Date: 12 JUL 1980 1342-EDT From: ALAN at MIT-MC (Alan Bawden) To: CUBE-LOVERS at MIT-MC Welcome to the cube-lovers mailing list. I don't know what we will be talking about, but another mailing list cannot hurt (too much). Mail to cube-lovers@mc or cube-hackers@mc whatever strikes your fancy.  Date: 15 July 1980 01:41-EDT From: Jef Poskanzer Subject: Complaints about :CUBE program. To: CUBE-HACKERS at MIT-MC This is a real-neat program, BUT: In solving the cube, the description it gives for the action it is about to perform can be ambiguous, I think. For example, can't "Turn TOP center to BOTTOM" be performed in two non-equivalent ways? I don't know what the display format is on color LISP machines, but on my terminal is sucks. How about something like this: +--------+ / YL YL YL \ / \ / YL YL YL \ / \ / YL YL YL \ ,-+------------------+-. ,-' | | `-. ,-' GR | RD RD RD | BL `-. +-' GR | | BL `-+----------+ | GR | | BL | OR OR OR | | | | | | | GR GR GR | RD RD RD | BL BL BL | OR OR OR | | | | | | | GR | | BL | OR OR OR | +-. GR | | BL ,-+----------+ `-. GR | RD RD RD | BL ,-' `-. | | ,-' `-+------------------+-' \ WH WH WH / \ / \ WH WH WH / \ / \ WH WH WH / +--------+ Maybe add FRONT, BACK, LEFT, RIGHT, TOP, BOTTOM labels; maybe compress it vertically so it fits on 24-line screens; maybe three-char abbrevs instead of two; maybe the back face should be shown reversed (i.e. from the inside of the cube looking out) to facilitate mental manipulations; the basic format is better, though, don't you agree? --- Jef  Date: 15 JUL 1980 1413-EDT From: ALAN at MIT-MC (Alan Bawden) To: CUBE-HACKERS at MIT-MC Since we have this mailing list I am tempted to put on record an interesting property of the cube that some of you might not be aware of: The cube has a degree of freedom that is rarely considered. The six center faces which are the pivots about which rotations are performed do not always return to their original orentation. In other words, if you were to paint little arrows on the center square of each face of a solved cube, randomized the cube, and then solved it, you might well find that the arrows no longer pointed in the same directions as they did before. (Now you have to get them back. And you thought you had already solved the damn thing!) This extended cube problem has been solved independently twice to my knowledge (not by me, I cheated and learned someone else's solution). The property was first shown to me by Spencer Love. Does anybody know if anybody else ever discovered it independently? Without considering this property we already knew that there were 43252003274489856000 = (8! * 3^8 * 12! * 2^12)/12 different rearrangements of the cube (all those numbers are obvious except the 12 in the denominator). Considering this property raises that number to 88580102706155225088000 = (8! * 3^8 * 12! * 2^12 * 4^6)/24 (the 24 being just as hard to explain as the 12 was).  CMB@MIT-ML 07/15/80 15:06:58 To: cube-lovers at MIT-MC I had known about the center faces being turnable for a while, and have a cube that I have marked up so I could work on solving the center face problem. In general, I solve the cube except for the center faces (since that was what I already knew how to do), and then have three transforms: one that will turn any center face 180 degrees and leave everything else unchanged, one that will turn one center face 90 in one direction, and an adjacent center face 90 in the other direction, and the last turns two adjacent center faces 90 in the same direction. My transforms are rather long and rep(it)*ous.  Date: 15 JUL 1980 2158-EDT From: ALAN at MIT-MC (Alan Bawden) To: CMB at MIT-ML CC: CUBE-HACKERS at MIT-MC The last two transforms you describe sound similar to the two I learned. Mine are also rather repititous. Perhaps it is the case that the two configurations are very distant using the obvious metric: smallest number of twists from one to the other. I wonder if anyone knows very much about the nature of that metric anyway? I understand that it is known that no two points are more than 94 (or is it 93?) twists apart (disregarding the extended problem). I don't know if that number is actually attained, or if it is only the currently known upper bound based on the best algorithm. (Or perhaps there isn't an algorithm that good yet, just a proof of the fact.) I believe that you and ACW and I once did the math to show that whatever that longest distance is, it has to be greater than something around 30, and for the extended cube problem it must be even bigger than that, so since I can do the transformations you speek of in about 28 (I think) moves, those must not be most distant points.  Date: 15 July 1980 2245-edt From: Bernard S. Greenberg Subject: Cube minima To: CUBE-LOVERS at MIT-MC I believe the 94 number comes from Singmaster's book. UNFORTUNATELY Singmaster doesnt seem to know what the word "canonical" means, and 180-degree twists count as single moves "too" in his religion. This makes his number kind of worthless.  ED@MIT-ML 07/15/80 23:21:07 Re: Singmeister who? To: cube-lovers at MIT-MC I've never heard of Singmaster's Book. Is it in the Old Testament or the New? Is it prophetic, historical or lackadaisical?  Date: 16 July 1980 09:35 edt From: Greenberg.Multics at MIT-Multics Subject: Singmaster To: cube-lovers at MIT-MC David Singmaster is a prof at an English university whose name escapes me, who publishes a 40-page pamphlet on cubing for about $5.00. It took me months to get; I have it for xeroxing in my posession. It contains many transforms, including many wierd solve-the-edge-cubes-first solutions. As I have pointed out, his noncanonical move counting is a problem.  ACW@MIT-AI 07/16/80 15:08:09 Re: Cubespeak To: CUBE-HACKERS at MIT-MC I would like to see some talk about a good language for describing cube manipulations. I know Bernie has one that he swears by, but I would rather not see the discussion START with everybody giving their favorite cubespraak... this can get confusing, dogmatic, counterproductive, nobody listens, etc. So let's keep the "My language is better than your language" to a minimum at first, and see what desiderata people consider as basic. My first contribution is: I think that turning the cube over, rotating it, etc., without performing any twists -- that is, all the things that you could do just as well to a solid block of wood -- SHOULD be considered manipulations in their own right. This includes performing a mirror reversal. Why? So that manipulations that only differ in starting orientation will not have representations that look completely different. ---Wechsler  Date: 16 JUL 1980 2051-EDT From: RP at MIT-MC (Richard Pavelle) To: CUBE-LOVERS at MIT-MC A simple transformation with a pretty resulting design: Let CS = center slice which faces you and is parallel to the axis of your body. Then CS up 90, rotate cube 90 in either direction so that the CS turns in the direcion of rotation. Performing these two operations 3 more times gives the desired result.  Date: 16 Jul 1980 2130-PDT (Wednesday) From: Lauren at UCLA-SECURITY (Lauren Weinstein) Subject: confusion To: CUBE-HACKERS at MC I am relatively new to the whole idea of cubing, in fact I do not yet even own a cube. I am a littel confused as to what the state of the art is in solving an actual randomized Rubik's Cube. Are these algorithms so complex that they are only usable with computer aid, or at least pencil and paper? Or are they such that, handed a random cube, you can just sit there and (given sufficient time) put it right? --Lauren-- -------  Date: 17 July 1980 01:22-EDT From: Alan Bawden Subject: confusion To: Lauren at UCLA-SECURITY cc: CUBE-HACKERS at MIT-MC Given a randomized cube it can take about 5 to 10 minutes to set it straight with no aid whatsoever. Pencil and paper or a computer can be a great deal of help when one is first learning to solve the cube (I used both), but I know of no one who uses such aids once they have learned how. A method of solution that required some computational aid to perform (some hairy calculation based on the current configuration of the cube, resulting in a single 259 twist sequence that brings it immediately back to the solved state) is not inconceivable, but most people have solutions composed of short, easily comprehended steps. Can anyone tell me who this Rubik character is? His name appears to be attached to the new American version of what some of us once knew as the "Hungarian Cube". Is Rubik the Hungarian who invented it? Has he done anything else? I heard this rumor that there was a 4x4x4 cube out there somewhere, anybody else heard about it?  Date: 17 JUL 1980 0846-EDT From: JURGEN at MIT-MC (Jon David Callas) To: CUBE-LOVERS at MIT-MC Yeah, sounds great, I always loved linear Alg. I'm not really COMPLETELY sure what you're doing, but count me in. It sounds like I might learn something. Thanx, Jurgen@mc  Date: 17 July 1980 09:40 edt From: Greenberg.Multics at MIT-Multics Subject: Re: confusion To: Alan Bawden cc: Lauren at UCLA-Security, CUBE-HACKERS at MIT-MC In-Reply-To: Message of 17 July 1980 01:23 edt from Alan Bawden For the record, Erno^H" Rubik is the hungarian teacher of architecture and design at some Budapest equivalent_high_school , who invented the Cube. Apparently he has a solution, but it is not a particularly good one. I would also like to add to ALAN's note that given my method or ALAN's method, I have never seen either take longer than 5 minutes (I promise clocked solutions in under 4 and have taken much less) and Singmaster has heard of solutions in 2 minutes, but I find this difficult to believe. I think I can also assert that most of the certified Cubemeisters did NOT use a computer in solving it, although it can be of great help. For the record for the purposes of this list, my algorithm is implemented in the :CUBE program that runs on all ITS's.  Date: 17 JUL 1980 1042-EDT From: RP at MIT-MC (Richard Pavelle) Subject: confusion but simplicity To: Lauren at UCLA-SECURITY CC: CUBE-HACKERS at MIT-MC I do not use a computer but I found it useful in keeping track of moves and testing various transformations. I can fix the cube now in 5 minutes or less. In fact, I have won cubes at stores by challenging the owner to a freeby if I can do it under 10 or 15 minutes. My transformations are very easy to remember because after one face is complete I need only 4 to fix the cube. The transformations are positioning the corners, toppling them, fixing edges and then toppling edges. These are not designed for speed but for simplicity. I have taught some people to solve the cube this way with a few hours of practice. To do it in 5 minutes, however, requires a few more transformations.  Date: 17 JUL 1980 1114-EDT From: ACW at MIT-MC (Allan C. Wechsler) Subject: Short Introductory Speech To: CUBE-LOVERS at MIT-MC Nobody else has made this kind of flame yet, so I guess I will. Welcome to CUBE-LOVERS. We are devotees of a certain mathematical puzzle variously called the Hungarian Cube, the Magic Cube, and Rubik's Cube. It is a hard puzzle. Very intelligent people often take weeks to learn to solve it. Once they have learned, though, they can solve it in a few minutes. The puzzle embodies mathematical sophistication and mechanical ingenuity in a pleasing and intriguing synthesis. I have forgotten the Hungarian inventor's name, but we should learn it: this person deserves our profound respect. For those who have not yet become Cube Solvers: you can only solve the Cube for the first time ONCE. After that, although there are a lot of problems to think about, the initial challenge is gone. So, in the words of Mr. Duffey: SPOILER WARNING! SPOILER WARNING! Messages to this list will often deal with particular solution techniques. If you haven't solved the cube yet, and want to do it on your own, reading these messages may ruin your fun. If there is any demand, I am willing to hack up a short introduction to Group Theory for Cubans. Group Theory gives us a mathematical language for talking about the cube. Also, if anyone out there knows Hungarian, there are some pamphlets we need to translate. Cubans should inform everyone on the list of any written material they know of, so that we can compile a bibliography. Happy cubing, ---Wechsler  Date: 17 JUL 1980 1420-EDT From: RP at MIT-MC (Richard Pavelle) Subject: the cross design To: CUBE-LOVERS at MIT-MC Does anyone know a transformation (or series of) which will produce the following pattern on each face where X and O are two different colors. |X O X| |O O O| |X O X| I think this may not be possible which brings me the question which is how one can say whether a particular arrangement is or is not possible.  Date: 17 July 1980 14:38 edt From: Greenberg.Multics at MIT-Multics Subject: Re: the cross design To: RP at MIT-MC (Richard Pavelle) cc: CUBE-LOVERS at MIT-MC In-Reply-To: Message of 17 July 1980 14:20 edt from Richard Pavelle There are two such sets of patterns known, one with three sets of paired colors (the Christman cross) and one with two triplets of colors (the Plumer cross). The Plummer cross is achieved by two orthogonal applications of the transform to the Christman cross. The transforms are fairly long and hairy, and I hesitate a little before attempting to describe them, but will if people want.  Date: 17 JUL 1980 1635-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: the cross design To: RP at MIT-MC CC: CUBE-HACKERS at MIT-MC Yes, the cross design is possible. I will let BSG try and describe it if people want. It is fairly straightfoward to tell if a position is reachable. I am thinking of onlining the proof that there are 12 equivalence classes of cubes (I think I have a fairly simplified version), and if I do it should contain an algorithm to tell if a position is reachable.  Yekta@MC (Sent by ___117) 07/17/80 16:45:00 Re: Checker board pattern... To: cube-lovers at MIT-MC Is the pattern in which evry face of the cube is a checkerboard of two colors possible?? ( I have not played with the cube at all, so my apologies if it is too trivial to get... )  Date: 17 July 1980 17:33 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Checker board pattern... To: Yekta at MIT-MC (Sent by ___117) cc: cube-lovers at MIT-MC In-Reply-To: Message of 17 July 1980 16:45 edt from Sent by ___117 Yes. This is absolutely trivial. Rotate each face 180. Si x total onehunred-eighty degree twists.  Date: 17 Jul 1980 1358-PDT (Thursday) From: Mike at UCLA-SECURITY (Michael Urban) Subject: Confusion To: lauren CC: cube-hackers at mit-mc Actually, dealing with the cube is a learn-as-you-go experience. The appeal of the cube, which makes it superior to Soma, or "Instant Insanity", etc, is that you actually have to analyze what's going on in order to approach the solution. For example, I began by learning how to put one face right; this required certain simple transformations that were useful later. As you go, you develop your own heuristics for moving the faces you need around without messing up what you've done so far. I can do the whole thing from an arbitrary starting position in around 20 minutes; I'm still not very adept at moving corners around. Even after you have solved it, there are still things to do with the cube, including improving your personal algorithm, as well as creating nifty patterns, etc. A Worthy Toy. According to "Games & Puzzles" magazine, Rubik is the Hungarian fellow that devised this evil little time-stealer. The article also impies a 2x2 version is in the works, which is even harder to understand mechanically than the 3x3 version. How the heck IS the thing put together, anyhow? Mike -------  Date: 17 July 1980 17:38 edt From: Greenberg.Multics at MIT-Multics Subject: Re: the cross design To: ALAN at MIT-MC (Alan Bawden) cc: RP at MIT-MC, CUBE-HACKERS at MIT-MC In-Reply-To: Message of 17 July 1980 16:35 edt from Alan Bawden There is a wonderful alogorithm for ANY pattern, which constitutes an existence proof. Given that you know how to "solve cubes", you can achieve a given pattern if it is achievable simply by "solving" to that position, which may in fact be faster than some set of arcane transforms. Before the "neat" algorithms for the Cruces Plummeri et Christmani were discovered, they were achieved by cubemeisters only by "solving" to that position, lambda-binding the target state (lessee, this guy wants to go here, etc.) to the desired pattern. I will burn some neurocomputrons tonight to describe the algorithms for the Crosses. Incidentallly, if you try to solve for some pattern and come to a roadblock of the form "this cant come here because its here" (we need two of these cubies ( a local jargon for the little cubes)), or a parity/trinity argument, you have proven that you can't achieve it.  Date: 17 July 1980 17:33 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Checker board pattern... To: Yekta at MIT-MC (Sent by ___117) cc: cube-lovers at MIT-MC In-Reply-To: Message of 17 July 1980 16:45 edt from Sent by ___117 Yes. This is absolutely trivial. Rotate each face 180. Si x total onehunred-eighty degree twists.  Date: 17 July 1980 21:18 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Confusion To: Mike at UCLA-Security (Michael Urban) cc: lauren at UCLA-Security, cube-hackers at MIT-MC In-Reply-To: Message of 17 July 1980 18:12 edt from Michael Urban The easiest way to see how the thing is put together is to take it apart. If you turn the "top" plane 45 degrees, you can pry out the top edge cube (any of the 4) fairly easily, and it becomes clear how the devilish thing is put together. In three lines or less, all the center-of-face pieces are on a 6-armed, solid, rigid cross. They can spin, but that's it. The EDGE pieces (i.e., not the corneres) have a rectangular projecction from their non-visible edge which gets them stuck behind whichever two center-of-face pieces they're currently visiting. While not rotating, it's stuck behind two. while rotating it (a edge piece) is stuc behind the one on its rotating face, and a groove that all edge pieces have on their sides, in this case on the edge pieces of the next plane back. The corner pieces have little cube-like projections on their invisible corners that basically wedge in behind the edge pieces, which are stuck as desribed above. No magnets, wires, universal joints or rubber bands. IF youshould decide to take one apart, be SURE to put it together SOLVED to ensure solvability.  