIT.EDU].952133.861010.RP> It is available at Toys-R-Us in the Boston area.  Received: from DIAMOND.S4CC.Symbolics.COM (TCP 20024231403) by AI.AI.MIT.EDU 4 Nov 86 17:11:18 EST Received: from KOYAANISQATSI.S4CC.Symbolics.COM by DIAMOND.S4CC.Symbolics.COM via CHAOS with CHAOS-MAIL id 24817; Tue 4-Nov-86 17:06:25 EST Date: Tue, 4 Nov 86 17:06 EST From: David C. Plummer Subject: rubiks magic To: "CLSTR1::BECK" , cube-lovers In-Reply-To: The message of 10 Oct 86 09:32 EDT from "CLSTR1::BECK" Message-ID: <861104170610.5.DCP@KOYAANISQATSI.S4CC.Symbolics.COM> Rubik's Magic is available at "Games people play" near Harvard Square for a little under $15. As with Beck, it took me about 4 days to solve it. My resulting solution is strickingly simple (3 macros), but I took a lot of interesting paths to get there. I think it could have been made harder be repainting the back surface, but doing so may take it out of the reach of the general public. (It would require solving it to an intermediate configuration that doesn't look anywhere near solved, and then frob it again.) It's a good puzzle for playing with and relieving the fidgets. Mathematically it isn't very interesting, nor are there any "pretty patterns" other than solved1 and solved2.  Received: from SUMEX-AIM.ARPA (TCP 1200000070) by AI.AI.MIT.EDU 4 Nov 86 17:48:42 EST Date: Tue 4 Nov 86 14:43:05-PST From: Haym Hirsh Subject: SF Bay area availability of Rubik things To: cube-lovers@AI.AI.MIT.EDU cc: hirsh@SUMEX-AIM.ARPA In-Reply-To: <861104170610.5.DCP@KOYAANISQATSI.S4CC.Symbolics.COM> Message-ID: <12252334415.75.HIRSH@SUMEX-AIM.ARPA> Rubik's magic was available at the local Sears (Palo Alto/Mtn View) for around $8, but they are out of stock now. It is currently available at Norney's in the same mall for around $13. A few 5x5x5 Rubik's cubes were available at Games and Things in the Stanford shopping center about a week ago for around $20. -------  Received: from WAIKATO.S4CC.Symbolics.COM (TCP 20024231532) by AI.AI.MIT.EDU 4 Nov 86 18:25:14 EST Received: from ROCKY-MOUNTAINS.S4CC.Symbolics.COM by WAIKATO.S4CC.Symbolics.COM via CHAOS with CHAOS-MAIL id 71267; Tue 4-Nov-86 18:23:59 EST Date: Tue, 4 Nov 86 18:22 EST From: Allan C. Wechsler Subject: rubiks magic To: DCP@QUABBIN.SCRC.Symbolics.COM, beck@clstr1.decnet, cube-lovers@MIT-AI.ARPA In-Reply-To: <861104170610.5.DCP@KOYAANISQATSI.S4CC.Symbolics.COM> Message-ID: <861104182207.9.ACW@ROCKY-MOUNTAINS.S4CC.Symbolics.COM> Date: Tue, 4 Nov 86 17:06 EST From: David C. Plummer Rubik's Magic is available at "Games people play" near Harvard Square for a little under $15. As usual, Carol Monica is charging unreasonable prices. Boycott this moderately obnoxious woman and buy your Magic from Zayre or Bradlee's at about $7-8. It's a good puzzle for playing with and relieving the fidgets. Mathematically it isn't very interesting, nor are there any "pretty patterns" other than solved1 and solved2. The "pretty patterns" are more sculptural than geometrical. Can you make a cube? Mathematically, there are a couple of interesting points. There are sixteen achievable 2x4 configurations, linked by a nice little rosette of generators. Unsolved stumper: The pattern XXX X X XXX appears unreachable, but we haven't been able to prove it. Can someone come up with a proof? (Or -- hope against hope -- has anyone achieved it?) I find the mechanical aspect more pleasing than the cube. "Magic" is more satisfying to manipulate than almost any of its predecessors. The linking principle can be generalized to any number of squares. I am considering breaking a couple and wiring them together to make Big Magic.  Received: from PROPHET.BBN.COM (TCP 20026200117) by AI.AI.MIT.EDU 4 Nov 86 19:23:09 EST Date: Tue, 4 Nov 86 19:12:58 EST From: Bernie Cosell To: cube-lovers@ai.ai.mit.edu cc: beeler@prophet.bbn.com, alatto@prophet.bbn.com, jr@prophet.bbn.com, lcosell@prophet.bbn.com Subject: Re: rubiks magic On the other side, GPP is generally the *only* comprehensive games-hackers store about and we would do well to go out of our way to patronize it. I hardly care about the $5.00 one way or the other on something like Magic, but where else can you get Monkey puzzles, and the Pentagle series (are they still in business? I haven't seen any new ones recently), go/shogi sets/books, imported jigsaw puzzles and on and on. If she doesn't get a decent profit on the things she expects to sell a fair number of, then she won't be able to stock the esoteric and fun stuff. /Bernie Bernie Cosell Internet: cosell@bbn.com Bolt, Beranek & Newman, Inc USENET: bbnccv!bpc Cambridge, MA 02238 Telco: (617) 497-3503  Received: from MC.LCS.MIT.EDU by AI.AI.MIT.EDU via Chaosnet; 6 NOV 86 15:06:54 EST Received: from nrl-aic (TCP 3200200010) by MC.LCS.MIT.EDU 6 Nov 86 15:02:30 EST Date: 6 Nov 1986 13:41:48 EST (Thu) From: Dan Hoey Subject: Magic To: Cube-lovers@mit-mc.ARPA Message-Id: <531686509/hoey@nrl-aic> I have three macros for transforming 2x4 rectangles. To solve the puzzle, I use two of them followed by a seven-flip macro that changes a 2x4 shape into a 3x3-1 shape (beats me how BECK can call this a ``2x3x3''). Took me a couple of half-days to solve it. I have found 32 different 2x4 rectangles. I think that is all of them, but I haven't got any proofs, nor even a decent mathematical model for deciding when a flip is possible. I am trying to understand how the strings work. First, it looks like there is twice as much string as necessary; each string is doubled. I guess that this duplication has no effect on the puzzle except for durability, but until I can dyke one out can't be sure. I'm concerned that the string may be one double loop, so I'm looking for a good way to make sure the thing doesn't unstring entirely when I cut one. Each side of each piece has four short channel segments and four long; half are occupied with string. If you continue each segment across each hinge to the next piece, you get eight channels composed of alternating long and short segments. Again, four of the eight channels are occupied. In the positions I've seen, each of the channels contains eight pairs of segments. But Magic is more complicated than that--the strings do not always follow a channel from piece to piece. On half of the pieces, there is an extra loop of string that wraps back onto the piece without following the channel to the next piece. I don't know what function this serves. If a good model of the string interactions can be developed, we may be able to make an attack on the doughnut problem based on the length of string channels. In the doughnut, there are four ten-pair channels and four six-pair channels. We may be able to show that the string wouldn't reach one, and would exceed the other. More likely, the model will prohibit the doughnut more directly. There is another string-related question I am wondering about. I have noticed some of the string-pairs getting twisted. I wonder how bad this can get. Does anyone have an operation that can be repeated to make the twists tighter and tighter? Are these puzzles built for obsolescence? I have been considering Magic metrics, but it's a difficult problem. Counting flips is easy enough, but how do you count a move that skews a parallelogram? Are such skew moves necessary? Dan  Received: from MC.LCS.MIT.EDU by AI.AI.MIT.EDU via Chaosnet; 6 NOV 86 18:17:21 EST Received: from WAIKATO.S4CC.Symbolics.COM (TCP 20024231532) by MC.LCS.MIT.EDU 6 Nov 86 18:15:02 EST Received: from ROCKY-MOUNTAINS.S4CC.Symbolics.COM by WAIKATO.S4CC.Symbolics.COM via CHAOS with CHAOS-MAIL id 71915; Thu 6-Nov-86 18:11:09 EST Date: Thu, 6 Nov 86 18:13 EST From: Allan C. Wechsler Subject: Magic To: hoey@NRL-AIC.ARPA, Cube-lovers@MIT-MC.ARPA In-Reply-To: <531686509/hoey@nrl-aic> Message-ID: <861106181326.6.ACW@ROCKY-MOUNTAINS.S4CC.Symbolics.COM> Date: 6 Nov 1986 13:41:48 EST (Thu) From: Dan Hoey I have found 32 different 2x4 rectangles. I think that is all of them, but I haven't got any proofs, nor even a decent mathematical model for deciding when a flip is possible. I think I have a proof. Wait a few paragraphs. If a good model of the string interactions can be developed, we may be able to make an attack on the doughnut problem based on the length of string channels. In the doughnut, there are four ten-pair channels and four six-pair channels. We may be able to show that the string wouldn't reach one, and would exceed the other. More likely, the model will prohibit the doughnut more directly. Here is my model. It might be wrong. The puzzle is a cycle of eight squares. Their underlying adjacency relationships never change. Each pair of squares is bound together by two loops of string (nylon fishing wire, actually). Those two loops are dedicated to holding those two squares together -- they never migrate to other squares, although parts of a loop may sometimes lie on one of the pair, sometimes on the other. I need a diagram of the channels: ---- ---- |/\/\|/\/\| |\/\/|\/\/| |/\/\|/\/\| |\/\/|\/\/| ---- ---- I offer the usual apologies about aspect ratio. Now, ignore the two sided nature of the puzzle. Imagine that the two squares form a 1x2 unit billiard table, and think of the channels as trajectories of billiard balls, and you will see that the channels form two disjoint "orbits", each (* 4 (sqrt 2)) units long. The two loops of string follow these two orbits, crossing from the obverse to the reverse sides of the puzzle and back again at every chance they get. If you work it out, you see that half the channels are empty at any given time. The details are too mindbending, but the result is clear: those two squares are bound together. They don't depend on synergy from the rest of the puzzle to bind them. Now close the pair of squares, putting the right on top of the left as if you were finishing reading a book. You can open the pair again vertically, but only in one direction. The front square can flip up, or it can flip down, but not both. Without loss of generality, let's say it can flip up. If you had folded the right square behind the left instead of in front, it would be able to flip down. Now gedankenfollow these gendankendirections. Put the squares back as above. Only the square on the right (the rotor) will