Date: 17 July 1980 2228-edt From: Bernard S. Greenberg Subject: The Higher Crosses To: CUBE-LOVERS at MIT-MC For the amusement of the experienced cubemeister, and the education of those desirous of learning the Art, I have here produced a Description of the Methods, as I have used, for the production of the Cruces Plummeri & Christmani. These are the elegant configigurations of lid Crosses of which we spoke earlier today. ********************************************************************** I herein describe the algorithms for construction of the higher (Plummer, Christman) crosses from a solved position. From an unsolved position, it is faster to "solve" directly to the desired configuration; by following the steps below, the eager cubist may learn exactly what these configurations are. Of the words and phrases I use: I call the faces front, back, right, left, top, bottom. A face has 9 cubies, viz., 4 corner cubies, 4 edge cubies, and its center cubies. Separating 2 opposite faces, is a "center slice", being of 4 center cubies and 4 edge cubies. As I hold the cube, I call three center slices: floor-parallel, body-parallel, body-slicing. For instance, the body-parallel and body-slicing centerslices meet in the front face. I name "double-swap" the transform which is performed as follows: Double-swap (front, back) ;parameter-faces Turn body-slicing centerslice 180. Turn bottom face 180. Turn body-slicing centerslice 180. Turn bottom face 180. Observe well what it has done, viz. swapped the two cubies of the turned centerslice on the front with those of the back. You will use it as needed during the following shenanigans: ---------------------------------------------------------------------- To achieve Christman's (DPC at MIT-MC) Cross, the simpler of the two: Rotate the body-slicing centerslice 180. Rotate the floor-parallel centerslice 90 either way (your choice). Stare hard at what you have. The CORNER CUBIES and CENTER CUBIES are in their final position for the Crux Christmani; all further hacking will be simply to move the EDGE CUBIES, IN PAIRS, into place. To achieve ANY Crux Plummeri or Crux Christmani configuration, learn how to do the initial rotations (see below for the CP) so that you get the center cubies to corners you want, and hack from there. I will now describe the edge-cubie moves for the CC given that the centerslices have been aligned to orient the center cubies as needed: Among the six faces you now have, find one of the two that have a solid stripe between two sides of the same color, i.e., x y x x y x x y x and align it like so, so that the stripe is vertical, and this face is the front. Note that the edge cubies of the y y y stripe want to be exchanged with the two x-showing edge cubies, i.e., x x x y y y x x x (Remember that the goal is x y x/y y y/x y x) You can tell that they want to be inthe horiz. positions by their non-showing faces, which you will observe match the center-cubies on the right and left sides. To do this: 1. Perform doubleswap on front-back. 2. Rotate the FRONT so that when you do (3), the two cubies we just moved to the back will come to such place so that when we undo this step (see 4), they will be in the right place. This will be either 90 deg. left or right. 3. Perform doubleswap on front-back. 4. Undo step 2, i.e., turn FRONT 90 deg the "other" way. Whehter you blew (2) or not, you will now find you have (x x x/y y y/x x x) on front. If you understood 2 and DIDNT blow it, you will have the sides of the y y edge cubies matching the side centers (if you blew it, doubleswaps on the side faces can fix you up). You will see the floor-parallel centerslice begin to form a band. We will now finish that band. The two appropriate cubies (to go in the two rear positions of the floor-parallel centerslice are now on the front plane, the x x cubies of the last step. Note that a simple doubleswap on front-back would move them to the back face, but the WRONG two places on the back face. Easy. So, turn the back face 90 degrees and do the doubleswap, and unturn the back. Choose which 90 such that these two cubies wind up in the right place. You will now find you have solid bands and solid crosses galore. The front and back should have solid crosses, and the floor-parallel slice should now be a solid band. Look at the top of the cube. Make it the front. Orient it so that it is (a b a/c c c/a b a). Do a front-back doubleswap, and now look at the remaining face pair we havent been thinking about. Do the appropriate doubleswap on them to get solid crosses, and then you should have the Crux Christmani. Study well what you have: three pairs of alternated crosses. ---------------------------------------------------------------------- The Crux Plummeri (after DCP at MIT-MC who first came up with it, altho by solving-to) is exactly equal to doing the entire above transformation twice, at 90 degrees. The following, however, is a direct route from solved that is more intuitive. Take the cube, turn the body-slicing centerslice up 90 deg. Turn the floor-parallel centerslice 90 deg clockwise as seen from the top. Note well the configuration of corners versus centers; it is the final one. Note that you will have two triplets of trebly-interleaved colors: that is the characteristic of the CP. Look at the TOP or BOTTOM. Let's say the TOP. Make it the front. Orient it so that you see x y x x y x x y x Only the top or bottom look like this; this is what you have to remember to look for after aligning centers to taste. We're gonna rotate the y y y band into the horiz position. Do this exactly as for the CC above, producing (x x x/ y y y/x x x) Next goal is again to complete the solid band of the floor-parallel centerslice by doubleswapping front/back so that the x x edgecubies,w hich would complete that band, go to the back. Of course, we must temporarily rotate the back during this doubleswap, so that they go to the side positions ofthe back when swapped. Do so, completing the solid color-band of the floor-parallel slice. Now consider the top and bottom. You note that exactly one appropriate doubleswap between top and bottom would give us solid crosses on both. Do it. Take what had been the top just now, and call that the front. Note that there are solid crosses on front and back, and the body-parallel plane is correct and complete. Think about the front: it looks like a b a b b b a b a Although it looks right fromt the front, the two vertical b-edge cubies want to be the two horizontal b-edge cubies, as a cursory inspectionof the top bottom and sides of the cube will show. This is true of the back, as well. Tofix up the FRONT do this: 1. Doubleswap front/back 2. Rotate the FRONT (temporarily) 90 degrees sothat the two vertical b-edge cubies are gonna come to the right place, 3. And doubleswap front/back 4. Undo 2. 5. Doubleswap front/back. Now you see all is right save the back. It wants the same thing done to it. Do it for it; Do this same thing just doNe in the last 5 steps for the back (viewing it as the temporary front). It is done. Consider it. An exquisite variant ont he CP is obtained by taking on of the trebly-bound sides and rotating the centers via the well-known center-cubie rotating algorithm. As the centers are rotated left or right, either a sextuple checkerboard or a stunning triply-rotated canon of centers , edges, and corners appears. The checkerboard is amusing insofar as it appears to a novice cubist to be the Pons Asinorum 6tuple checkerboard made by 6 twists (described earlier today), but cannot be fixed (solved, or produced) without the consummate hair of the CP that only true cubemeisters can execute. The application of the Pons Asinorum checkerboard transform to the CP (as well as the CC) produces interesting and suprising results. ---------------------------------------------------------------------- The Higher Crosses are fascinating insofar as they appear to be very simple edge-cube hacks, but are in fact quite "far" from home; the CP being exactly twice as "hairy" (far) as the CC (discovered by ALAN) is in itself a source of wonderment.  Date: 17 JUL 1980 2245-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: The Higher Crosses To: Greenberg at MIT-MULTICS CC: CUBE-HACKERS at MIT-MC Date: 17 July 1980 2228-edt From: Bernard S. Greenberg ... The Higher Crosses are fascinating insofar as they appear to be very simple edge-cube hacks, but are in fact quite "far" from home; the CP being exactly twice as "hairy" (far) as the CC (discovered by ALAN) is in itself a source of wonderment. I wish I could really say something was "exactly twice as far" from home as something else. Unfortunately, as I complained before, the metric by which one measures the distance of one configuration from another is not well enough understood to be able to make claims like this. It might well be that the CP is less than twice as far as the CC, all we can really be sure is that it is not any further than that.  Date: 17 July 1980 22:52 edt From: Greenberg.Multics at MIT-Multics Subject: Postscript to above To: cube-hackers at MIT-MC I should have noted in the above flamage (btw, will all of those of us who are paid by our respective employers to hack this lunacy please let me know at once) that there is room for opitimization and lookahead in the final doubleswaps in correcting the top/bottom, and the doubleswap preceding it, but I do not do this, so that I might better keep track of what I am doing.  Date: 17 July 1980 22:54 edt From: Greenberg.Multics at MIT-Multics Subject: Bug in above To: cube-hackers at MIT-MC In the terminOlogy section, note that the body-slcing and FLOOR-PARALLEL centerslices meet in the front face, not the body-parallel and body-slicing as given.  ED@MIT-AI 07/18/80 00:12:53 Re: Patterns, designs &c. To: cube-lovers at MIT-MC To jump the gun slightly on the group-theoretic explanation, any sequence of rotations of any number of faces can be thought of as an atomic "transformation" for the purposes of group theory. One of the precepts of this theory is that any such transformation, repeated often enough, will return the cube to the original state. For instance, given the transform "rotate top ccw 90", 4 iterations suffice to return the cube to the original state. Mike Speciner, a fellow Camexian, claims that no transformation can be created that requires more than 216 (=6^3) iterations to return to the virgin state. (He doesn't yet have a cube, but has been stealing his daughter's blocks and modeling the cube with them.) Where does this number come from, and is it true? I have been playing with various transforms, and have found at least one reasonably trivial one that requires the 216 iterations: rotate a face 90, then turn the cube 90 and repeat. The transform here is 4 twists in a band around the axis of cube rotation. The patterns generated in the process are interesting, too, though none of them are as unique as the cruciform or center-face patterns.  Date: 18 JUL 1980 0205-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: Patterns, designs &c. To: ED at MIT-AI CC: CUBE-HACKERS at MIT-MC Er, perhaps I don't understand the move you describe, but in any case, by my calculations it would take 1260 repetitions to come back home doing that one (1260=2^2*3^2*5*11). It is certainly the case that 1260 > 216 so that number must be wrong (I could be miscalculating, but I don't see how). 1260 is rather similar in appearance to 216, perhaps you spazzed somehow?  Date: 18 JUL 1980 0351-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: 1260 To: ED at MIT-MC CC: CUBE-HACKERS at MIT-MC I now have a proof that no element of the cube group can be of order greater than 1260. Since you have so thoughfully provided me with an element of order 1260, I must conclude that this element is indeed the maximum, as you claimed (but where did the number 216=6^3 come from?). My proof contains no nice derivation of the number 1260, you will be dissapointed to see where it comes from, it is just all that is left after a number of cases have been eliminated. Perhaps someone can devise a "nice" proof of this fact. Is this fact in the literature? (Bernie?)  Date: 18 JUL 1980 0932-EDT From: RP at MIT-MC (Richard Pavelle) To: CUBE-LOVERS at MIT-MC CC: ZIM at MIT-MC The file which contains all cube-lovers mail is ALAN;CUBE MAIL on MC.  ACW@MIT-AI 07/19/80 01:42:56 Re: Where to get them in the Boston Area, Cube Language. To: CUBE-LOVERS at MIT-MC "Games People Play" on Mass Ave, near the fork where Mt. Auburn St. branches off, carred them the last I knew. According to rumor, the domestic product ("Rubik's Cube"), selling for about $7, is mechanically inferior to the Hungarian import, costing $15. I don't know how everyone feels about money, but to me, not having to fight with the thing would be worth the extra $8. Bernie's explanation of how to achieve the Plummer and Christman Crosses is a prime example of why we need a cube language. Since no one has proposed anything, I will jump into the fray. Center the cube at the origin of a 3-d coordinate system. The axes of the coordinate system protrude from the centers of the faces. Make a hitch-hiker's gesture with your right hand and point the thumb along the X axis. Imagine rotating the WHOLE CUBE one quarter turn in the direction your curled fingers are pointing. I call this operation "I". (The X axis is the horizontal axis that does not skewer you.) If the cube was lying on a table, "I" would roll it toward you. Now point up, along the Y axis, with your thumb. A quarter-rotation in the direction your curled fingers point is the operation I call "J". The Z axis goes right through your belly. A quarter turn around it I call "K". Actually, K=IIIJI. To simplify things a little, we define I'=III, J'=JJJ, and K'=KKK. In general, M' is M done backwards. If we call the do-nothing manipulation "1", then MM'=M'M=1. For my own nefarious reasons I define "H" as the operation (unachievable in real life) of REFLECTING the cube right-to-left through the YZ plane. We note H'=H. Twisting the front (Z=1) face 90 degrees counter-clockwise I call "T". One more piece of notation: For any manipulations M and N, I write M'NM as N[M], reading this as if N were a function: "N of M". Note M[1]=M. Examples: T[I] means "Twist the top face" T[II]=T[JJ] means "Twist the back face" T[I'] means "Twist the bottom face" T'[J] means "Twist the left face CLOCKWISE" T[I] T'[I'] J' means "Rotate the floor-parallel center-slice a quarter turn counter-clockwise as seen from above." Note that (MN)'=N'M', and (N[M])'=N'[M]. Also notice that (MN)[P] = M[P] N[P]. To do the Pons Asinorum checkerboard: Set Q= (TT)[J] (TT)[ZJ] "Half turn body-slicing center-slice." Then the Pons is Q Q[J] Q[K]. Does anybody see what I'm getting at or am I a lone, mad genius? Set Q= T[J] T[J']. Then (Q Q[J])^3 is quite pretty. ---Wechsler  Date: 19 JUL 1980 0249-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: 1260 vs. 2520 To: ED at MIT-MC, CUBE-HACKERS at MIT-MC Sorry folks, there was a slight error in my last message about the maximum order of any element in the cube group. To understand why, let us examine the transformation that ED described. I like to label the cube this way: A0 AE A1 AB AX AD A3 AC A2 B0 BA B3 C3 CA C2 D2 DA D1 E1 EA E0 BE BX BC CB CX CD DC DX DE ED EX EB B7 BF B4 C4 CF C5 D5 DF D6 E6 EF E7 F4 FC F5 FB FX FD F7 FE F6 (I won't bother to describe the features of this labeling, and I will presume that the correspondence between this unfolded labeling and an actual 3D cube is obvious.) If I understand ED's directions this is the way the cube looks after applying his transformation: A3 AB A0 AC AX AE D1 DA D6 C3 CA E1 A1 AD F6 E6 EA E0 B0 BA B3 CB CX BF FB DX DE ED EX EB BE BX BC C2 EF E7 B7 CF B4 C4 DC C5 D5 DF D2 F7 FC F4 FE FX CD A2 FD F5 Note that the center faces BX, CX, DX and EX have all moved. My proof that the maximum order of any element is 1260 assumes that these center faces remain fixed. There is nothing wrong with that assumption, it simply relects the intuition that picking a cube up and putting it down again on a different face doesn't change the configuration at all. If we decide that the orentation of the cube DOES matter, then we get transformations like ED's here. The group we are dealing with is also 24 times larger than before. And my proof now shows that no element has an order greater than 2520 (twice as big). Could it be that ED's transformation is not actually a maximal one? Could there be one with higher order? Or can my proof be tightened up some, so that even in the larger group the maximum order remains the same? To answer these questions (hopefully) I present a transformation that I claim has order 2520: D2 AB A0 DC AX AE D5 AD A1 C2 CD C5 F5 DA D1 E1 EA E0 B0 BA A2 CA CX CF FC DX DE ED EX EB BE BX AC C3 CB C4 F4 DF D6 E6 EF E7 B7 BF A3 B4 FD F6 BC FX FE B3 FB F7 (This transformation, like ED's, is not a member of the usual group, but the larger one where the center faces are allowed to move.) It is easy to check that this permutation has order 2520, the real question is whether you can get to this configuration from a solved cube. I haven't actually tried to do that, but I am fairly certain that it is possible. If anyone would like to try it, and report to me their success or failure, I would be very grateful.  Date: 19 JUL 1980 0346-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: OOPS To: ACW at MIT-AI CC: CUBE-LOVERS at MIT-MC Let me suggest the following corrections to your cube language message. If any of these are not actual errors, but are actually misunderstandings on my part, then I apologize for causing undue confusion, but I thought it would be helpfull to people trying to understand you system to have these corrected as soom as possible. ACW@MIT-AI 07/19/80 01:42:56 ... One more piece of notation: For any manipulations M and N, I write M'NM as N[M], reading this as if N were a function: "N of M". I think you must intend N[M] to mean M N M' . ... T[I] T'[I'] J' means "Rotate the floor-parallel center-slice a quarter turn counter-clockwise as seen from above." T[I] and T'[I'] don't touch the floor-parallel center-slice at all and J' rotates it clockwise (as seen from above), my guess is that the description should simply read: "Rotate the floor-parallel center-slice a quarter turn clockwise as seen from above." ... Set Q= (TT)[J] (TT)[ZJ] "Half turn body-slicing center-slice." This must be a typo or some discarded notation. My guess is that this should read Q= (TT)[J] (TT)[J']. I don't think that the description here ("Half turn body-slicing center-slice") fits, but it works in producing the pattern anyway. If you had said Q= (TT)[J] (TT)[J'] II then you could get the pattern and fit the description too!  Date: 19 Jul 1980 11:27 PDT From: McKeeman.PA at PARC-MAXC Subject: Re: Where to get them in the Boston Area, Cube Language. In-reply-to: Your message of 07/19/80 01:42:56 To: ACW@MIT-AI cc: CUBE-LOVERS at MIT-MC, Lynn.ES, Horning, Sturgis Your proposal for a language leads me to suspect that you have not seen: NOTES ON THE 'MAGIC CUBE' David Singmaster Mathematical Sciences and Computing Polytechnic of the South Bank London, SE1 0AA, England which you can get by sending him $3. The fourth printing runs 36 pages. It is definitely worth the money. Cube quality: I have handled about a dozen Hungarian import versions of the cube with enormous variation in quality. They were all the colortab-on-black variety. They sell out here in the discount stores like K-Mart for less than $9. I have also seen a colortab-on-white version which is not substantially different in quality from the Hungarian versions. My advice is to pick and choose as best you can from the ones on the shelf, regardless of what kind you get. As for your language proposal, I like the RLUDFB abbreviations given by Singmaster and generally used in Europe to your T plus whole cube moves. They mean rotate the (Right, Left, Up, Down, Front, Back) surface 90o clockwise. I also use rludfb for the inverses of them. However since (fortunately) these two notations have a null intersection, there is no disadvantage in mixing them. For consistency, however, I suggest ijk for counterclockwise moves, and IJK for clockwise ones. This does little violence to the quaternion origins where lower case is generally used anyway. The ' for inverse notation is fine; in fact I frequently prefer R' to r when I write down algorithms. The mappings then are, substituting f for your T: F = f' R = F[J] r = f[J] = R' L = F[j] l = f[j] = L' U = F[i] u = f[i] = U' D = F[I] d = f[I] = D' B = F[ii] b = f[ii] = B' It also solves the problem that I was having of saying IJK in English since there is no way to say it in "Singmaster". I didn't follow your Pons Asinorum checkerboard solution because I didn't understand the Z in Q, but in the mixed notation I think it is: rrllffbbuudd or, if Q = rrll; QjQkQ Your "quite pretty" is (lrfb)^3 = (lr (lr)[j])^3 I propose the following grammer for RCML (Rubik Cube Manipulation Language): algorithm ::= definition* move* definition ::= letter = algorithm ; move ::= letter | move' | move^digit | move[algorithm] | (algorithm) digit ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 where the letters RLUDFBrludfbIJKijk are predefined as noted above. --Bill  Date: 19 July 1980 1455-edt From: Bernard S. Greenberg Subject: General remarks To: CUBE-LOVERS at MIT-MC While watching these various "linguae cubisticae" make the rounds, I note one categoric deficiency, whose perception may indeed be due to the idiosyncracies of my own meta-algorithms. In specific, my algorithms (including my 200-line procedure of the other day) include DECISIONS of the form "find a face that has a pattern like-so" or "look around for the target cubie, and categorize its place like so". Now clearly, a general solution algorithm must include some expression of such decisions, although I find Singmaster's procedures deficient in this regard (he tends to say "Use enough of FOO transform to get all the..." often). However, the generation of specific patterns from another canonical state NEED not involve "decisions", although most of my procedures (and hence the lisp-macro language of :cube) do. I offer that looking for certain patterns during the course of even one of these pretty-pattern-generations leads to more rememberable (^= memorable) algorithms than even a well-subroutinized set of absolute moves. On the Physical Reality of cubes: I have seen three species of the genus Cubi, the original Hungarians (C. Hungaricus), which is black faced, cubes sold by Ideal (C. Americanus), the American Black-faced cube, and the American Whiteface (C. Albus). The last-mentioned was first shown to me by the lady in the Cambridge cube-store, who called me at home to make me aware of them when they first appeared (she says they are done by some Friend of Rubik in Virginia). C. Americanus seems a good deal lighter in weight and build than C. Hungaricus. While C. Hungaricus takes two to three weeks of constant use before they are loose enough for Concert performances, C. Americanus can be turned with one hand, yea, one finger while held by the rest of the hand WHEN NEW. C. Americani do not seem to bind at all; the Hungarians not only bind, but decompose and bind more. I confess a certain sentimental attachment to C. Hungaricus, on which I mastered the Art. The stiff, clean solidity of a new Hungarian is indeed reassuring, but speed of movment and non-binding is something else. I have not subjected Americanus to truly extensive use, and I do not know how it ages, but so far, they seem to show less CHANGE per time than the Hungarians, and are usable from the start. I think they will be a win. Of the American White-faced Cube, I have little to say; C. Albus is a curio item, and has nothing to recommend it over either of the others.  Date: 19 July 1980 1524-edt From: Bernard S. Greenberg Subject: :cube feature To: CUBE-LOVERS at MIT-MC Ok, everybody wants a way to tell :CUBE "I have a cube like so and so, solve it". The reason I have avoided doing this (other than laze) is that you may specify a non-reachable state. (Duplicate cubies, or purple faces, etc. are easy to check for). I am not convinced that there is any better or faster check for an inconsistent/unreachable cube than trying to solve it and blowing out: lest we find such a check (other possibility: run the program once silently and once loudly; if the first time fails (no, i will not make it list all cubes it cant solve) give an error), all internal breakpoints become user errors. The second issue is what is the best language to specify a configuration in? Comments?  Date: 19 JUL 1980 1548-EDT From: ALAN at MIT-MC (Alan Bawden) To: Greenberg at MIT-MULTICS CC: CUBE-HACKERS at MIT-MC Date: 19 July 1980 1524-edt From: Bernard S. Greenberg ... I am not convinced that there is any better or faster check for an inconsistent/unreachable cube than trying to solve it and blowing out: ... You mean I haven't convinced you of that yet? Show me how your cube is represented and I will write one for you. What would be so bad about blowing out anyway? What happens when you hand a person an impossible cube? his algorithm "blows out" eventually! There is no shame in being tricked into trying to solve a bad cube.  Date: 19 July 1980 15:53 edt From: Greenberg.Multics at MIT-Multics To: ALAN at MIT-MC (Alan Bawden) cc: CUBE-HACKERS at MIT-MC In-Reply-To: Message of 19 July 1980 15:48 edt from Alan Bawden I certainly believe you have shown me sufficient conditions for inconsistency, but until now I wa not aware that you claimed that they were necessary...  Date: 19 July 1980 16:31-EDT From: Ed Schwalenberg Subject: Nope, I guess we don't understand To: ALAN at MIT-MC cc: CUBE-LOVERS at MIT-MC First of all, I was sitting here inventing a GOOD cube notation, when McKeeman's message about his notation came in. I begin to fear that ACW was right, and we will all die before agreeing on a notation. I dislike all proposed notations so far, for the following reasons: ALAN's labeling of the cube for describing a POSITION is excellent; however it is not useful for describing transforms, which are operations and not positions. ACW's language I find baffling, principally because it lacks verbosity. I would much rather have a notation like [doubleswap] than F[X,[Y']] since it is descriptive. An analogy with Life is useful here; I think that the adoption of names like "traffic lights" is far more usable in the long run than any of [the-oscillator-of-period-two-composed-of-three-blots-placed- orthogonally-adjacent] or | . ... | . | . or "if you put 3 blots in a row, the center dot remains alive forever while the two outer dots appear first horizontally then vertically". Creating names like "The Christman Cross" is fun, and makes for interesting wordplay, even if you don't resort to Latin. So my proposed language is Augmented English, which has the great feature of being able to put in comments (a feature notably absent in the others). I urge people to describe the transform and its result in any message describing a nifty transform or pattern (provided both are known, of course!). RP's pretty pattern may not be what I think it is, since "pretty" is not a good description. I propose that this be called "swapping-centers-in-triplets" (procedural notation) or Twelve Squares (which is not a movie by Mel Brooks, but the positional notation). Bernie's comment about decision-making I think is important: it seems to me that cubemeisters do in fact approach the problem with a set of "subroutines", which are defined to NOT contain conditional branches. Doubleswap and checkerboard-all-faces are examples of subroutines. Conditional branching is generally simply the matter of selecting an orientation of the cube before applying a transformation, "setting up" the subroutine if you will. I think that in the case of generating patterns from a solved cube, branches are unnecessary but helpful: rather than say "doubleswap then rotate the cube 90 clockwise in Z and doubleswap again" saying "doubleswap, then doubleswap the remaining solid face" much more clearly indicates what is going on. (the examples above are spurious). I herein announce two patterns I have independently invented; I do not know if they are elsewhere available. The first is called Ten Squares by analogy with Twelve Squares, since it is highly related. Twelve Squares causes the 3 centercubies of three mutually adjacent faces to move to an adjacent face; three iterations of "swapping- centers-in-triplets" suffice to return to the solved state. Ten Squares, on the other hand, is a configuration wherein two opposite faces are solid, while the other 2 sets of opposing faces each possess the centercubie belonging to its opposing face. This is created by first swapping-centers-in-triplets, then swapping-centers-in-triplets again, only with the cube rotated 90 degrees away from you. Note that this results in the final centerslice rotation of the first transform and the first centerslice rotation of the second effectively combining into a single 180-degree centerslice rotation. To resolve the cube, simply do swapping-centers-in-triplets without regard to the orientation of the cube, then you are back to the trivially soluble Twelve Squares. The second I call Laughter, after the use of the string \/\/ in comlinks to signify laughter. It leaves the top and bottom faces solid, while causing pairs of opposing faces to have a diagonal stripe of the opposing color. (I propose that the color of the face opposite a given face be called the complementary color; my cube has complementary pairs of red-white, orange-yellow and blue-green.) To create Laughter, select a top-bottom pair, which will remain solid. Rotate the left and right sides clockwise (in "opposite" directions as viewed from the top, thus resulting initially in the top being 3 differently-colored stripes). I call this "splaying". Then rotate the cube 90 degrees while preserving the top/bottom orientation (i.e., rotate it about the Z axis.). Six iterations of splay/rotate suffice. Laughing the cube again solves it. Laughing the cube, then Frowning it (same as laughing, only rotate the faces anticlockwise) results in Four Crosses: 2 complementary pairs of crosses with the top and bottom solid.  Date: 22 Jul 1980 1211-PDT (Tuesday) From: Mike at UCLA-SECURITY (Michael Urban) Subject: Yet another checkerboard To: cube-lovers at mit-mc Nobody has mentioned (or discovered?) the fully-interlocked checkerboard pattern obtained by applying the Pons Asinorum transformation to the Plummer's Cross as suggested by Greenberg during his description thereof, and then performing a few triple-center swaps (unfortunately, I was playing rather randomly with the 3c swaps at the time and must leave them as an "exercise for the reader"). The six colors are interlocked in pairs of AB, BC, CD, DE, EF, and FA on each checkerboard. Quite mysterious. Mike -------  Date: 23 Jul 1980 4:10 am PDT (Wednesday) From: Woods at PARC-MAXC Subject: Xerox cube-lovers To: Cube-Lovers at MIT-MC This message is of (possible) interest only to Xerox members of Cube-Lovers, but there's no easy way for me to send it only to them; apologies to the rest of you. I've created a Laurel-format message file containing the Cube-Lovers mail (from the archive at MC) up through approximately last weekend. The message is in chronological order (instead of reverse chronological the way MIT's stupid mailer defaults) and has otherwise been cleaned up slightly (e.g., to get rid of lines just barely too wide). If you're interested in looking through this archive, it's in [maxc]Cube.mail. It'll probably be around for several days at least. -- Don.  Date: 23 JUL 1980 1044-EDT From: RP at MIT-MC (Richard Pavelle) To: CUBE-LOVERS at MIT-MC There is a short article on the cube written by Singmaster. The reference is Mathematical Intelligencer, Vol.2, #1, pp.29, 1979.  Date: 23 Jul 1980 1640-PDT (Wednesday) From: Mike at UCLA-SECURITY (Michael Urban) Subject: Clarification To: cube-lovers at mit-mc My description of the checkerboard pattern was clearly inadequate, due to haste and the fact that my cube was "borrowed" and scrambled before I had a chance to see precisely what was going on. The pattern in question is formed from Crux Plummeri by applying the "Pons Asinorum" transformation; from the resulting almost-checkerboard, a simple "12-squares" transformation will provide the 6 checkerboards; it isn't hard to inspect the almost-checkerboard to find a trio of center cubies to rotate. The resulting checkerboards are a completely interlocked set. In the following unfolded cube, A/B means that the center and corners are color A, and the edges are color B. A/B C/A B/D D/E F/C E/F I don't THINK anyone has mentioned this pattern; the checkerboard pattern mentioned by Greenberg consists of two cycles of three colors each, something like A/B B/C C/A D/E E/F F/D although I may have the handedness wrong. Mike -------  Date: 23 Jul 1980 5:23 pm PDT (Wednesday) Sender: Woods at PARC-MAXC From: Woods at PARC-MAXC, Boyce at PARC-MAXC Subject: 216 vs 1260 vs . . . To: Cube-Lovers at MIT-MC Regarding finding a sequence of twists that requires a maximal number of repetitions to restore the original state, it is indeed true that 1260 is the maximum, but the sequence suggested by ED (on July 18) is not it. (He thought its period was 216; ALAN thought it was 1260. It's actually 315.) This assumes that we don't consider the orientation of the center-face cubies nor the orientation of the cube as a whole. First off, note that the transformation given by ED was ill-specified, since he didn't say which way to rotate the cube relative to the rotations of the faces. In Singmaster's notation the two operations would be, say, LBRF and LFRB. It turns out these are each period-315, though it's not immediately obvious that the two should be at all similar. Having done (LBRF)^1, you'll find there are 7 edge cubies that have cycled through each other's positions, and the other 5 edge cubies have done likewise, 5 corner cubies have rotated 120 degrees in place, and the other 3 corner cubies have cycled through each other's positions and one of them has rotated. Hence, if we do (LBRF) a multiple of 9 times, the corner cubies will be back where they started, and if we do it a multiple of 35 times, the edge cubies will be restored, so if we do (LBRF)^315, all of the cubies will be back to the solved state. Perhaps the reason ALAN thought this was period 1260 was because it takes 1260 twists to restore the original cube. But since the transformation that we are repeating is a sequence of four twists, it actually has period 315. If you count total twists, you can obviously get "identity" sequences with lengths much greater than 1260. Or perhaps ALAN was considering the transformation to be LJ (twist left face, rotate cube about vertical axis). This indeed has period 1260, but it "cheats" in that we're not supposed to consider reorienting the entire cube as a legitimate operation. If we do, then it is again possible to get periods greater than 1260. To get a period-1260 transformation, we need to observe whence the period arises. If we do a sequence of twists, and then look at the cycles of the form "face X moved to where face Y used to be, face Y moved to where face Z was, . . . , face moved to where face X was", we take the least common multiple of the lengths of those cycles. If we repeat the sequence of twists that many times, the faces will moved around those cycles an integral number of times and end up where they started; if we stop short of that many repetitions, at least one of the cycles will be left stuck in the middle. (Apologies to any readers who are familiar with group theory and are bored by this verbosity.) To get a period of 1260, we need to have a cycle of 7 edge cubies, another cycle of 2 edge cubies where one of them is also rotated (so that the four faces on those cubies form a single period-4 cycle), a cycle of 5 corner cubies, and a cycle of 3 corner cubies where again one of them is rotated (forming a face-cycle of length 9). If, instead of the 2 edge cubies, we could get 4 edge cubies cycling with one of them rotated, that would be a period-8 cycle and the overall transformation would have period 2520, but due to the fact that it's impossible to swap exactly two cubies it turns out you can't get that combination of subcycles. Based on this analysis, it's not hard to construct a period-1260 sequence, although this one is almost certainly not the shortest such (it's 24 twists): F U F^2 u r f b l r f D^2 F b R U (R^2 F^2)^3 u r B  Date: 23 Jul 1980 1757-PDT From: DDYER Subject: impossible cubes To: cubr-lovers at MIT-MC Remailed-date: 23 Jul 1980 1758-PDT Remailed-from: Dave Dyer Remailed-to: cube-lovers at MIT-MC According to Singmaster, there are 12 disjoint orbits (his term) of cube positions. The article I read doesn't give a derivation, but the basic idea is clear. It should be possible to characterize the orbit of a given position as a function of the transformations that would have had to be done on each cubie ( as if it could be moved alone ) to bring it to its proposed position. Given such a charaterization, one could declare positions fed to :CUBE illegal without trying to solve the problem. This relates to my thoughts on the problem of inversion of an unknown transformation. Suppose you are given a Cube that is N unit moves from home, determine the correct sequence to get it there. I think a representation of the Cube in terms of combined tranformations on the cubies is the place to start. -------  Date: 24 JUL 1980 2115-EDT From: RP at MIT-MC (Richard Pavelle) To: CUBE-LOVERS at MIT-MC Many of you have seen this message before. However, since the mailing list of CUBISTS is growing rapidly I shall repeat this on occasion. A file of past cube mail is on ALAN;CUBE MAIL on MC.  