move. Keep the left square (the stator) fixed. Fold the rotor in front of the stator. Flip the rotor up, and over, and behind the stator. Now it will open to the left. Flip the rotor around the left edge of the stator until it is in front again. Then it will flip down. Do so, until it is in back again. Now it will open to the right. You have moved the rotor all the way around the stator. At some point, each edge of both served as the hinge. This amazing orbit is the basis for the bewilderingness of the puzzle. The path is a bizarre three-dimensional cloverleaf. I think that I have now given all the "laws of motion" of the puzzle. Since the laws are all local, governing the motion of one adjacent pair of squares, there is no obvious invariant that forbids the doughnut. There is another string-related question I am wondering about. I have noticed some of the string-pairs getting twisted. I wonder how bad this can get. Does anyone have an operation that can be repeated to make the twists tighter and tighter? Are these puzzles built for obsolescence? I am convinced not. I have been considering Magic metrics, but it's a difficult problem. Counting flips is easy enough, but how do you count a move that skews a parallelogram? Are such skew moves necessary? Dan Your 32 configurations are characterized as follows: Turn the puzzle so the unlinked rings are in front. Rotate it so Rubik's signature is in the bottom row. The signature square could be in any of four positions. The "matchbox" square could be clockwise from Rubik, or counterclockwise. And Rubik himself could be in any of four orientations. This determines the orientations of all the other squares. There are no more degrees of freedom. It is more instructive to consider the "supergroup", which includes 180-deg rotations but excludes turning the puzzle over. This group has sixty-four elements in thirty-two similar pairs. I have a table.  Received: from nrl-aic (TCP 3200200010) by AI.AI.MIT.EDU 13 Nov 86 15:00:34 EST Date: 13 Nov 1986 13:16:44 EST (Thu) From: Dan Hoey Subject: Magic To: acw%WAIKATO.S4CC.Symbolics.COM@scrc-stony-brook.ARPA, jr@bbncc5.ARPA, alatto@bbncc5.ARPA, cube-lovers@mit-AI.ARPA In-Reply-To: <861106181326.6.ACW@ROCKY-MOUNTAINS.S4CC.Symbolics.COM> Message-Id: <532289805/hoey@nrl-aic> Date: Thu, 6 Nov 86 18:13 EST From: Allan C. Wechsler ... Here is my model. It might be wrong. From what I have seen of the puzzle, your model seems to correctly specify necessary conditions for puzzle movement. I have some additional constraints, below. Each pair of squares is bound together by two loops of string (nylon fishing wire, actually). This is acceptable for a model, though the physical puzzles I have seen are have twice as much wire on half of the squares as on the others. I don't know why, and I am nearly ready to take one apart just to verify my several curiosities about the construction. ... Only the square on the right (the rotor) will move. Keep the left square (the stator) fixed. Flip the rotor .... You have moved the rotor all the way around the stator. At some point, each edge of both served as the hinge. This amazing orbit is the basis for the bewilderingness of the puzzle. The path is a bizarre three-dimensional cloverleaf. This corresponds with my understanding of two-square interaction. Thanks for nailing it down. I think that I have now given all the "laws of motion" of the puzzle. There is something more, because each square has *two* neighbors. Consider a stator with two rotors. There are two ``bizarre three- dimensional cloverleaves'' about the stator, but both rotors move in the same b3dc. Furthermore, they cannot pass each other in their common b3dc. Thus if one goes around more than a complete revolution, the other must make a net advance in the same direction. Since the laws are all local, governing the motion of one adjacent pair of squares, there is no obvious invariant that forbids the doughnut. I think I can demonstrate one, given the fact that the pieces are joined in a cycle. Let us tie each pair of adjacent squares together as in your model. We notice that in the starting position, each of the occupied channels is occupied by two wires, one for each neighbor of the square. Suppose we dyke out a loop from the left neighbor, and splice the left neighbor's connections into the right neighbor's loop. Then do this all the other duplicate loops. I believe we end up with a puzzle held together by four pieces of wire, each (* 8 (sqrt 2)) units long. One of them is drawn below, assuming transparent squares. I believe that this puzzle will move in the same way as the original one. ---- ---- ---- ---- |/\ | |/\ | | |\ \ | /| \ | | | \ \| / | \| | | \ |\/ | |\ | ---- ---- ---- ---- | \| | /\| \ | | |\ | / |\ \ | | | \ |/ | \ \| | | \/| | \/| ---- ---- ---- ---- Now put away the dykes and try to fold the puzzle into a doughnut. If you succeed, you will have formed two (* 10 (sqrt 2)) wires and two (* 6 (sqrt 2)) wires: ---- ---- ---- ---- ---- ---- |/\ | |/\ | | |/\ | | |\ \ | /| \ | | /| \ | | | \ \| / | \| | / | \| | | \ |\/ | /| | / | |\ | ---- ---- ---- ---- ---- ---- | \| | / | |/ | | \ | | /| | / | |\ | | \ | | / | |/ | | \ | | \| | / | |\ | | \ | | /| ---- ---- ---- ---- ---- ---- |/ | /\| \ | | \| | / | |\ | / |\ \ | | |\ | / | | \ |/ | \ \| | | \ |/ | | \/| | \/| | | \/| | ---- ---- ---- ---- ---- ---- Changing the string lengths is a little more magic than I expect from Rubik. My thanks to John Robinson for encouraging me to believe that the string-length criterion has merit. He tells me Andy Latto has a similar proof. Dan  Received: from MC.LCS.MIT.EDU by AI.AI.MIT.EDU via Chaosnet; 30 NOV 86 15:36:27 EST Received: from cc2.bbn.com (TCP 1000200022) by MC.LCS.MIT.EDU 30 Nov 86 15:36:55 EST Date: Sun, 30 Nov 86 15:34:25 EST From: Buz Owen Subject: 5 cube? To: cube-lovers@mc.lcs.mit.edu Cc: ado@cc2.bbn.com Can someone on this list supply me with the address of a store that can sell me a 5x5x5 cube? I am not on the list so please reply directly. Thanks/Buz  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 3 Dec 86 10:27:14 EST Date: 3 Dec 86 08:55:00 EST From: "CLSTR1::BECK" Subject: 5X5X5 AVAILABILITY To: "cube-lovers" Reply-To: "CLSTR1::BECK" RE: Buz Owen's request for a source for 5x5x5 cubes 1. Jerry Slocum 257 South Palm Drive Beverly Hills, CA 90212 213/273-2270 has both 5x5x5s and skewbs available for $20 each. 2. Jerry Slocum has also authored a book "Puzzles Old and New, How to make and Solve Them" which he sells for $22 outside of CA, autographed if requested. The table of contents is: chap 1 PUT-TOGETHER PUZZLES - Putting the object together is the puzzle. chap 2 TAKE-APART PUZZLES - Opening or taking the object apart is the puzzle. chap 3 INTERLOCKING SOLID PUZZLES - Disassembly and assembly is required to solve the puzzle. chap 4 DISENTANGLEMENT PUZZLES - The puzzle is to disentangle and re-entangle the parts of the puzzle. chap 5 SEQUENTIAL MOVEMENT PUZZLES - The puzzle is to move parts of the object to a goal (The cube, et al) chap 6 PUZZLE VESSELS - Drinking or pouring liquid, or filling the vessels without spilling is the puzzle. chap 7 DEXTERITY PUZZLES - Manual dexterity is required to solve the puzzle. chap 8 VANISH PUZZLES - The puzzle is to explain a vanishing or changing image. chap 9 IMPOSSIBLE OBJECT PUZZLES - The puzzle is to discover how the object is made. chap 10 FOLDING PUZZLES - The puzzle is to achieve a specified gola by folding. 3. Richard Hess 4100 Palos Verdes Dr. East Rancho Palos Verdes, CA 90274 has available for sale many of the puzzles from Jerry Slocum's book. 4. Another book of interest is: 536 Puzzles & Curious Problems by Henry Ernest Dudeney, 1967 by Charles Scribner's Sons Library of Congress Catalog Card NUmber 67-15488. 5. There are sources for other variations of the cube, if anybody is interested I will compile a list. CUBING IS FOREVER --- ------  Received: from REAGAN.AI.MIT.EDU by AI.AI.MIT.EDU via Chaosnet; 3 DEC 86 17:04:45 EST Received: from PIGPEN.AI.MIT.EDU by REAGAN.AI.MIT.EDU via CHAOS with CHAOS-MAIL id 14155; Wed 3-Dec-86 17:04:49 EST Date: Wed, 3 Dec 86 17:04 EST From: Alan Bawden Subject: BITNET Redistribution of Cube-Lovers To: Cube-Lovers@AI.AI.MIT.EDU Message-ID: <861203170452.2.ALAN@PIGPEN.AI.MIT.EDU> I'm looking for someone on BITNET to volunteer to maintain a BITNET redistribution list for the Cube-Lovers mailing list. This is necessary because the machine that's acting as the Internet/BITNET gateway, WISCVM.WISC.EDU, is swamped with mailing list mail. List administrators (like me) are being asked to set up redistribution lists to help lighten the load. If we can set one up for Cube-Lovers then WISCVM will only have to relay each message to one host instead of the current 20 or so. If you're willing and able to take this on, please let me know.  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 5 Dec 86 09:20:22 EST Date: 5 Dec 86 09:06:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" Reply-To: "CLSTR1::BECK" From: CLSTR1::SYSTEM 4-DEC-1986 11:27 To: CLSTR1::BECK Subj: Undeliverable mail ----Transcript of session follows---- "mit" is an unrecognized hostname/address ----Unsent message follows---- Date: 4 Dec 86 11:18:00 EST From: "CLSTR1::BECK" Subject: CUBE AVAILABILITY To: "cube-lovera" Reply-To: "CLSTR1::BECK" 1. MY MSG TO BUZ OWEN was not delivered (RE: 5x5x5 availability). If anybody out there knows him or how to forward the msg to him (he is not on the cube lovers list) please do so. 2. It will take a bout a week but I will compile a list where and what is available in the world of cubes. It will be a compilation of Helm's , Bandelow's, Hess's and Beck's supplies. Please be patient. If you are looking for a specific cube let me know I can get you that answer faster. ------ ------  Received: from Xerox.COM (TCP 1200400040) by AI.AI.MIT.EDU 7 Dec 86 19:21:54 EST Received: from CheninBlanc.ms by ArpaGateway.ms ; 07 DEC 86 16:21:13 PST Date: 7 Dec 86 16:21:12 PST (Sunday) From: Hoffman.es@Xerox.COM Subject: Puzzle show To: Cube-Lovers@AI.AI.MIT.EDU Reply-To: Hoffman.es@Xerox.COM Message-ID: <861207-162113-1746@Xerox> PUZZLES OLD AND NEW: Head Crackers, Patience Provers, and Other