Date: 25 JUL 1980 0845-EDT From: ZIM at MIT-MC (Mark Zimmermann) Subject: "blindfold" cube? To: CUBE-HACKERS at MIT-MC How long does it typically take for one to be able to work with only a mental image of the cube? (As in"blindfold" chess, soma, pentominoes....)  Date: 25 JUL 1980 0906-EDT From: RP at MIT-MC (Richard Pavelle) Subject: "blindfold" cube? To: ZIM at MIT-MC CC: CUBE-LOVERS at MIT-MC Date: 25 JUL 1980 0845-EDT From: ZIM at MIT-MC (Mark Zimmermann) How long does it typically take for one to be able to work with only a mental image of the cube? (As in"blindfold" chess, soma, pentominoes. I do not know of anyone who can do it blindfolded (I cannot). However, a few days ago mine fell into a swimming pool and I did it underwater with five gulps of air.  Date: 25 July 1980 10:53 edt From: Greenberg.Multics at MIT-Multics Subject: Re: "blindfold" cube? To: RP at MIT-MC (Richard Pavelle) cc: ZIM at MIT-MC, CUBE-LOVERS at MIT-MC In-Reply-To: Message of 25 July 1980 09:06 edt from Richard Pavelle This interchange very elegantly points out the difference between the Mathematician's view of the Cube and the Hacker(System Programmer-type)'s view. While an algorithm that looked at the initial state and categorized it ("Perform the folliwng 152 steps for configuration 106xy205a (left-handed) and it will be solved") would thrill mathematicians, practical cube-solving ALGORITHMS require iteration, conditionals, subroutines, and other program-like techniques. Ineed, many non-cognoscenti have watched me solve the cube, noting that I look at it only a very small percent of the time (seeing which subroutine to invoke, as you will, ) but they don't know that, and conclude that I "solve cubes basically without looking". The whole notion of algorithmic cubism is based upon "I will do this hairy thing and it will have this desired effect, and I need not thinK about the intermediate states, unless, god forbid, the phone rings, etc." To contain a cube map and intermediate states of transforms in one's head woud, in my mind, involve a greater skill than blindfold chess; it is not a normal human-memory capability, although doubtless institutiOnalized freaks with pathological intelligence/visulization problems (in the positive direction) exist who could conceivable do such a thing.  Date: 26 Jul 1980 1315-PDT From: Davis at OFFICE-3 Subject: Some (perhaps well-known) Transformations To: cube-lovers at MIT-MC I have just recently joined the cube-hackers mailing list, but I have read over all the old mail. I suspect, however, that some of what I have to say will be well-known already; on the other hand, I suspect that some of it is new. When I first got my cube, I had the advantage and disadvantage of working in a vacuum. Hence, my original solution was more complicated than it should have been, but I did go about it another way, and discovered a few interesting facts along the way. Having only seen one cube, I foolishly assumed that the color pattern was the same on all cubes, and my notation for a move was simply a color. A R (or Red) move meant that you look at the red face, and turn it 90 degrees courter-clockwise. I wrote a little hack based on that notation, and since all my notes are in that form, I will use it here. It should be trivial enough to convert it to the Left, Right, ... form. For reference, here is my color pattern in the "solved" condition: GGG GGG GGG OOO RRR YYY WWW OOO RRR YYY WWW OOO RRR YYY WWW BBB BBB BBB When I purchased my cube, the store was out of all but the demonstration model, so I never got a chance to mess around with a virgin cube, and I approached the problem as follows: My program would simply take moves and print out the condition after so many moves. It would also, given a sequence of moves, find the order of the move as a group element. I then tried a number of patterns, some from my experience with the cube, and some completely at random, and found the orders of those moves. If the order came out to be, say, 90, then I would do 30 and 45 moves to see what the cube looked like after that many moves. These half- and third- way patterns were often fairly simple, and after trying aout about 20 or 30 likely candidates, I generated most of the primitive transformations I used to solve the cube originally. I also generated (as half way positions) a large number of nice patterns, which are pretty to look at, but not of much use for solving a cube. In particular, laughter, mentioned by ALAN, is gotten by: 3(OYRW) I find 3(RYYR) a useful solving transformation, as well as 3(RYRRRYYY) and 3(RYYYRRRY) (these last two being commutators). I have not yet gotten a look at any of Singmaster's stuff, but these last two may be well known. Believee it or not, one of the transformations I used to solvee the cube originally was 45(RGWY), which has the net effect (after 180 moves) of flipping the R-Y and G-Y side cubies in place. I'm glad I found a better one than that. In fact, if you haul out any book on group theory, it is interesting to find group equations, and plug in random cube moves as the group elements to see what happens. Needless to say, things which group theorists find interesting usually tend to do interesting things to the cube. One other thing I discovered which also does interesting things to the cube is the following transformation: Leave the body-slicing center slice in place, and rotate the other two sides, one toward and one away from you. Then rotate the top and bottom 180 degrees, and rotate the sides back. This transformation also gives a cube configuration which looks like something in the center-slice group, but is not. -- Tom Davis -------  Date: 26 July 1980 22:25 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Some (perhaps well-known) Transformations To: Davis at OFFICE-3 cc: cube-lovers at MIT-MC In-Reply-To: Message of 26 July 1980 16:15 edt from Davis Out of curiosity (I am responsible for :cube), I'd be interested in seeing your code. Is it someplace accessible? -Bernie Greenberg  Date: 27 Jul 1980 1004-PDT From: Davis at OFFICE-3 Subject: Re: Some (perhaps well-known) Transformations To: Greenberg.Multics at MIT-MULTICS cc: cube-lovers at MIT-MC, DAVIS In response to your message sent 26 July 1980 22:25 edt Hi, You're welcome to look at the code, although it is hardly fit for human consumption. I wrote it essentially in one sitting, and stopped as soon as I solved the cube. There are some poorly thought out design features, etc. Anyway, you can get it at SAIL. It is written in PASSCAL, and is called MCUBE.PAS[1,TRD]. I scribbled in a few comments this morning, so maybe there is enough information to figure out how to use it. Unfortunately I am at home on a typewriter terminal, and I refuse to try to edit without a display. Good luck. By the way, I have never used :cube, and have no idea exactly what it does. Is there some documentation associated with it somewhere? Thanks, -- Tom DAvis -------  Date: 31 JUL 1980 1006-EDT From: RP at MIT-MC (Richard Pavelle) To: CUBE-LOVERS at MIT-MC IS IT POSSIBLE? I just got a note from Martin Gardner who plans an article about the cube in Sci.Am. sometime. But he claims that a person names THISTLETHWAITE has proved a restoration procedure of 50 moves and believes he can reduce it to 41. Gardner also says that a 2x3x3 solid is selling in Hungary, also by Rubic.  Date: 31 July 1980 10:52 edt From: Greenberg.Multics at MIT-Multics To: RP at MIT-MC (Richard Pavelle) cc: CUBE-LOVERS at MIT-MC In-Reply-To: Message of 31 July 1980 10:06 edt from Richard Pavelle OK, your'e on the spot..... you'd better produce these artifacts or we're all not gonna let you log out....  Date: 31 July 1980 13:06-EDT From: Alan Bawden To: RP at MIT-MC cc: CUBE-HACKERS at MIT-MC Date: 31 JUL 1980 1006-EDT From: RP at MIT-MC (Richard Pavelle) IS IT POSSIBLE? The Singmaster notes claim that Thistlethwaite had an 85 twist algorithm in an addenda dated November 30, 1979. I presume that since then Thistlethwaite has continued to cube-hack, so why not 50 (or even 41)? It should be noted that Singmaster insists on counting a 180 twist as ONE twist, so I presume that the 85 number is measured that way. How is Gardner counting? It is certainly possible. If you count twists Singmaster's way, you can show that there are positions at least 18 twists away from home. There is nothing to suggest that this might not in fact be the maximum. So there might be room for Thistlethwaite to lower his number all the way to 18! (If you count 180 twists as TWO twists, then a similar proof shows that there are positions 21 twists away from home. In a past message I reported that some of us had proved the existence of positions as far away from home as around 30. I believe that the reasoning that led to such a high number was incorrect. (Although I cannot prove that there AREN'T positions that far away, I now believe that I have never seen a proof that there ARE.))  Date: 31 Jul 1980 10:34 am PDT (Thursday) From: Woods at PARC-MAXC Subject: Re: 180 degree twists In-reply-to: ALAN's message of 31 July 1980 13:06-EDT To: CUBE-HACKERS at MIT-MC It appears that Singmaster, Thistlethwaite, and just about all the cube hackers I know at and around Stanford consider anything you can do with one wrist motion to be a single twist. Since this gives a more accurate measure of how complicated a sequence is, I'm happy with it. Why do you folks at MIT insist that you're right and the world is wrong? (I admit it complicates the notation. My own cube notation uses two chars per twist, one being the face and the other being the direction: left-arrow for counterclockwise, right-arrow for clockwise, down-arrow for 180 -- isn't extended ASCII wonderful?) -- Don.  Date: 31 July 1980 1413-EDT (Thursday) From: Sandeep.Johar at CMU-10A Subject: cube group theory. To: cube-lovers at mit-mc Message-ID: <31Jul80 141352 SJ61@CMU-10A> In one of the previous letters on the subject of cubing there was a letter from some one offering to put together a piece on group theory and cubing, I for one would certainly be interested in seeing it. When I took my cube apart and tried to put it together I wondered as to how many ways there are of putting it together so that the cube is unsolvable, i.e. how many equivalence classes are there which are 'wrong'. Any one worked it out, I would but group theory is unfortunately not my strongest subject. Sandeep  ACW@MIT-AI 07/31/80 14:28:38 To: cube-lovers at MIT-MC Yeah, we have a prejudice against regarding 180-degree twists as atomic. I understand your feeling that a 180-degree twist is intuitively a single operation. Many of the cube-hackers at MIT became interested in the mathematical aspects of the cube, and the preference for counting quarter twists arose from this (admittedly rather Spartan) mathematical viewpoint. When the cube first appeared, the mathematicians among us instantly exclaimed, with great delight, "Wow, here we have a group, whose elements are possible manipulations of the cube, and whose binary operation consists of following one manipulation with another." We immediately got interested in group- theory questions like, "What is the order of this group?" "Does it have any interesting subgroups?" and, in general "What kind of object is this group? Does understanding it help us solve the cube better?" There are several common ways of representing groups. One is as a subgroup of a permutation group. This doesn't really help in the case of the Hungarian Cube, because it is too close to what the cube really is: few new facts or insights are revealed. Another way is with generators and relations. This means, to list a few basic group elements from which the whole group may be derived by multiplying them together. We soon figured out (along with hundreds of other mathematically-inclined cube-hackers) that the whole group of possible manipulations could be generated from six elements: the quarter-twists of each of the six faces. This observation later turned out to be crucial in calculating the order (number of possible states) of the group. Hence our predilection for counting quarter-turns. The half-turns were already accounted for, and we thought of them as two juxtaposed quarter-turns. I guess some of us believe that the mathematical structure of the cube group is built on quarter- turns. Those whose delight in the cube is not mathematical will not agree: after all, a half-twist is as easy as a quarter-twist to perform. But you will miss things like the fact that many useful manipulations are 8, 12, or 24 quarter-turns long. If you count half-turns, you get a whole spectrum of random move counts, thus missing some fundamental (and as yet little-understood) kinship between these manipulations. Of course, if you are not interested in such things, any measure of complexity (why not count equator twists? why not penalize for counter-clockwise twists, since they are marginally harder for right-handed people to do?) will suffice. ---Wechsler  Date: 31 JUL 1980 1431-EDT From: RP at MIT-MC (Richard Pavelle) To: CUBE-LOVERS at MIT-MC Sorry to spoil your day, but..... As you may know Ideal Toy has international rights to the manufacture and distribution of the cube. I just spoke to a fellow from Ideal who is writing up a solution booklet for them and he told me the following: 1) When Ideal first bought the rights (and by the way Rubik gets nothing- his "institute" gets it all) they discussed various national promotion schemes. One of them was to offer 1000K bucks to the first person to solve it less than 1/2 hour- that right 1 million. 2) One of the first stores in NY to carry it put adverts in the paper offering $50 to anyone who could solve 1 face in 1/2 hour. They lost $3,000 before they ended the offer the next day. 3) A fourth grade girl from Florida solved the cube in 3 weeks and can now do it in under 5 minutes. 4) The 2x3x3 solid does exist and will be marketed by Ideal later this year. However, instead of colors it has "dominoe" like faces, whatever that means.  Date: 31 Jul 1980 16:44 PDT Sender: McKeeman.PA at PARC-MAXC Subject: Re: The shortest solution? In-reply-to: ALAN's message of 31 July 1980 13:06-EDT To: Alan Bawden From: (Bill) McKeeman cc: CUBE-HACKERS at MIT-MC, Lynn.ES A lower bound on the number of twists can be derived as follows: There are 4.3*10^19 distinct reachable arrangments of the cube. Suppose the moves are restricted to the (more than sufficient) set RLFBUD. Then there are at most six independent choices at each step and the number of reachable places is bounded by 6^n. That gives 6^25 < 4.3*10^19 < 6^26, or 26 moves as the (probably unachievable) minimum. If all single-hand-motion twists, R RR RRR L LL .... DDD are allowed, there are 18 choices, giving 18^15 < 4.3*10^19 < 18^16, or 16 moves as a minimum. This isn't very interesting since Singmaster has examples 18 twists away. If the orientation of the center squares is also considered, then the combinatoric is 8.8*10^22, and the minima are, respectively, 30 and 19.  Date: 31 Jul 1980 5:13 pm PDT (Thursday) From: Woods at PARC-MAXC Subject: Re: The shortest solution? In-reply-to: McKeeman's message of 31 Jul 1980 16:44 PDT To: CUBE-HACKERS at MIT-MC You can do better than that for a lower bound! Say you consider all single-hand-motion twists to be okay. Then yes, there are 18 such, but there's no point in twisting the same face twice consecutively, so after the first twist the tree branching factor is only 15. In fact, there's no point in twisting a face twice if the only intervening twist was done on the opposite face; if we look at the operations of the form "twist face X thusly and the opposite face thusly", there are 45 initial such operations, and 30 at each succeeding branch, but since some branches now represent two twists and some only one twist, it's much harder to compute the minimum depth of the tree. -- Don.  Date: 31 JUL 1980 2159-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: The shortest solution? To: McKeeman at PARC-MAXC CC: CUBE-HACKERS at MIT-MC Date: 31 Jul 1980 16:44 PDT Sender: McKeeman.PA at PARC-MAXC In-reply-to: ALAN's message of 31 July 1980 13:06-EDT From: (Bill) McKeeman A lower bound on the number of twists can be derived as follows: There are 4.3*10^19 distinct reachable arrangments of the cube. Suppose the moves are restricted to the (more than sufficient) set RLFBUD. Then there are at most six independent choices at each step and the number of reachable places is bounded by 6^n. That gives 6^25 < 4.3*10^19 < 6^26, or 26 moves as the (probably unachievable) minimum. This is not an improvement on my result. I (and the rest of the cube hackers I know) consider a unit move to be a 90 degree twist in EITHER direction. You are only considering CLOCKWISE 90 degree twists. Let me point out that if we were to count twists your way, we would no longer have a metric. Both the quarter twist method and Singmaster's method result in a measure of distance that is a true metric. Date: 31 Jul 1980 5:13 pm PDT (Thursday) From: Woods at PARC-MAXC Subject: Re: The shortest solution? In-reply-to: McKeeman's message of 31 Jul 1980 16:44 PDT To: CUBE-HACKERS at MIT-MC You can do better than that for a lower bound! Say you consider all single-hand-motion twists to be okay. Then yes, there are 18 such, but there's no point in twisting the same face twice consecutively, so after the first twist the tree branching factor is only 15. In fact, there's no point in twisting a face twice if the only intervening twist was done on the opposite face; if we look at the operations of the form "twist face X thusly and the opposite face thusly", there are 45 initial such operations, and 30 at each succeeding branch, but since some branches now represent two twists and some only one twist, it's much harder to compute the minimum depth of the tree. -- Don. Singmaster's notes are aware of these factors, that is how he improves on the 16 count computed by McKeeman to arrive at 18. Similarly I used the same factors to improve on the 12^n argument for quarter twists (which gives 19 as a lower bound) to arrive at the number 21. I also compute 24 as the quarter twist lower bound for the extended cube (considering orentations of the center faces).  