Tactile Teasers [including more variations on Rubik's cube than I've ever seen before] EXHIBITION TOUR: Craft and Folk Art Museum, LA Nov. 26, '86 - Feb. 22, 1987 MIT Museum April 6 - June 15, 1987 Hudson River Museum July 22 - Sept. 27, 1987 Science Museum of Minnesota Oct. 19, '87 - Jan 3, 1988 Ontario Science Center Jan. 25 - March 6, 1988 Accompanied by the book PUZZLES OLD AND NEW: HOW TO MAKE AND SOLVE THEM by Jerry Slocum and Jack Botermans ($20). The exhibit discusses The Art in Puzzles The Social Experience Cultural Values Puzzles in the Industrial Age Puzzles in Education Puzzles and Science It aims "to explore the history, meanings, and design of mechanical puzzles." The extensive displays categorize puzzles according to their object: Put-Together Puzzles Take-Apart Puzzles Interlocking Solid Puzzles Disentanglement Puzzles Sequential Movement Puzzles Puzzle Vessels Dexterity Puzzles Vanish Puzzles Impossible Object Puzzles Folding Puzzles Puzzles have been lent by collectors and museums around the world. The exhibit consultants are Benjamin Kilborne and Martin Gardner. There are hands-on puzzles, but too simple and too few. It's a wonderful and tantalizing display of puzzles. I was SOOO frustrated not to be able to handle all the beautiful, enticing pieces. Of course, if I had been permitted to, I would never leave.... -- Rodney Hoffman  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 9 Dec 86 14:05:07 EST Date: 9 Dec 86 13:53:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" Reply-To: "CLSTR1::BECK" From: CLSTR1::SYSTEM 9-DEC-1986 09:22 To: CLSTR1::BECK Subj: Undeliverable mail ----Transcript of session follows---- "mit.ai" is an unrecognized hostname/address ----Unsent message follows---- Date: 9 Dec 86 09:18:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" Reply-To: "CLSTR1::BECK" From: CLSTR1::SYSTEM 8-DEC-1986 15:52 To: CLSTR1::BECK Subj: Undeliverable mail ----Transcript of session follows---- "ai.ai.mit.edu" is an unrecognized hostname/address ----Unsent message follows---- Date: 8 Dec 86 15:32:00 EST From: "CLSTR1::BECK" Subject: slocum's book To: "cube-lovers" Reply-To: "CLSTR1::BECK" RE: Slocum's book. I have not read it yet but I have ordered a copy and hope to be able to answer your questions soon. In issue #6 of "World Game Review" (avaialble for $8 for 4 issues from Michael Keller, 3367-I North Chatam Road, Ellicott City, MD 21043) Slocum's book is reviewed. It says that this book is a collaboration with Jack Botermans author of "creative puzzles of the world" which was the best puzzle book but is now out of print. The new book has much of the older books material on mechanical puzzles plus new material. The book was liked in general. ------ ------ ------  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 9 Dec 86 14:27:43 EST Date: 9 Dec 86 13:56:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" Reply-To: "CLSTR1::BECK" From: CLSTR1::SYSTEM 8-DEC-1986 15:52 To: CLSTR1::BECK Subj: Undeliverable mail ----Transcript of session follows---- "mit.ai" is an unrecognized hostname/address ----Unsent message follows---- Date: 8 Dec 86 15:35:00 EST From: "CLSTR1::BECK" Subject: buz owens To: "cube-lovers" Reply-To: "CLSTR1::BECK" 1. MY direct MSG TO BUZ OWEN was not delivered (RE: 5x5x5 availability). If anybody out there knows him or how to forward the msg to him (he is not on the cube lovers list) please do so. 2. It will take a bout a week but I will compile a list where and what is available in the world of cubes. It will be the highlights from Helm's , Bandelow's, Hess's and Beck's supplies. If you want a complete classification guide request it separately and I will mail you a copy of Bandelow's collection or if you are more patient wait for Cecil Smith to complete his book. Please be patient. If you are looking for a specific cube let me know I can get you that answer faster. ------ ------ ------  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 10 Dec 86 08:41:01 EST Date: 10 Dec 86 08:28:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" cc: beck Reply-To: "CLSTR1::BECK" RE: Slocum's book. I have not read it yet but I have ordered a copy and hope to be able to answer your questions soon. In issue #6 of "World Game Review" (avaialble for $8 for 4 issues from Michael Keller, 3367-I North Chatam Road, Ellicott City, MD 21043) Slocum's book is reviewed. It says that this book is a collaboration with Jack Botermans author of "creative puzzles of the world" which was the best puzzle book but is now out of print. The new book has much of the older books material on mechanical puzzles plus new material. The book was liked in general. ------ ------ ------  Received: from MC.LCS.MIT.EDU (CHAOS 3131) by AI.AI.MIT.EDU 10 Dec 86 17:15:52 EST Received: from navajo.stanford.edu (TCP 4402000060) by MC.LCS.MIT.EDU 10 Dec 86 17:00:36 EST Received: by navajo.stanford.edu; Wed, 10 Dec 86 12:56:25 PST Received: from jack-jr.edsel.uucp by edsel.uucp (2.2/SMI-2.0) id AA27267; Wed, 10 Dec 86 12:51:04 pst Received: by jack-jr.edsel.uucp (1.1/SMI-3.0DEV3) id AA12942; Wed, 10 Dec 86 12:52:32 PST Date: Wed, 10 Dec 86 12:52:32 PST From: edsel!jack-jr!jeb@navajo.stanford.edu (Jim Boyce) Message-Id: <8612102052.AA12942@jack-jr.edsel.uucp> To: navajo!cube-lovers%mit-mc.ARPA@navajo.stanford.edu Subject: Magic: Construction A friend on mine applied too much force in an odd direction and produced a tangled mess. I was nominated to fix it. My approach was to take it apart completely and then put it back together. Tools required: Well, I used a paper clip. Components: A Magic is made out of 16 loops of string (actually nylon fishline) and 8 panels. The loops of string are all the same length. The loops of string are not tangled in any way. The panels each decompose into two clear plastic covers and a piece of paper (actually plastic). The panels are held together by the string. Construction: Each loop of string is twisted around three panels. It follows a path like these: ---- ---- ---- ---- ---- ---- |/\ | /\|/\ | | /\|/\ | /\| |\ \ | / /|\ \ | or | / /|\ \ | / /| | \ \|/ / | \ \| |/ / | \ \|/ / | | \/|\/ | \/| |\/ | \/|\/ | ---- ---- ---- ---- ---- ---- For each string, there is another string that lies in the same channels. When stringing a loop through the channels, there is a choice at the points where the string passes from one panel the next: Which string is closer to the center of the panel? That question is answered differently for the two strings running throught the same channels. [I believe that this is done so that the net force trying to twist the toy at that point is near zero. The redundancy also probably strengthens the toy.] The panels can be divided into two sets of four: The panels that are centers in these triples and the panels that are ends. Each "end panel" is an end for two different triples. Strings don't lie in crossing channels on the same side of a panel. (That describes how the two pairs of loops go on the same triple and how two triples interact on their common end panel.) -jim boyce  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 11 Dec 86 09:03:53 EST Date: 11 Dec 86 08:58:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" Reply-To: "CLSTR1::BECK" From: CLSTR1::MAILER 3-DEC-1986 10:50 To: BECK Subj: [TCP/IP Mail From: <@AI.AI.MIT.EDU,@ardec-lcss.arpa:beck@clstr1.decnet>] 5X5X5 AVAILABILITY Return-Path: <@AI.AI.MIT.EDU,@ardec-lcss.arpa:beck@clstr1.decnet> Received: from AI.AI.MIT.EDU by ARDEC-LCSS.ARPA.ARPA ; 3 Dec 86 10:49:48 EST Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 3 Dec 86 10:27:14 EST Date: 3 Dec 86 08:55:00 EST From: "CLSTR1::BECK" Subject: 5X5X5 AVAILABILITY To: "cube-lovers" Reply-To: "CLSTR1::BECK" RE: Buz Owen's request for a source for 5x5x5 cubes 1. Jerry Slocum 257 South Palm Drive Beverly Hills, CA 90212 213/273-2270 has both 5x5x5s and skewbs available for $20 each. 2. Jerry Slocum has also authored a book "Puzzles Old and New, How to make and Solve Them" which he sells for $22 outside of CA, autographed if requested. The table of contents is: chap 1 PUT-TOGETHER PUZZLES - Putting the object together is the puzzle. chap 2 TAKE-APART PUZZLES - Opening or taking the object apart is the puzzle. chap 3 INTERLOCKING SOLID PUZZLES - Disassembly and assembly is required to solve the puzzle. chap 4 DISENTANGLEMENT PUZZLES - The puzzle is to disentangle and re-entangle the parts of the puzzle. chap 5 SEQUENTIAL MOVEMENT PUZZLES - The puzzle is to move parts of the object to a goal (The cube, et al) chap 6 PUZZLE VESSELS - Drinking or pouring liquid, or filling the vessels without spilling is the puzzle. chap 7 DEXTERITY PUZZLES - Manual dexterity is required to solve the puzzle. chap 8 VANISH PUZZLES - The puzzle is to explain a vanishing or changing image. chap 9 IMPOSSIBLE OBJECT PUZZLES - The puzzle is to discover how the object is made. chap 10 FOLDING PUZZLES - The puzzle is to achieve a specified gola by folding. 3. Richard Hess 4100 Palos Verdes Dr. East Rancho Palos Verdes, CA 90274 has available for sale many of the puzzles from Jerry Slocum's book. 4. Another book of interest is: 536 Puzzles & Curious Problems by Henry Ernest Dudeney, 1967 by Charles Scribner's Sons Library of Congress Catalog Card NUmber 67-15488. 5. There are sources for other variations of the cube, if anybody is interested I will compile a list. CUBING IS FOREVER --- ------ ------  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 11 Dec 86 09:27:35 EST Date: 11 Dec 86 08:53:00 EST From: "CLSTR1::BECK" Subject: SOME CUBES To: "cube-lovers" Reply-To: "CLSTR1::BECK" ------ HIGHLIGHTS OF CUBE TYPES WITH AVAILABILITY ( Bandelow's complete compilation is 9 pages. Supposedly both Slocum and Cecil Smith have more extensive lists.) I will follow Bandelow/Helm classification guide, ie, A. 3x3x3 cubes of standard size (57mm edge) B. size variations of 3x3x3 cube C. shape variations of the 3x3x3 cube D. NxN (N not 3) cubes and their relatives E. other magic polyhedra F. some other logical puzzles G. various cube rel;ated products SOURCES: PB - Peter Beck, 54 Richwood place, Denville, NJ 07834, add $1 shipping per order CB - Christoph Bandelow, Haarholzer Strasse 13, D-4630 Bochum 1, W Germany