Date: 1 Aug 1980 11:53 PDT From: Lynn.ES at PARC-MAXC Subject: Re: cube group theory. In-reply-to: Sandeep.Johar's message of 31 July 1980 1413-EDT (Thursday), <31Jul80 141352 SJ61@CMU-10A> To: Sandeep.Johar at CMU-10A cc: cube-lovers at mit-mc Singmaster ("Notes on the 'Magic Cube'", p. 12 of edition 4) gives 12 classes or "orbits" of assemblies, each one such that the other 11 cannot be reached without disassembling the cube. To quote Singmaster, "This also means that if we reassemble the cube at random, there is only a 1/12 chance of being able to get back to START," and, "It is advisable to reassemble in the starting pattern." Singmaster's notes have several pages of relevant group theory for those interested. /Don Lynn  Date: 1 Aug 1980 12:20 PDT From: Lynn.ES at PARC-MAXC Subject: Re: Sorry to spoil your day, but..... In-reply-to: RP's message of 31 JUL 1980 1431-EDT To: RP at MIT-MC (Richard Pavelle) cc: CUBE-LOVERS at MIT-MC 2) Another TRUE story of rubiking: The May Company department stores in the Los Angeles area held contests at four of their stores when they started carrying the cube, a couple of months ago. The object was to solve one face in three minutes with their scrambled cube. Reward was a 50 buck gift certificate. They allowed about 6 people every fifteen minutes to enter for two days. The store where I won (and my 10 year old daughter, my wife, and three of my next door neighbors) had 21 winners. I assume the other three stores had similar numbers. They also gave away lots of cube tee shirts for "good tries". I got my cube the night before the contest. But it took me another month to solve the whole thing. I might have worked faster at it if the megabuck prize had been offered. But then you all would have also. 4) Singmaster (edition 4, p. 34) reports on the Rubik domino (2x3x3 cubies), and says each cubie has spots like a domino (1 up to 9). They are to be lined up in numeric order. Top and bottom (the 3x3 faces) are two different colors. Unfortunately, Singmaster laments, magic square type patterns (all directions add to same number) are not possible, since all edges are even, all corners odd. The mechanics of the device are said to be "more complicated than for the Magic Cube." /Don Lynn  Date: 1 Aug 1980 15:47 PDT From: McKeeman at PARC-MAXC Subject: Re: The shortest solution? In-reply-to: ALAN's message of 31 JUL 1980 2159-EDT To: ALAN at MIT-MC (Alan Bawden) cc: McKeeman, CUBE-HACKERS at MIT-MC Alan, Sorry I missed your earlier words of wisdom on the subject. Anyway, I am interested in how you show any of these is, or is not, a metric. Can you forward same? Thanks, Bill  Date: 2 August 1980 01:55-EDT From: Alan Bawden Subject: a metric for the cube group. To: CUBE-HACKERS at MIT-MC, McKeeman at PARC-MAXC First off, a metric is a function (call it D) that assigns a non-negitive number to every pair of points in some set. This number is to be thought off as the distance between those two points. The function must satisfy the following: For all a, b and c 1) D(a,b) >= 0 2) D(a,b) = D(b,a) 3) D(a,b) = 0 if and only if a = b 4) D(a,b) + D(b,c) >= D(a,c) (Number 4 is usually called the "triangle inequality". It is the constraint that most makes D act like a distance, and not something random.) We wish to construct a metric on the set of all attainable cube configurations. So from now on, those lower case letters will represent cube configurations. Now we have recently been talking a lot about methods of counting the number of "twists" that it takes to perform certain manipulations of the cube. We have been looking for a function (call it T) that assigns a non-negitive integer to each manipulation. I claim that it is obvious that any such function should satisfy the following: For all M and N 1) T(M) >= 0 3) T(M) = 0 if and only if M = I (I is the identity manipulation) 4) T(M) + T(N) >= T(MN) (We adopt the convention of using upper case letters to represent manipulations. Also we shall use M' to denote the inverse manipulation from M.) Now manipulations can be applied to configurations to yeild other configurations. We use aM to denote the configuration resulting from applyint the manipulation M to the configuration a. (Note that (aM)N = a(MN), so we may omit the parens and simply write aMN.) Now how may we use our twist measuring function T to obtain a metric on the configurations? Again I think it is obvious that we wish the relationship D(a,aN) = T(N) to be true for all configurations a, and all manipulations N. It is easy to show that given that D(a,aN) = T(N), metric property number 1 is equivalent to twist measure property number 1. Similarly for numbers 3 and 4. But what about metric property number 2? Well, if T(N) = D(a,aN), and D(a,aN) = D(aN,a) (property 2!), and a = aNN', then we have that T(N) = D(aN,aNN') = T(N'). So the missing property of twist measures must be that T(N) = T(N'). So this means that if we agree that T(L) = 1, and we like metrics (how can we use words like "distance" unless we have a metric?), then T(LLL) = T(L') = T(L) = 1. We can argue about T(LL) some other time!  Date: 2 Aug 1980 12:26 PDT From: McKeeman at PARC-MAXC Subject: Re: a metric for the cube group. In-reply-to: ALAN's message of 2 August 1980 01:55-EDT (yawn) To: Alan Bawden cc: CUBE-HACKERS at MIT-MC Alan, I enjoyed your presentation. I am convinced that counting the RLFBUD manipulations will not give a metric. I do not, however, see an easy way to compute twists T(M). I think one gets a metric only if one takes the minimum over some set of manipulations. That is, take a set, AM, of atomic moves including their inverses, let AM* be the strings of AM, and |M| be the length of M in AM*. Then D(a, b) = min |M| such that a = bM defines a metric. D(a,b) would sometimes be undefined if AM did not generate the whole group. The recent discussion on shortest solutions is in fact about the maximum of such a T(M) for all M in some AM*. Bill  Date: 3 August 1980 02:20-EDT From: Alan Bawden Subject: more on metrics To: McKeeman at PARC-MAXC cc: CUBE-HACKERS at MIT-MC Yes, it is true that the four conditions I gave for twist measures don't guarentee that the function will behave anything like the kind of complexity measure we are looking for. I was only trying to show how some of the properties you might expect of a twist measure could be used to generate a metric, so I didn't actually need strong enough properties to ensure reasonable twist measures. The additional property I have been using to assure reasonability is the following: 5) For all M, if T(M) > 1, then there exists an N such that 0 < T(N) < T(M) and T(N) + T(N'M) = T(M). Note that N'M has the property that 0 < T(N'M) < T(M) (easy to show) so the two manipulations N and N'M are both "simpler" than M. We can thus easily show that any manipulation M can be expressed as the product of T(M) twists (where a twist is defined as a manipulation such that T(N) = 1).  Date: 4 Aug 1980 1156-PDT From: Dave Dyer Subject: cube permutations To: cube-lovers at MIT-MC I'm not quite satisfied with the numbers that have been quoted for various cube groups. I believe they are probably correct, but I haven't seen anything resembling a satisfactory sketch of a proof. Number of Reachable Positions: The standard calculation runs like : we have 8 corners that can be in all 8! arrangements, and 12 sides that can be in all 12! arrangements, except that the total permutation must be even. ( I'll talk about orientations later) I can accept this figure as an upper bound, but can anyone demonstrate that all the positions not ruled out can actually be reached? The only way I can think of is an actual construction, which means finding a generator for the group (for instance of corner cube positions) and showing that its order is 8!. This is somewhat less than elegant, and requires an unspecified bit of magic to 'find' a generator for the group. I also don't like the recourse to geometric arguments ( the corners and sides can't be interchanged ) but I am willing to accept it. Number of Reachable orientations: A similar line of reasoning is used to argue the number of reachable orientations : the amount of twist on all corners and (independantly) on all sides is a multiple of 360 degrees. I haven't seen a demonstration that all the not-forbidden orientations are actually reachable. Finally, I haven't seen a demonstration that the orientation and permutation subgroups are indepentant, that is, that you can get to an arbitrarily selected location and an arbitrarily selected orientation at the same time. This assumption is the basis for Singmasters claim that there are 12 orbits of cubes : ( even-spacial-permutation X even-side-orientation X one-of-three-corner-orientation = 2 X 2 X 3 = 12 ) -------  Date: 4 August 1980 20:36-EDT From: Alan Bawden Subject: cube permutations To: DDYER at USC-ISIB cc: CUBE-LOVERS at MIT-MC The part of the proof that shows that you can actually reach all the configurations in a particular equivalence class is not particularly elegant. Basically, you have to appeal to the details of a particular cube solving algorithm. For example, I have a tool that "flips" two edge cubies in place, without desturbing anything else. This tool shows that I can orient the edge cubies in ANY even permutation of the edge cubie faces. The fact that I cannot obtain any odd permutations is a result of the fact that a quarter twist is itself an even permutation of the edge cubie faces. Most people can examine their own cube solving tools and see that in fact, they are capable of obtaining all the configurations not forbidden by the familiar constraints.  Date: 4 Aug 1980 2204-PDT From: Davis at OFFICE-3 Subject: Cube Permutations To: ddyer at USC-ISIB cc: cube-lovers at MIT-MC The following two transformations are useful in demonstrating that all of the claimed elements of an equivalence class of positions can be reached. Using the RLFBUP notation, the transformation RB'R'B'U'BU reverses two of the corner cubies and leaves all other corner cubies in place (It does, however, shuffle around the side cubies, and does some random twists to the corners). If the above transformation is repeated four times in a row, everything is left exactly fixed, except that three corner cubies are each rotated one-third of a turn. By then performing the inverse of this operation on two of the three corners which were turned and a new corner, it is not hard to see that any two corners can be the only ones moved, and that they are each rotated one-third of a turn in opposite directions. If we look at just the corners alone, and ignore in-place rotation, since we can exchange any adjacent pair, we can obviously get to all permutations of the corners. A similar argument can be made to show that all the edge cubies can be arranged arbitrarily (permuted arbitrarily, that is). An easy transformation rotates three of them (among themselves) on a face, and since we are also allowed to rotate the face, it is easy to generate a transposition of any pair. Unfortunately, the two operations described above are not independent. If we just look at the blocks and label them ignoring color, a primitive (one-quarter turn) transformation moves four corners into four corners, and four edge cubies into four edge cubies. If this is viewed as a member of a permutation group, it is obviously even (the set being permuted is all the movable cubes). Thus, at least half of the positions are impossible. If we ignore the corner cubies, and look at the colors of the edge cubies, every primitive rotation rotates the four front colors, and the four colors around the outside, again, an even permutation. Since one of the above used coloring and no corner cubes, and the other did not, there must be at least a factor of 4 impossible positions. A much more complicated argument shows the necessity of a factor of three in the set of impossibles. (Does anyone know of a simple way to see this? I just did the obvious thing of defining "standard" orientations of every cube in every corner, and showed that all the primitive transformations caused the total rotation away from standard to be a multiple of 2 pi.) Using the transformation which flips any two in place, and the two discussed above, it is not hard to see that the factor is at most and at least 12. Boy, It sure is hard to prove things on a computer. On myy notes with diagrams and all, this is perfectly clear, but I get confused trying to read my online proof. I hope that someone can make some sense out of it. -------  Date: 6 AUG 1980 0939-EDT From: JURGEN at MIT-MC (Jon David Callas) Subject: Random Notes on Cubism To: CUBE-LOVERS at MIT-MC CC: JURGEN at MIT-MC I am beginning to feel competant enough to jump into the fray, so ... 1) language: I guess that RLUDFB has become the de facto lg. for cubing. (by the way, how IS 'RLUDFB' pronounced?) Is this to include the IJK rotations that ACW described on 7-19? They are very handy, but I know that there is a desire tokeep the lq. brief. Also what (if anything is being done with the center slices that Bernie used while describing the higher crosses? Given RLUDFB+IJK notation, they are easy algorithms (subroutines?) that can be described as: FPC {floor-parallel center} := udK BPC {body-|| center} := fbI BSC {body-slicing center} := rlJ without IJK, they're much more complex. Perhaps this is a case for using RLUDFB & IJK. 2) I have been playing a little with the group of rotations, twists and so forth. I thought that it mighht be very wieldy until I remembered that Don Woods has given us a transform with period 1260. (Is it just my imagination or is 1260 important? it seems like a familiar number) This means (by la Grange's thm ) that the order of this group is some integral multiple of 1260 (ugh!) That eliminates hand-enumerating (at least by me) it. How much does anyone else know about this, and are thre any interesting subgroups? It also seems that this thing might be related in some way to the dihedral gps. but I'm not sure. 3)Somewhat connected with (2) above, it would be nice if we compiled "Famous Transforms I Have Known" or something to that effect. Not only would it help in finding my desired sbgp, but it would help keep some of the folklore out of cubing (while there is nothing wrong with folklore, there are problems when falsities (especially VERY inobvious ones) creep into the body of lore. Folklore also does tend to cause duplication of effort and that most of us have probably been wasting time re-inventing the Pons Ansinorum (which, by the way, IS 'rrllffbbuudd', isn't it?) Since I'm suggesting this nonesense, I guess it's only fair to volunteer to do this compendium (unless someone else is dying to). 4) Would it be possible to modify :cube to work on a printing tty? I really don't care if it produces large volumes of paper. I am avoiding implementing my own form of a Cube program on my APPLE ][ (see my notes above on duplication of effort), if I can get away with using :cube, but at least the Apple display will be pretty. 5) There MUST be a better way to check for an impossible cube than solving for it. While there is no shame in solving a bad cube, there IS a little embarassment, and there certainly is no honor in it , either. Besides, the anti-intuitionist part of me finds solving-as-proof distasteful. 6) Has any one taken 'Rubik's Cube ' apart? I have tried prying the edge cubie as mentioned, and all I did is hurt my thumb. I'm afriad to use any sort of tool, lest I break the little beast. 7) Can we get a Cubing Bibliography, including Singmaster's articles, and the Games & Puzzles article and anything else anyone finds? (ONce again, since I suggested it, I'll do it if need be.) 8) Are Singmaster's 'orbits' REAL permutation-group type orbits, or just equivalence classes? I'm confused by his choice of words. -happy cubing, Jurgen@mc  Date: 6 August 1980 12:22-EDT From: Alan Bawden Subject: Reply to Random Notes on Cubism To: JURGEN at MIT-MC cc: CUBE-LOVERS at MIT-MC 1) I guess the RLUDFB notation is pretty standard, but the part about using lower case letters to represent inverses isn't. I find it pretty distasteful and much prefer using "'". This has the added feature that you can write: (RLU)' to indicate the inverse if RLU. (I KNOW that it is just U'L'R', but sometimes one wants to make the distinction.) 2) Yes, lots of things are known about the group and its subgroups. Start by reading old cube-hackers mail while you wait for your copy of Singmaster to arrive. 5) Yes, there is an easy way to tell if a cube is solveable without actually trying it. Here it is: Three conditions must be true: a) The sum of the rotations of the corner cubies must be a multiple of 360 or 2 pi (depending on how you count!). b) The permutation of the faces of the edge cubies must be even. c) The permutation of all the cubies must be even. 8) Yes, they are real orbits.  Date: 6 Aug 1980 09:50 PDT From: Lynn.ES at PARC-MAXC Subject: Re: Random Notes on Cubism In-reply-to: JURGEN's message of 6 AUG 1980 0939-EDT To: JURGEN at MIT-MC (Jon David Callas) cc: CUBE-LOVERS at MIT-MC I have taken my cube apart several times (several friends wanted to see its innards), with knife, screwdriver, letter opener, etc. It doesn't seem to hurt it. I assume you are referring to the "twist the top 45 degrees, then pry up on an edge" instructions, which I used. The springiness in the way the 6 centers are held together seems to allow this prying without damage. I think the compilation of transforms and bibliography is a great idea, and I will back your candidacy for the job. Several references of both types are buried in the previous messages to cube-lovers. I would favor using Singmaster's extensions of the RLUDFB to describe the results of a transform. Combinations of two of the set RLUDFB specify edges, and three give the corners. Then statements like "this transform moves URF to DBL" (or briefly "URF => DBL") imply not only the position within the cube, but the rotation of the cubie also. That is, the U face of URF corner moves to the D face of position DBL. /Don Lynn  Date: 6 Aug 1980 1131-PDT (Wednesday) From: Mike at UCLA-SECURITY (Michael Urban) Subject: Ideal Toys To: cube-lovers at mc Has anyone learned whether Ideal will be "supporting" Rubik's Cube with some vehicle like Parker's "Soma Addict" newsletter of some years past? They will certainly be missing a bet if they don't. Just the number of people who would pay for a solution... ...hmm...maybe we shouldn't suggest it to Ideal after all.. Mike -------  Date: 6 August 1980 1453-edt From: Bernard S. Greenberg Subject: Parsing cubes To: CUBE-LOVERS at MIT-MC I had been taking my cubes apart by hand, as described. I have found that they INDEED seem to suffer from it, at least the C. Hungaricus (see earlier letter, I have taken neither of the other species apart). They become disconnected and loose, and turn into a clattering collection of colligenous junk, to paraphrase the famous wizard. A fitting solution to answering visitors' queries about the inside of cubes lies in a retired cube (which broke due to impact with the floor, the central cross shattered) which lies in a little box in dissociated cubies for perusal of passers-by. This cube which served me well in life, now serves even in retirement, by saving me from having to take apart others.  Date: 6 Aug 1980 1223-PDT From: Dave Dyer Subject: 'cube lovers digest' To: cube-lovers at MIT-MC A U.S. Mail Cubists publication, in the grand traditon of Lifeline and the Soma newsletter, is a good idea. Some Cubist with time on his hands could be in business instantly, if he could convince Martin Gardner to plug it in his (allegededly forthcoming) Mathematical games column. -------  Date: 6 Aug 1980 1247-PDT From: Steve Saunders To: RP at MIT-MC, Cube-Hackers at MIT-MC We ought to adopt a distinctive name for cube-devotees. Several terms have been used in messages to this list; unfortunately, most of them are already taken by other groups: a "Cuban" is a citizen of Cuba, a "Cubist" is an artist of a certain school. I propose a new word for us: a "Cubik" (from Cube and Rubik) is interested in Rubik's Cube. This should be taken to mean anyone strongly interested in, or addicted to playing with, the cubes in question. a "Cubemeister" is a full expert in the manipulation of R'sC. This term, also used on this list, should remain for the restricted meaning of "one who has truly mastered the art of Rubik's Cube". Steve -------  Date: 6 Aug 1980 1241-PDT (Wednesday) From: Lauren at UCLA-SECURITY (Lauren Weinstein) Subject: Random Notes on Cubism In-reply-to: Your message of 6 AUG 1980 0939-EDT To: JURGEN at MIT-MC CC: CUBE-LOVERS at MC I have taken my cube apart. Simply rotate the "top" surface so that it is offset by 45 degrees to the rest of the cube. Then pry up one of the edge cubies (I used a small, thinbladed screwdriver.) The cubie pops right up, and the rest of the cube will come apart easily. It is really a spectacular design... it is so elegant and yet so simple. --Lauren-- -------  Date: 6 August 1980 16:46 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Random Notes on Cubism To: Lauren at UCLA-Security (Lauren Weinstein) cc: JURGEN at MIT-MC, CUBE-LOVERS at MIT-MC In-Reply-To: Message of 6 August 1980 16:14 edt from Lauren Weinstein I do not feel I am being overcautious or overobvious in warning the potential cube dismantler to only reconstruct the cube SOLVED after dismantling, lest it be placed (11 to 1 odds against you) in an unsolvable orbit by accident!!!  Date: 6 August 1980 16:48 edt From: Greenberg.Multics at MIT-Multics To: Steve Saunders cc: RP at MIT-MC, Cube-Hackers at MIT-MC In-Reply-To: Message of 6 August 1980 15:47 edt from Steve Saunders I have generally been willing to award the designation "cubemeister" to anyone who can reproducibly solve the cube from randomness , in an organized fashion (i.e., some small number of minutes). I feel it is reasonable to grant this title to those who have achieved this, even if their methods are highly unorthodox, or they have not familiarized themself with any other persons cube knowledge, etc.  Date: 6 August 1980 16:52 edt From: Greenberg.Multics at MIT-Multics Subject: Re: 'cube lovers digest' To: Dave Dyer cc: cube-lovers at MIT-MC In-Reply-To: Message of 6 August 1980 15:23 edt from Dave Dyer I am really interested in when the _ Gardner is going to get his act together and publish THE cubing article-- each month for the last n I have eagerly taken the cover of SciAm and been disappointed. This is clearly THE mathematical game (siNce the inception of SciAm, with the possible excpetion of Conway's LIFE), and I wonder what he's waiting for. About a week ago, I actually dreamt that I opened the SciAm wrapper and found the Cube on the cover, introducing a WHOLE ISSUE about it (Social implications, Ancient Cubing, Cubing in the Soviet Union, etc...)  Date: 6 Aug 1980 1:51 pm PDT (Wednesday) From: Woods at PARC-MAXC Subject: Singmaster to talk at Stanford To: Cube-Lovers at MIT-MC For those Stanford/Xerox/nearby people who aren't on the "AFL" mailing list (and haven't heard about this from Frances Yao, who arranged it): ------------------------------------------------------------ Date: 5 Aug 1980 1626-PDT From: CSD.ULLMAN at SU-SCORE Subject: AFL To: Algorithms for Lunch Bunch: ; The speaker on Aug. 12 will be David Singmaster. He will talk about the manipulation of the Hungarian magic cube. ------------------------------------------------------------ (AFL meets 12:30 Tuesday in Margaret Jacks Hall room 301. Bring a lunch.)  Date: 6 Aug 1980 1909-PDT From: CSD.VANDERSCHEL at SU-SCORE Subject: Orbit Classification To: CUBE-LOVERS at MIT-MC cc: CSD.VANDERSCHEL at SU-SCORE Having read the CUBE-LOVERS mail, I note a recurring theme of dismay that there is no clearly stated procedure for deciding whether a given configuration is reachable (or solvable) or not. Furthermore, I share Dyer's view that there has been no persuasive presentation of the fact that there are precisely 12 equivalence classes of permutations under the transformations permitted by the cube. In this note, I intend to clear up this situation for the benefit of the less sophisticated readers of this material. I should apologize at the outset for my ignorance. I have not read Singmaster's pamphlet, nor have I communicated with any of the knowledgeable people around here at Stanford on the subject of cubes. My knowledge stems entirely from my own personal efforts at solving cubes and what I have gleaned from CUBE-LOVERS mail. So I hope readers will be tolerant of any lengthy explanations of well-known cube-lore. COMMENTS ON NOTATION I will start by offering my two-bits-worth on notation. It is relevant because I will use some of it later. I prefer to call the cubies in the centers of the faces "face cubies", because "center" and "corner" both start with a "C". (Also, I have worked with other 3X3X3 cubes and tended to use "center" for the one in the middle that you cannot see.) I do not agree that there is any need for notation to describe orientation of the whole cube or reorientation of it. For discussion purposes, you should pick one orientation of the cube (based on the direction in which its face cubies are oriented) and leave it that way. All transformations can be described relative to that orientation. Rotating a center-slice should not be considered to be a move, since it changes the orientation of the cube. Since it is not the case that all cubes have the same color pattern, no direct reference should be made to color. Instead the exposed faces of any cubie should be identified in terms of the direction (LRFBUD) they will face when the cubie is in its home (or at START) position relative to the chosen orientation of the cube. When I was learning to work cubes, the concept of complementarity played a critical role. (I do not claim my methods are good. It takes me about 20 min.) It is certainly handy to have a way of talking about pairs of opposing faces. I think most would agree that the top and bottom can be referred to as "horizontal" faces, and left and right can be called "lateral" faces. For front and back, I really do not know a good word; I suggest calling them "extremal", since their face cubies are the closest and most distant from the observer's point of view. ORBIT CLASSIFICATION (for the uninitiated) Introduction - It is possible to define some "parity" concepts that simplify stating the characterization of the equivalence classes of cube configurations. The precise definitions will follow; but to start with, we will name them: Edge Permutation Parity (EPP = 0 or 1) Edge Orientation Parity (EOP = 0 or 1) Corner Permutation Parity (CPP = 0 or 1) Corner Orientation Parity (COP = 0, 1, or 2) A configuration is reachable if COP and EOP are zero and CPP=EPP. More generally, setting TPP=CPP+EPP mod 2, we can define Total Permutation Parity (TPP = 0 or 1). Then the Parity Vector, defined by (TPP, EOP, COP), can be used to represent the equivalence class of a configuration. Permutation Parities Defined - Background: A permutation of [1,n] is considered to be even (0) or odd (1) depending on whether an even or odd number of pair swaps is required to restore the set to original order. You can compute it by counting the number of reversals in the sequence - ie., the number of pairs, (p,q) such that p>q and p precedes q. For the cube, you can assign a position number (1-8) for each of the corners and also a number (1-12) for each of the edge positions. For each cubie, you can the speak of its home position number and its current position. Writing down home position numbers in order of current position gives a permutation of natural numbers in which you can count reversals to see if it is odd or even. This is done without regard to orientation. Properties: A quarter turn is an odd permutation of the four edges involved and also of the four corners. Thus a quarter turn changes both EPP and CPP but leaves TPP unchanged. TPP is preserved by all twists. Edge Orientation Parity Defined - For each edge cubie consider its Oriented Distance from Home, defined to be the smallest number of quarter turn twists required to put it in its home position with correct orientation. It is no bigger than 4. As an example, an edge cubie at home with the wrong orientation is at an oriented distance of 3. EOP is the sum modulo 2 of the Oriented Distances from Home for all edge cubies. Properties: For each edge cubie affected, a quarter turn either increases or decreases its Oriented Distance from Home by 1. Since 4 edge cubies are affected, the net effect must be even. Thus all twists preserve EOP. Corner Orientation Parity Defined - Looking head-on at the apex of any corner you can consider twisting it to any one of three positions. For any given corner cubie we define its individual orientation parity to be the number of 120 degree counter-clockwise twists required to bring the horizontal face of the cubie into horizontal position relative to the whole cube. COP is the sum modulo 3 of the individual parities. (Since it is three-valued, it could be argued that COP ought not to be called a "parity", but that fouls up the parallelism in the discussion.) Properties: Twisting a horizontal face does not change the orientation of any corner. Twisting an extremal or lateral face does alter orientations. Consider a counter-clockwise quarter turn of either kind of vertical face. For each of the two corner cubies that remain in the same horizontal face, the orientation parity decreases by 1 modulo 3. For the other two, it increases by 1. Again, the net effect is zero. COP is preserved by all twists. Equivalence Classes - There must be at least 12 orbits, since all transformations preserve the Parity Vector, (TPP,EOP,COP), and it has 12 possible values. To show there cannot be more, you must show, given two configurations with the same Parity Vector, that one can be transformed into the other. The first paragraph of the note from Davis to DDYER, dated Aug. 4, indicates how the corners can be permuted and reoriented. We need to be a little more careful about the interaction between the processes that rearrange corners and edges. Suppose we are considering going from a configuration A to a configuration B, and consider the intermediate stage C at which we have permuted and reoriented the corners to agree with the target configuration B. First we observe that the EPP of C must agree with that of B, whether or not A and B have the same CPP. Thus there is an even permutation of the edges of C that will put them in the desired target positions of B. This permutation can be generated by swapping appropriate pairs of edge cubie pairs. Consider the intermediate stage D achieved after all edge cubies are in their desired positions. Now the EOPs of D and B agree, so the number of edge cubies with wrong orientation must be even. This can be corrected by flipping an appropriate set of edge pairs in place. The Extended Problem - If the orientations of face cubies are to be considered then we introduce Face Orientation Parity (FOP = 0 or 1). It is the number of quarter turns modulo 2 required to get all the face cubies back to starting position. Adding this fourth component to the Parity Vector yields 24 equivalence classes. Existence of appropriate transformations for moving about within equivalence classes was indicated by ALAN and CMB in their notes of Jul. 15. Complementary Configurations - A configuration may be said to be complementary if every color exposed on a face of the cube either agrees with that of the face cubie on that cube face or comes from the parallel opposing face. There seems to be a fair amount of interest in such positions. It is easy to see for such configurations that COP and EOP are 0. Thus the configruation is reachable if and only if its TPP is 0. This can be determined easily by counting "wrong" cubie faces - ie., edge or corner cubie faces with the color complementary to that of the face cubie on the same cube face. TPP is 0 if and only if this number is a multiple of four. (It is always even.) If you restrict yourself to 180 degree twists, you can generate only complementary configurations. Thus the more interesting complementary configurations tend to be those that require quarter turns to achieve. A REDUCED PROBLEM ? Suppose you lock up four of the axles and consider turning just two adjacent faces. This Two-Faced puzzle deals with only 15 of the cubies. I got the bright idea of playing with a logically jammed cube after I had already developed a crude methodology for working cubes. I figured that with a simpler problem I might gain some new insights. Actually, I got confused. In retrospect, this is not so surprising. I think that most of us would grant that if you can work Two-Faced cubes, you can certainly work the whole thing. The point is that you tend to encounter many of the same sorts of problems, but you have fewer degrees of freedom for dealing with them. The argument already given shows that the Two-Faced cube must have at least 12 equivalence classes. It is not clear to me that there are not more in this case. I have not discovered all the tools to show that two members of the same equivalence class (in the regular sense) can be transformed into one another. Can anyone out there resolve this issue? -------  Date: 7 August 1980 0930-edt From: Bernard S. Greenberg Subject: Singmaster Talk To: CUBE-LOVERS at MIT-MC I was wondering, given Woods' message, if anybody on either coast would like to corral Singmaster into speaking in Massachusetts, and if someone on the West coast would let us know what he says......  Date: 7 Aug 1980 09:49 PDT From: McKeeman at PARC-MAXC Subject: Proposed Glossary To: Cube-lovers at MIT-MC Cube -- Any interesting cube-shaped puzzle. To cube -- Doing anything with a Cube. Cubology -- The science of Cubes. Cubologist -- One who (seriously) studies Cubology. cuber -- One who cubes. cubnik -- A fanatik cuber. cubelet -- A (conceptually) cubic subpiece of a Cube. cubie -- A cute cubelet. Cubemaster -- An exceptional cuber or Cubologist. Cubophobia -- A mental disease found in mates and co-workers of cubniks. Cubomania -- A mental disease found in cubniks. Cubotomy, Cubectomy, Cubotherapy -- Cures for Cubophobia, Cubomania.  Date: 7 Aug 1980 1246-PDT From: Dave Dyer Subject: improve your cube To: cube-lovers at MIT-MC I just squirted a tiny amount of dry graphite lubricant into the cracks of my cube, and it now works MUCH more smoothly. -------  Date: 7 Aug 1980 1345-PDT From: Stuart McLure Cracraft Subject: Where to find them? To: cube-lovers at MIT-MC Does anyone know where I can find one of these cubes? The old cube mail mentions May Co. and K-Mart but we don't have any of those around here. -------  Date: 7 Aug 1980 1438-PDT From: CSD.VANDERSCHEL at SU-SCORE Subject: Re: Where to find them? To: McLure at SRI-KL, cube-lovers at MIT-MC cc: CSD.VANDERSCHEL at SU-SCORE In-Reply-To: Your message of 7-Aug-80 1345-PDT A sure way, but not so fast and not so cheap (about $14), is MARKLINE in Mass. Call 1-800-225-8390 and use your plastic money. -------  Date: 7 August 1980 1751-EDT (Thursday) From: Sandeep.Johar at CMU-10A Subject: Re: Where to find them? To: Stuart McLure Cracraft CC: cube-lovers at mit-mc Message-ID: <07Aug80 175123 SJ61@CMU-10A> In-Reply-To: Stuart McLure Cracraft's message of 7 Aug 80 15:45-EST The cube is available via mail order from: LOGICAL GAMES, Inc 4509 Martinwood Dr. Haymarket, VA 22069 I have not tried it, so there is no guarantee. I found an ad in some magazine (i do not remember, but could find it if some one wants to know the type of mag.). They ask you to send $9 for the cube, and/or $4 for "notes on the magic cube". VA residents to add sales tax. Or you can do what I did, telephone every toy store in the area, till you find one that stocks it. - Sandeep  Date: 7 Aug 1980 2:58 pm PDT (Thursday) From: Woods at PARC-MAXC Subject: Re: Where to find them? In-reply-to: McLure's message of 7 Aug 1980 1345-PDT To: Stuart McLure Cracraft cc: cube-lovers at MIT-MC Also try: Stanford Bookstore and Gemco.  