GH - Georges Helm, 22A rue Bommert, 4716 Petange, Luxembourg RH - Richard Hess, 4100 Palos Verdes Dr. East, Rancho Palos Verdes, CA 90274 A. 3x3x3 cubes of standard size (57mm edge) ORIGINAL HUNGARIAN RUBIK'S CUBE (CB $8) DELUXE CUBE, bonded plastic instead of stickers, Ideal (PB $3) SIAMESE CUBES, 2 standard cubes joined by having one edge in common, Ideal (CB $15) THERE ARE MANY STICKER VARIATIONS; some are one to one subsitions of cubie stickers, some are pictures that cover a whole cube face, some are puzzles (like the calendar cubes), some divide the coloring of cubies so that the 3x3x3 simulates other cubes (see Hofstadter's articles). I have not seen any references to Penrose Tiling cubie designs. NUMBERS, FRUITS, CARDS, FLOWERS, SPORTS BALLS,DIE 1, FEMALE NUDES wonderful, taiwan (PB $2) B. size variations of 3x3x3 cube 50mm, 38mm, 33mm, 31mm, 30mm, 25mm, 20mm - made into keychains and necklaces. C. shape variations of the 3x3x3 cube because the cubies always maintain the same in-out orientation relative to the center of the cube, cubies can have any surface extension desired. Therefore there are cubes are spherical cubes, cubes with corner pieces trimmed to equilateral triangles, some edge pieces are trimmed so that they make a single plane presentation (eg, makes a barrel), I HAVE NOT HEARD OF ANY THAT MAKE A SOLID SCULPTURE OF THE CUBE, eg, a bust of RUBIK. OCTAGONAL PRISM - four parallel edges are trimmed giving an octagonal prism, (CB $8) MEDIUM MAGIC GLOBE - Sold as Rubik's World, (CB or GH $10) MEDIUM MAGIC BALL - sp-herical cube, (PB $2) D. NxN (N not 3) cubes and their relatives 56mm 2x2x2 RUBIK'S POCKET CUBE, (CB $8) OCTAHEDRON 2X2X2 each rotation axis runs through two opposite corners of the octahedron, (CB $8) TETRAHEDRAL 2X2X2, each rotation axis runs through the centers of two opposite edges, (CB $8) STAR PUZZLE 2X2X2, similar to tetrahedron with a unicolored tetrahedron attached to the center of the big tetrahedron. TRICKHAUS 2X2X2, four edges are trimmed leaving a prism, can be arranged to look like a house, ie, square base and and roof, (CB $8) 4x4x4, RUBIK'S REVENGE, if you are lucky $2 at KAYBEE TOY STORES 5x5x5, ($20 from jerry slocum, see previous msg) E. other magic polyhedra SKEWB PYRAMINX MEGAMINX ALEXANDER'S STAR IMPOSSI-BALL ORB HUGARIAN GLOBE, FROM NATURE CO., NIT A CUBE MISSINK LINK F. some other logical puzzles G. various cube related products ------  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 11 Dec 86 15:15:58 EST Date: 11 Dec 86 15:06:00 EST From: "CLSTR1::BECK" Subject: slocums book To: "cube-lovers" Reply-To: "CLSTR1::BECK" RE: Slocum's book. I have not read it yet but I have ordered a copy and hope to be able to answer your questions soon. In issue #6 of "World Game Review" (avaialble for $8 for 4 issues from Michael Keller, 3367-I North Chatam Road, Ellicott City, MD 21043) Slocum's book is reviewed. It says that this book is a collaboration with Jack Botermans author of "creative puzzles of the world" which was the best puzzle book but is now out of print. The new book has much of the older books material on mechanical puzzles plus new material. The book was liked in general. ------ ------  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 24 Dec 86 11:41:58 EST Date: 24 Dec 86 11:38:00 EST From: "CLSTR1::BECK" Subject: magic review article To: "cube-lovers" Reply-To: "CLSTR1::BECK" I have written a review of MAGIC for the "World Game Review", edited and published by Michael Keller, 3367-I north Chatam Road, Ellicott City, MD 21043, $8 for 4 issues. It as you can tell is based on the CUBE-LOVERS list dialog. If anybody would like to comment or make recomMend any additions or deletions please forward your suggestions to me ,. A REVIEW OF RUBIK'S MAGIC BY Peter Beck, Dec 24, 1986 The Hungarian Government two years ago approved Rubik's plans for a private business, Rubik Studio, to develop designs for what Mr. Rubik hopes will be a wide range of items including puzzles. Rubik's Magic is its first commercial venture. It is being manufactured and marketed by Matchbox and is generally available at prices ranging from $9-$15, $7 on sale. Matchbox's deal with Rubik was based on a three-to-five year plan that includes the development and marketing of more advanced versions of Magic. Like the cube, MAGIC is good for playing with and relieving the fidgets. It is palm-sized and made up of eight 2"x1/4" squares ,of impact-resistant transparent plastic, folding up into a 1"x2"x4" block which easily fits into a shirt or jacket pocket. Unlike the cube it can be maneuvered into a plethora of different geometrical shapes which makes it more fun and pleasureful to manipulate than the cube. Any of the various geometrical variations look good on the coffee table and your guests can make magical discoveries as they play with it. The object of the puzzle is to manipulate the squares from their original pattern (henceforth known as pattern #1) to pattern #2. In pattern #1 the squares form two equal rows (i.e., 2x4 arrangement) and spread across one side is a depiction of three unconnected rainbow-colored rings printed on a black background. By folding, flipping, flopping and flapping the squares, which are linked by an ingenious hinge, we arrive at pattern #2 which is three intersecting rings on the reverse side. Even though Magic has many interesting 3-dimensional geometric shapes, e.g., cube, A-frame house, 1x2 box with lid, both named patterns occur when the puzzle is in its planar or flat state. To go from pattern #1 to #2 there are two operators necessary in the 2x4 arrangement to position the squares for the operator that transforms the puzzle into the 3x3 minus a corner arrangement that displays the three intersecting rings pattern. When the puzzle is in the 2x4 arrangement with pattern #1 correct, the squares can be rearranged by either folding it on the long axis to make a loop which can be rotated or by flipping the 2 ends towards the center and then by un-flipping the puzzle on the opposite side in a perpendicular direction to the flipping. In order to display the solved puzzle it is now necessary to flip and flap and flop the puzzle (6 moves) into the 3x3 minus a corner arrangement. It should be noted that it is possible to be in a 2x4 arrangement where you cannot get to pattern #1 with only the two 2x4 operators of above. You will either need to use the operator that changes the puzzle to the 3x3 or develop another operator. For those of you who would like to take it apart and then put it back together here are some hints. Tools required: a paper clip. Each loop of string is twisted around three squares in a path like this: ---- ---- ---- ---- ---- ---- |/ \ | / \ | / \ | | / \ |/ \ | / \ | |\ \ | / /|\ \ | or | / /|\ \ | / /| | \ \|/ / | \ \| |/ / | \ \|/ / | | \ /|\ / | \ /| |\ / | \ /|\ / | ---- ---- ---- ---- ---- ---- 16 loops of string (actually nylon fishline) are used to make the hinges that hold the squares together. The loops of string are all the same length.and are not tangled in any way. For each string, there is another string that lies in the same channels. When stringing a loop through the channels, there is a choice at the points where the string passes from one square to the next: Which string is closer to the center of the square? That question is answered differently for the two strings running throught the same channels. Strings don't lie in crossing channels on the same side of a square. (That describes how the two pairs of loops go on the same triple and how two triples interact on their common end square.) Now that you have decomposed the puzzle into its component parts why don't you customize it before reassembly. (The squares each decompose into two clear plastic covers and a piece of paper with the design printed on it which are held together by the strings and not glue.). So what about some original designs, maybe even Penrose tiling? How about adding additional squares? For the more mathematically inclined it has been noted that the sameness of the string length is what restricts the arrangements of the squares. Can you prove that the Donut, 3x3 with center missing, is impossible? Can you invent a nomenclature and metric for counting moves? The future exists, first in the imagination, then in the will and fianlly in reality. ACKNOWLEDGEMENT: The above was written with the passive participation of with CUBE-LOVERS computer bulletin board at MIT; MILNET ADDRESS . Thanks to all who participated in the MAGIC dialog to date. ------  Date: Mon, 29 Dec 86 00:31:25 EST From: Marty Connor To: CUBE-LOVERS@AI.AI.MIT.EDU cc: BOTTLENECK@AI.AI.MIT.EDU Message-ID: <134690.861229.MDC@AI.AI.MIT.EDU> There is a new puzzle named the "Bottleneck" from a company called Groton Ltd. in Fitchburg, MA. It looks impossible to solve but they claim it can be done. It consists of an ordinary glass bottle and a solid plastic rod which fits closely within the bottle. There is a marble in the bottle but the rod is held in the bottle by a screw and nut as shown below: | | | | | | || || || || || || / | | \ / | | \ / | | \ | | | | | | | | | | | | | |==|===|=|| | ---- Threaded screw and nut go thru threaded rod | | | | | | | | | | | __ | | ( ) | ---- Marble --------------- The object is to remove the marble. It is fairly easy to remove the nut but it is very difficult to turn the screw. It apparantly can be done without inserting a tool between the bottleneck and the rod. It costs about $10.00 and is available from Games People Play in Cambridge and Miscellanea in Acton. Has anyone tried it?  Received: from ARDEC-LCSS.ARPA.ARPA (TCP 30003004013) by AI.AI.MIT.EDU 6 Jan 87 14:57:48 EST Date: 6 Jan 87 14:43:00 EST From: "CLSTR1::BECK" Subject: To: "cube-lovers" Reply-To: "CLSTR1::BECK" From: CLSTR1::SYSTEM 5-JAN-1987 16:10 To: CLSTR1::BECK Subj: Undeliverable mail ----Transcript of session follows---- "ai.ai.mit.edu" is an unrecognized hostname/address ----Unsent message follows---- Date: 5 Jan 87 16:02:00 EST From: "CLSTR1::BECK" Subject: bottleneck To: "mdc" Reply-To: "CLSTR1::BECK" The puzzle is trivial. It takes about 5 minutes to disassemble and 10 to assemble. No sopecial tools are needed. ------ ------