Date: 8 Aug 1980 1702-PDT From: Isaacs at SRI-KL Subject: where to buy them To: cube-lovers at MC In the Palo Alto/Stanford area, the cheapest place to buy the cubes seems to be Toys-R-Us (in Redwood City) at $7.85. They say they sell the dIdeal version. Logical Games sells the US made version; they are the little guys, and perhaps should be supported. L.G. price is actually $7.50, plus $1.50 postage. Two, therefore, are 'only' $16.50. They were, I believe, the first U.S. importer; the owner is Hungarian and had bought two for his children after visiting Hungary. The children didn't like them, but he did, and he imported 1000 more and sent them around (including 10 to Stanford Book Store). When Ideal took over the rights (by offering the Hungarians more money, I assume), he started manufacturing them here - the white plastic versions. He is also a source for the Singmaster pamphlet, and an 'Anginvine' solution. -------  Date: 9 Aug 1980 1610-PDT From: CSD.VANDERSCHEL at SU-SCORE Subject: Two-Faced Cubing To: CUBE-LOVERS at MIT-MC Having posed the question about Two-Faced cubes, I could not resist thinking about it some more and I have pretty well resolved it. If you want to try your hand at Two-Faced Cubing, DON'T READ THIS ! Suppose we only allow twists of the Right and Back faces. Edge Orientations: You cannot change the orientation of any edge cubie. So there are 2^7 different ways to orient them. Edge Permutations: Any permutation is possible. (Must be even if corner permutation is.) Corner Orientations: Just like the complete cube. Corner Permutations: The twists generate a subgroup containing only one-sixth of the 6! possible permutations of the 6 corners. I do not know a simple way to see why this has to be so. (I have a messy argument that is no more persuasive than enumerating them.) Solving It - Do whatever it takes to put the two right-front corners in their home positions (without regard to orientation). Rotate the back face to bring (say) the URB corner to correct position. Of the six possible permutations of the remaining three corners, you must have the correct one. (That is, assuming you started from a reachable configuration. Otherwise, this is a way to find a unique representative of the equivalence class of corner permuations.) (RB')^3(R'B)^3 may be used to fix the orientations of the corners. (RRBB)^3 may be used to permute the edges. (You can always get the two edge cubie pairs you want to swap in vertical opposition.) Counting Things - There are 6!X3^6X12!/(6X3X2) reachable configurations. There are 6X3X2X2^7 equivalence classes. Anyone for Three-Faced Cubing? -------  Date: 10 Aug 1980 1709-PDT From: Davis at OFFICE-3 Subject: A solution subroutine To: cube-lovers at MIT-MC Someone in an earlier message suggested that it might be a good idea to keep a list of various interesting move combinations. My particular set of algorithms for solving the cube includes one which flips two side cubies in place, and leaves all the others fixed. The method I had been using to do this up until now was 26 twists long, and I think that it was fairly standard -- find a set of moves which flips one edge in place and randomizes the rest of the faces, then move another side into the same position and invert it. Others that I talked to since then have indicated that they used the same method. Anyway, today I found a 22 quarter-twist version which does the same thing. I would be interested if anyone has found shorter ones; I am also interested in seeing the various other primitive transformations that others use to solve the cube. Using the FBUDRL notation, here is my sequence: (F'U')^3(FU)^2L'U'LR'FFL'RU'LFF -- Tom Davis -------  Date: 10 AUG 1980 2154-EDT From: MJL at MIT-MC (Matthew Jody Lecin) To: CUBE-HACKERS at MIT-MC In the September issue of OMNI, in the "Games" section is a 2 page introduction to the cube, complete with a few sketches showing a solved cube, an example of rotating one side, and an example of a really RANDOMIZED cube. For those of us not familiar with the cube (I still can't find one...sigh) this article was very enlightening. It also includes the address from which one can get Singmaster's "Notes on the Magic Cube". {Matt}  Date: 10 August 1980 2252-edt From: Bernard S. Greenberg Subject: Edge-cubie flipping algorithms To: Davis at OFFICE-3, CUBE-LOVERS at MIT-MC Here is my (and my apprentice :cube's) 2-edge-cubie-flipping subroutine, which was named the "Spratt Wrench technique" after the fellow I was talking to while cubing when I discovered it. It is 24 quarter turns, but is frumiously easy to perform rapidly blindfolded. It is impossible to express it in FDRUBL (or FBURDL or FLUDBR or whatever we're calling Singmaster-notation; let the euphony of the acronym be the issue). Let F be the usual, front clockwise. Let Q (please dont pounce on me for introducing a notation) be rotating the floor-parallel centerslice clockwise as viewed from heaven above. (It is clearly not IMPOSSIBLE to express it in FLURBD or BRULFD or whatever, but it would be like analyzing a 4-voiced composition in terms of frequencies instead of harmonies.) Formula: (FQ)^4(QF)^4 The sequence (FQ)^4 is called a right-handed Spratt Wrench; (FQ^-1)^4 is a left-handed one. Studying the effect of both of these (which are used non-paired in other phases of my cube solution (the sequence given above is used only during the final cadenza)) is worthwhile; the equivalence of the formula above to ((rh wrench)(turn the cube over)(lh wrench)), the form in which I first used it, is due to ALAN. The elegant formula above is best executed by a right-handed person, who will grasp the Cube from above with his/her left hand, and use his/her right hand to alternately turn the FRONT (which we will allow to face right, for ease of manipulation), and the lower layer and two layers alternately (while holding the top) to accomplish the Q moves. This procedure can be done with great grace and rapidity, esp. on the Ideal cubes.  Date: 11 Aug 1980 09:51 PDT From: McKeeman at PARC-MAXC Subject: Re: Edge-cubie flipping algorithms In-reply-to: Greenberg's message of 10 August 1980 2252-edt To: Bernard S. Greenberg cc: Davis at OFFICE-3, CUBE-LOVERS at MIT-MC Bernard, Actually, using Alan's IJK convention and the macro facility I proposed, your 24 move edge-flip is: Q=U'DJ'; -- horizontal center slice 90o cw macro (FQ)^4(QF)^4 and (FQ')^4 You didn't introduce notation, you just weren't very formal. You may some trouble getting the metric people agree that it only takes 24 moves however. As for what we call the extended Singmaster notation, (1) I do not know that it originated with him, (2) I do not know that he cares, (3) it could be called RubikSong, CubeTalk, TwistSpeak, etc. More to the point, who bothers to name the notation of calculus, or group theory, or tensors. Once standarditis sets in, a notation just gets used, not named. For time being, I nominate RubikSong, giving a bow to both known sources of our pleasures. Bill  Date: 11 August 1980 15:17 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Edge-cubie flipping algorithms To: McKeeman at PARC-Maxc cc: Davis at OFFICE-3, CUBE-LOVERS at MIT-MC In-Reply-To: Message of 11 August 1980 12:51 edt from McKeeman Yes, indeed, you can express Q as U'DJ', that's legit. I can't see any way you can count (FQ)^4 as more than 24 quarter turns, for thats all it is. In the eyes of the people who count turns, not quarter turns, the QQ in the middle (4 quarter turns) can be expressed as U^2D^2, where U^2 and D^2 each count as one, which could be thus contributory to counting the formula as 22 turns, but surely no more than 24 by any kind of metric. By the way, my description of the left-handed wrench in the annotations was in error, it should have been (F'Q)^4, not (FQ')^4. (The formula (FQ)^4(QF)^4 is correct as stands).  Date: 12 Aug 1980 0823-PDT From: Davis at OFFICE-3 Subject: The Spratt Wrench To: greenberg at MIT-MULTICS cc: cube-lovers at MIT-MC Thanks for sharing the wrench with us. It is a truly wonderful tool. Another question I had had is now answered. It was basically: Once I get near the end of a solution, I often have more than one set of edge flips to do. Is there some scheme which will easily allow me to do more than one. After staring at how the wrench works it is clear that in many cases, with one or two preliminary flips, or flips between the applications of the two halves of the wrench, wonderous things can be accomplished. Also, having worked on my own for quite awhile, and being more of a mathematician than a computer scientist, I decided that a standard rotation was CCW, and most of my algorithms favor turns in that direction. In converting the wrench to my own particular quirks (I can't turn things CW very well anymore), I discovered that essentially any combination of F and F', and of Q and Q' work just fine. In other words, if "f" stands for F or F' and "q" stands for Q or Q', then (fq)^4 (qf)^4 is a wrench. (Better leave f and q fixed through the whole transformation, however.) This next comment is off the subject, but I had been meaning to ask it for some time. I have one of the white cubes, and after reading the message some time ago about underwater cubing, my cube "accidentally" fell into a pool too. Well, the white ones float, which I found very annoying. Does anyone know if the specific gravity of the hungarian or Ideal cubes is greater than 1? I'm also not so sure that I would recommend that others try to teach their cubes to swim -- mine now has a disturbing squeak as it turns -- I fear that something has rusted inside. Tom Davis -------  Date: 12 Aug 1980 15:10 PDT From: McKeeman at PARC-MAXC Subject: Singmaster's talk at Stanford To: Cube-lovers at MIT-MC David talked at ICME (Int. Conf. Math. Ed.) in Berkeley on 8/11 about using the cube to teach group theory. He talked to some Rubniks at Stanford this noon. Among the tidbits: Notes on Rubik's 'Magic Cube', Fifth Edition, Preliminary Version, 75 pgs., $5. Including: A Detailed Step-By-Step Solution, Thistlewaite's Best Algorithm (52 moves), Conway's Monoswop, Rubik's Duotwist and much more. Write: David Singmaster, Polytechnic of the South Bank, London, SE1 0AA, UK. He brought a 3x3x2 domino version, and a 2x2x2 Stanford homebrew which is apparently nearly identical to a Japanese patent showed up. The 2x2x2 is conceptually just a pasting of big overlapping corners on the standard 3x3x3 version although the one we saw was nicely machined in brass and some kind of ivory-like material. Singmaster counts double twists, e.g., R^2, as single moves. He doesn't see much use for the IJK whole-cube moves. The Thistlewaite algorithm goes from subgroup to subgroup as follows: Starting with a random cube, reachable by closure(F,B,R,L,U,D) = the full group 7 moves to a cube reachable by closure(F,B,R,L,U^2,D^2) 13 moves to a cube reachable by closure(F,B,R^2,L^2,U^2,D^2) 15 moves to a cube reachable by closure(F^2,B^2,R^2,L^2,U^2,D^2) 17 moves to a cube reachable by the identity ---- 52 moves total. Singmaster expects the 17 to be 15 by the time he returns to London. The Hungarians have cube races. A contestant take his/her cube out of its box and unscrambles the judges' randomizing in about 50 seconds. Apparently they file and lubricate their cubes with loving care to reach such speeds. The U.S. made white cubes violate no patents because Rubik never applied for foreign rights. Hmmmm. Is that ethical? Bill  Date: 15 Aug 1980 1558-PDT From: CSD.VANDERSCHEL at SU-SCORE Subject: Re: A solution subroutine To: Davis at OFFICE-3, cube-lovers at MIT-MC cc: CSD.VANDERSCHEL at SU-SCORE In-Reply-To: Your message of 10-Aug-80 1709-PDT I am surprised about all this talk about edge flipping algorithms. Once I learned the concept of mono-ops, I immediately proceeded to invent a very simple, and quite obvious, edge monoflip. You can do it by manipulating only the front and the horizontal center slice (hereinafter referred to as the "slice"). A virtue of the procedure I am going to describe is that you can think of it in geometrical terms and explain it that way. The result is that it is easy to remember and invert, and you do not need any notation. Hold the cube with the two edge cubies you want to flip in the U face. Have one of them, the current "target", in the UF position. You can think of its "socket" as moving with the F face. Now turn the target cubie over to the left side.(F') Then move it to LB position by turning the slice. Next flip its socket over to the right.(FF) Now put it back in its socket by giving the slice a half turn. Finally, return it to the U face. Don't worry about the fact that you have changed the orientation of the cube, as that will be fixed when you reverse the process for the other edge you want to flip. Conceptually, this is a 5 move sequence. Since two moves are slice moves, you have to count it as 7. If you want to count quarter-turns, it would be 10. In any case, it is simple. (Singmaster has published even shorter monoflips.) But to me the most important thing is that it is obvious on inspection that it has to do what you want. The way I look at it, you are using the slice for manipulation and storage. You remove the target from between its neighboring corners and put it back with the opposite orientation. (Is this as obvious to everyone else as it is to me?) The same way of looking at things allows you to create edge monoswaps. Just put the two edge cubies you want to swap in diagonally opposite corners of the slice and give it a half turn before putting them back. You can arrange it so that they will or will not be flipped. Such moves are their own inverses. Using the D face for storage and manipulation, you can also easily invent monotwists and monoswaps for corners in the U face. For all such moves that I have created, it is apparent to me what needs to be done and why it will work. No memorization of obscure move sequences with magical effects. -------  ACW@MIT-AI 08/15/80 19:27:57 Re: "Monoflips" To: CUBE-LOVERS at MIT-MC M. van der Schel has come up with a very elegant move, indeed. Easy to do, easy to think about. But he sets no records. A 22-qtw diflip was reported some time ago to this mailing list. It is true that the monoflip takes only 10 qtw. Unfortunately, this does not allow you to do a 20-qtw diflip, since the monoflip is only useful as a tranform: in other words, when accompanied by its inverse. Monoflip U Monoflip' U' still has 22 qtw. I suspect this to be a minimum. ---Wechsler  Date: 15 Aug 1980 2347-PDT From: Stuart McLure Cracraft Subject: Undocumented position? To: cube-lovers at MIT-MC I've been attempting to use the 'method' Singmaster offers in his 5th edition pamphlet for solving my cube, and have arrived at an undocumented position. Perhaps I'm overlooking something and a cubeophile out there can offer an explanation of where I'm going wrong. I have reached the position described after step 6 just before proceeding to step 7. The bottom and top faces are one color, the center slice cubies are corrrectly colored and the upper edge cubies (the + pattern) are correctly oriented. Two of the upper adjacent corners are correctly oriented but the opposite two are not. At this point Singmaster deals with putting the rest of the U corners in their correct slots although not orienting them correctly necessarily. If we represent my U corners with 1 2 3 4 then the cubies want to be rearranged as follows 1 and 2 want to be exchanged I don't see any way of getting this operation out of the cases documented by Singmaster. What gives? -------  Date: 16 Aug 1980 0010-PDT From: Mclure at SRI-KL Subject: Aha. To: cube-lovers at MC Looks like one of his alternatives does appear when the cube is turned upside down. -------  Date: 16 AUG 1980 0338-EDT From: MJL at MIT-MC (Matthew Jody Lecin) To: CUBE-HACKERS at MIT-MC Seems the ever popular CUBE is getting places. In this month's issue of one of America's popular men's entertainment magazines there is a picture of Zsa Zsa Gabor holding a cube, and a small explanation of what the cube is, and who's marketing it, etc. (Isn't Zsa Zsa Hungarian?) The way she is holding her cube you see 2 faces - orange and green - and it is right. {Matt}  Date: 16 August 1980 12:45 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Undocumented position? To: Stuart McLure Cracraft cc: cube-lovers at MIT-MC In-Reply-To: Message of 16 August 1980 02:47 edt from Stuart McLure Cracraft Would somebody out there who has Singmaster version 5 benefit us with a brief (or as long as you can stand) description of the algorithm for solution contained therein, as the rest of us wait for International Snail Mail? Thanks, -Bernie.  Date: 16 Aug 1980 1053-PDT From: Stuart McLure Cracraft Subject: Singmaster's method To: Greenberg.Multics at MIT-MULTICS cc: cube-lovers at MIT-MC In-Reply-To: Your message of 16-Aug-80 1245-PDT Well, it does have a copyright on the method itself so I doubt it would be reasonable to type the thing in. It is rather long too. Briefly, he first puts the U edges in place and then the U corners. One face is a single color now. He turns over the cube so that it is on the bottom and puts the middle layer edges correcty in place and then makes the U edges into an even permutation. Then he puts the U edges in place and then the U corners. Finally he orients the U corners. And supposedly it is solved at that point. I haven't had much luck with the method though. He claims that this method takes less then 200 moves. -------  BSG@MIT-AI 08/17/80 17:22:24 To: CUBE-LOVERS at MIT-MC OK, you guys. You asked for it, you got it. :CUBE will now accept typed-in cube configurations. The ^F command (control f) to installed :cube prompts for a standard ITS file name. If you give ? as the file name, it tells you the thing I'm about to tell you. The hack is as follows. You copy AI:BSG;CUBE TEMPLT into some choice name of your own, and follow what it says inside of it, editing it to look like your cube configuration. Then enter :cube, and give the file name in response to ^F. We do only the most rudimentary checking, for syntax, center duplications, only six colors, overambitious file munging. We have not yet put in conservation of cubies, let alone ALAN's necessary and sufficient solubility conditions. So it's quite possible to make :cube go blarmy in this way, so be careful. Occasionally, :CUBE may give an ambiguous description of a 180-degree center-slice twist: to disambiguate it, let it do it, and see what it did, and do it on your cube. Go nuts in mid-August, -bsg  BSG@MIT-AI 08/18/80 06:58:28 To: cube-hackers at MIT-MC The ^F command (read in file) of :CUBE has now been augmented to make cubie conservation checks; it will tell you by way of what cubie is missing if there are duplications. The Bawdenian solvability criteria have not yet been implemented. There is also a brief form of the input file in AI:BSG;CUBE BFTPLT.  BSG@MIT-AI 08/18/80 19:47:49 To: cube-lovers at MIT-MC As per request of CLIVE, :CUBE ^F now accepts R for RED, Y for YELLOW and so forth. See AI:BSG;CUBE TEMPLT for reference.  Date: 19 AUG 1980 0927-EDT From: ZIM at MIT-MC (Mark Zimmermann) Subject: Singmaster 5th ed. via LOGICAL GAMES To: CUBE-HACKERS at MIT-MC Bela Fzalai at LOGICAL GAMES (Haymarket, VA) told me last night that he is out of the 4th edition booklets, but that he has a bunch of the 5th on order. Maybe it would be quicker to get them via him than from England directly; I don't know.... Mark  Date: 21 Aug 1980 17:14 PDT From: Wastebasket at PARC-MAXC (not authenticated) Subject: Re: please add me to your mailing list In-reply-to: CSD.CVW's message of 21 Aug 1980 1536-PDT To: CSD.CVW at SU-SCORE cc: cube-lovers at MIT-MC Please send requests like "please add me to your mailing list" only to the maintainer of the list, not to everyone on the list. Thank you.  Date: 24 AUG 1980 1623-EDT From: RP at MIT-MC (Richard Pavelle) Subject: Plumbers Cross? To: CUBE-LOVERS at MIT-MC There have been discussions of Plummer's Cross which I believed to be unique. Lets have an almost visual representation such as the following for one face: X Y X def Y Y Y === (Y,X) X Y X (O,B) Then Plummer's Cross looks like (W,G) (G,R) (B,Y) (Y,O) (R,W) (and this gives the coloring of my cube as well). The point about this configuration which I am stressing is that opposite sides have no colors in common. Now I find there is a second cross for which the point above is not valid. This cross takes a form (Y,G) (R,B) (O,W) (W,O) (B,R) (G,Y) Is this known by any Cubists out there?  Date: 24 August 1980 17:52 edt From: Greenberg.Multics at MIT-Multics Subject: Re: Plumbers Cross? Sender: BSG1.SIPB at MIT-Multics To: RP at MIT-MC (Richard Pavelle) cc: CUBE-LOVERS at MIT-MC In-Reply-To: Message of 24 August 1980 16:23 edt from Richard Pavelle That is Christmans crross, describedin my earlier letter in which I described Plummers cross.  Date: 25 AUG 1980 0907-EDT From: ZIM at MIT-MC (Mark Zimmermann) Subject: anecdotes from visit to Logical Games Inc. To: CUBE-HACKERS at MIT-MC I visited Bela Szalai on Saturday; his country home near Manassas battlefield ("Bull Run") is LGI, and he and his family are the employees. I learned a few interesting things: --he saw the cube in 1978 Aug, during a trip to visit relatives in Hungary; after many delays was able to get some wholesale from the Hungarian gov't., but they wanted $1 million for exclusive rights to distribute the things in the Western world. Ideal may have learned of the cube from Bernie DeKovon, GAMES magazine editor who is also a toy consultant for them; Ideal paid the $10^6 in 1979 Sep. --the cube is not patentable in the USA bnecause it was sold publicly for over a year in Hungary before patents were applied for; in England, however, it is "copyrighted" (equivalent to US patent+trademark) and Ideal has a legal monopoly. --Ideal will run nationwide TV advertisements for the cube beginning in a month or so; something to do with Newton, and including an animated cube which solves itself.... --Bela took out a second mortgage on his home to pay for the plastic molds for his cube parts. He uses white plastic so that it will be possible to print the colors on ("pad printing", the same process whereby labels are put on some shampoo bottles). If all goes well, LGI will start making printed-color cubes within a month. --Bela ordered 300 copies of the 4th edition of Singmaster's booklet in 1980 Jun; Singmaster informed him in July that the 4th edition was out of print, but that he could have 300 of the 5th edition for the same price as soon as they come out. LGI has >90 orders already pending, and while the remaining <210 copies of the 5th edition last, Bela is willing to sell them for $4 each... whenever they arrive. --LGI wholesale prices start at orders for 12 cubes, $6 each plus shipping; individuals may want to consolidate their orders to save money. --Rubik developed the cube partly as an aid to teaching 3-dimensional visualization in students. --cube manufacturing is VERY labor-intensive: 4 minutes to tap in and glue the caps to cover the internal 6th face of edge and corner cubes for the 20 pieces necessary to make one 3x3x3 1 minute to assemble with screws and springs 5 out of six swivels (central cube faces) onto the middle cross 1 minute to assemble the pieces and screw in the last central face 4 minutes to perform final adjustment: silicone grease, torque up all screws evenly, rotate every which way to test each cube out, tap in and glue 6 central face caps over screws. 6 minutes to apply the color squares to the faces Bela can read while performing most of the final assembly, now that his hands have had several thousand cubes' practice. The color application step will be eliminated if/when they begin to use pad printing for face coloring. --about 4% of cubes are rejected for mechanical reasons, so far --could work without gluing in internal faces of sub-cubes, but then about 1 cube in 20 would fail...so, they don't --READERS' DIGEST phoned to confirm some data, presumably for a story someday....  Date: 26 Aug 1980 1729-PDT From: CSD.VANDERSCHEL at SU-SCORE Subject: [CSD.VANDERSCHEL: Re: "Monoflips"] To: cube-lovers at MIT-MC Date: 25 Aug 1980 2041-PDT From: CSD.VANDERSCHEL Subject: Re: "Monoflips" To: ACW at MIT-AI cc: CSD.VANDERSCHEL In-Reply-To: Your message of 15-Aug-80 1927-PDT When I offered my monoflip, I tried to make it clear that I was claiming no superlatives except, perhaps, that of conceptual simplicity. Nor can I claim originality, for I recently noticed that Singmaster lists essentially the same move I thought of in his first supplement. It is presented as FUD'LLUUDDR and attributed to David Seal. (Besides, if I claimed originality, it would refute my claim of obviousness.) You indicated that you believe 22 qtw is the best one can do for a diflip. The only sense I know of in which your claim could be valid would be a diflip generated by a monoflip and its inverse, where that monoflip preserves the set of cubies in a face. B'UUBBUB'U'B'UUFRBR'F' is a 16 qtw process that flips a pair of adjacent edges, and Singmaster attributes it to Morwen Thistlethwaite's computer program. The simplest mono-ops are those that preserve the set of cubies in a center-slice. For example, FF could be viewed as a monoswap of the right and left front edge cubies in the horizontal center-slice. If we denote by "S" a quarter turn of that slice, then FFSFFS' will produce any three cycle you might like of edge cubies in the slice while leaving the rest of the cube intact. It also becomes more clear how FFSSFFSS produces the well-known double swap through opposing faces. U'FR'UF' is a 5 qtw monoflip that Singmaster attributes to Frank Barnes. It preserves the set of cubies in the RL center-slice. I think it is clear how it works and that you could not possibly do it in fewer moves. Using this monoflip, you can generate a 14 qtw diflip for an opposing pair of edge cubies. I still prefer the move I thought of because I seem to be less likely to make a mistake using it, it is more readily adapted to any pair of edge cubies, and it is about as easy to perform. David Vanderschel ------- --------------- -------  Date: 27 Aug 1980 0128-PDT (Wednesday) From: Dal at UCLA-SECURITY (Doug Landauer) Subject: Notation, transforms I like to use ... To: cube-lovers at mit-mc,alan at mit-mc After 5 beers and two Drambuies, finally, here we go ... Here is all of my knowledge and notation regarding the cube ... SPOILER warning these transformations are enough to solve any cube. I look forward to anyone publishing a compendium; please go ahead and use anything in here. Two notes about my particular cube ... it worked fine until I decided to lubricate it (as is recommended in one of these cube-lovers messages ... they say to use a graphite lubricant (I believe) ... do so.) DO NOT use "dri-slide". It is a Molybdenum Disulfide lubricant for use with metals and it tends to eat plastic, making it harder to turn. Also (this may be unrelated), three weeks or so after I lubricated mine (an Ideal from Toys-R-Us in the San Fernando Valley (in LA)) one of the facies popped off and now it's out of commission until I try to glue it back together. I found that the descriptions of its inner workings do not do it justice. It is indeed a work of 3-dimensional mechanical genius. To elaborate on the descriptions ... the face cubies are about 7 mm thick, and they are what holds the other pieces in. The corner cubies have a growth on their "opposite" corners which is roughly cubical but the "inside" corner is rounded to allow them to rotate. The side cubies (sidies?) have a 7-mm thick disk-like thing sticking out of the center of their least visible edge (the one opposite the one that the sidie's two colors border on). It's actually more like a thick square with one rounded corner (the inside one). Anyway, here's a description of my notation followed by a set of transformations I use; this is indeed sufficient to fix a cube and if you wish to remain self-sufficient in this matter, read no further; but then what're you doing on this list anyway? happy cubing .... -dalgorf- flubrd Notation: (pronounce it `flubberd') Object names: Faces: f l u b r d stand for the front, left, up, back, right and down faces respectively. (up==top and down==bottom) Little cubies The 6 face cubies (one at the center of each face) are seldom referred to; use the face name (e.g the f-face cubie) Side (edge) cubies: fl fu fr fd ul dl bl bu br bd ur dr Corner cubies: flu fur bul bru fdl frd bld bdr For side and corner cubies, the order of its faces only matters in permutation descriptions (see below), but by convention, the corner cubies are all named clockwise starting in front or back. Center Slices H (`Horizontal'==`floor-parallel') runs through faces f,r,b,l P (`Parallel' for `body-Parallel') runs through faces u,r,d,l S (`Slicing' for `body-Slicing') runs through faces u,f,d,b Moves: The ambiguity between object names and moves is cleared up by always surrounding a move in angle brackets (<>). A move can be any of the following: - a single letter, meaning twist a single face or centerslice, or rotate the entire cube one quarter-turn (see below); - a move followed by a move indicates those two moves in sequence; - a number n followed by '(', a move, and a ')' is the same as repeating the move n times; - a move followed by a prime ("'") indicates the inverse of the move --- by convention also all one-letter moves are defined so that changing the letter's case also inverts the move. A move may be grouped by parentheses, so that e.g. <(FUDRB)'> is equal to (ie, undo each twist in reverse order to undo this whole move). twists f clockwise; similarly for ,,,, & l,u,b,r,d twists f counter-clockwise; similarly for ,,,,. Note =, =, and so on. Extended moves (centerslices and cube-turning) cube-turning: I,J,K=front-top-right clockwise move cube's front to top or top to front move top to right side or right side to top move right side to front or front to right side centerslice moves (`Horizontal') move H clockwise as seen from above =

move P clockwise as seen from the front = move S clockwise as seen from the right side = Move combinations are two-letter names beginning with C, E or A; C1-C9 and (CA-CZ, Ca-Cz if needed) describe corner-combinations; E1-E9 and EA-EZ (Ea-Ez if needed) describe edge-combinations; A1-A9, AA-AZ & Aa-Az describe combinations that mess with both. I also use V,W,X,Y,Z and v,w,x,y,z as temporary macros Permutations = { list of cubies -> list of new spots }. The permutation described puts the first list into the new LOCATIONS described by the second list, moving each face of each cubie in the order specified. E.g., does {flu fu fur fr frd fd fdl -> fur fr frd fd fdl fu fdl} Useful basic corner combinations: C1: <6(RurFU)> :: {fdl flu bul -> dlf luf ulb} twist corners clockwise C2: <6(rFRuf)> :: {fdl flu bul -> lfd ufl lbu} " " ccw (lf2 + bul) C3: <3(RUru)> :: {frd fur bul bru -> rfu rdf ubr ulb} switch those pairs of corners, twisting each C4: <3(fuFU)> :: {frd fur flu bul -> rfu rdf bul flu} (ditto) C5: <3(FrfR)> :: {frd fur flu bru -> rfu rdf bru flu} (ditto) compound ones: C6: = :: {frd fur fdl flu -> rfu rdf luf lfd} switch those pairs of corners all in the front face C7: = <6(RurFU)M6(rFRuf)m> :: {fdl bld -> dlf dbl} twist fdl cw and bld ccw. C8: = <3(FrfR)3(RUru)> :: {flu bul bru -> bru ufl ulb} cycle three top corners (left two & bru) "cw" C9: = <3(FrfR)3(fuFU)> :: {flu bul bru -> bul bru flu} cycle three top corners (left two & bru) "ccw" Useful basic edge (side) combinations: E1: <4(FH)4(HF)> :: {fl fr -> lf rf} twist two sides in place E2: <2(ssdd)> :: {fu bu fd bd -> bu fu bd fd} "doubleswap" swaps front for back sides on top and bottom E3: <3(FRRF)> :: {fl fr ru rd -> fr fl rd ru} E4: <3(ffrr)> :: {fu fd ru rd -> fd fu rd ru} E5: <3(URRU)> :: {ul ur rf rb -> ur ul rb rf} E6: <3(ffuu)> :: {ul ur fl fr -> ur ul fr fl} E7: <3(FLLF)> :: {lu ld fl fr -> ld lu fr fl} E8: <3(ULLU)> :: {lf lb ul ur -> lb lf ur ul} E9: = <3(FRRF)3(ffrr)> :: {fl fr fu fd -> fr fl fd fu} EA: = :: {fu ul ur -> ul ur fu} cycle top (left right and front) sidies clockwise EB: = :: {fu ul ur -> ur ul fu} cycle top (left right and front) sidies counterclockwise ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Pons Asinorum: HHSSPP (180 degrees on each centerslice) Laughter: <4(lrfb)> = <6(lrK)> (but in the latter the cube is KK'd)... Another laugh resolves the cube; a K first produces the 4 crosses. Crux Christmani: or, call E2 (the doubleswap) `Z', . This is 6 crosses in 3 pairs: colors go top with bottom, front with right side and back with left side. Crux Plummeri (the fancy one): ; or (again): or if Y := , it's simply ! This is 6 crosses in 2 sets of three each: Front, Right and Top each with each other; and Back, Left and Bottom each with each other. Greenberg's "well-known center-cubie rotation algorithm" has 12 varieties: Pick two centerslices (say X and Y) and a direction (cw or ccw): does ccw and does cw. Details: (xyz cw) means the center (face) cubies of the faces x y and z are rotate clockwise; ... ,ccw means counterclockwise. fur cw :: or or , ccw :: or or frd cw :: or or , ccw :: or or fdl cw :: or or , ccw :: or or flu cw :: or or , ccw :: or or Outstanding questions mentioned in the note about Singmaster's talk: (I.e., I ain't seen these yet ... ) 1. Conway's Monoswop 2. Rubik's Duotwist (ignore the two lines above beginning with Pick ... and ending with ... Details: they're wrong. I'd go edit them but my Unix is out of space on /tmp and on my filesystem and I only have 300-baud access to it anyway). Have fun DalGorf  Date: 27 August 1980 1:58 pm PDT (Wednesday) From: Woods at PARC-MAXC Subject: Lexicon To: Cube-Lovers@MIT-MC We need a term to refer to the fear that someone will randomise your cube. Any suggestions? -- Don.  ACW@MIT-AI 08/29/80 13:24:04 To: Cube-hackers at MIT-MC Date: 29 AUG 1980 1316-EDT From: ACW at MIT-AI (Allan C. Wechsler) On this coast we have a colorful slang term for randomizing the cube. Hint: the command to randomize the cube in Bernie's :CUBE program is "F". ---Wechsler  Date: 29 Aug 1980 10:55 PDT From: Lynn.ES at PARC-MAXC Subject: Re: Lexicon In-reply-to: Woods's message of 27 August 1980 1:58 pm PDT (Wednesday) To: Woods.PA cc: Cube-Lovers@MIT-MC Incongruphobia seems nice, but may imply more of a fear of lack of order, rather than of the change from order to lack thereof. It doesn't specify the objects that are disordered either. Cubic discombobulaphobia might be a little better in these respects. /Don Lynn  MJA@MIT-AI 08/29/80 21:45:41 Re: Knowledge-based Rubik's Cube solver? To: cube-hackers at MIT-MC Please excuse the previous badly munged versions of this message i accidently sent. I am taking an artificial intelligence course at the University of Illinois at Urbana-Champaign and I need to do a term project for the course that involves developing and/or using a knowledge based system. Since I am interested in the Rubik's cube and since it seems like there are human expert cube solvers (as is evident from msgs on this mailing list) that use heuristic knowledge (for example the various macro's (as I like to call them, they are more properly called composite elements of the group, i guess) that people have come up with to do some particular operations on the cube), it seems natural to me that a knowledge based system would be just right for this task. I would like to hear: (1) Pros/cons on why this can/can't be done and how effective such a system would be. (2) Considering that I have but one semester to work on this project (while concurently taking 5 courses, incl. this AI course) is it reasonable to attempt this for a term project? (The actual system would not have to be 100% complete at the end of the term, but I would at least like to have something to show for a semester's worth of work.) (3) If this is too ambitious for a term project for a course, what about the use of this as a topic for a master's thesis. (I would have until the summer of '81 to finish since thats how long my company will wait for me to get a master's degree if im going to continue working for them.) (4) Of course in addition to all of the above trivia, I would like some ideas for such a system (even if it isnt feasable to do in this limited amount for time, im still interested in at least thinking about such a system with the idea in mind that someday it can be implemented). I dont know how interested the majority of the people on this mailing list are in knowledge based systems, so we might consider a seperate mailing list for knowledge based cube solvers to spare those on the current mailing list from mail they dont want. / MJA  ZILCH@MIT-AI 08/30/80 01:39:38 To: CUBE-LOVERS at MIT-MC Subjects covered:random notes on the cube and cubing 1> To enable new and old cube-lovers to communicate on an equal basis I propose that a file be established that describes FBLRUD (or whatever),the I,J,K and centerslice move and any other concepts that might be usefull in describing transformations.(I can not do this as I know nothing about establishing files or editting them, and as a tourist I am unsure of my status as file-creater.) 2> Today having nothing better to do, I fiddled with the 3x3x2 version of the cube (actually I just didn't allow certain moves on my 3x3x3). At this point I have two transforms which would enable me to solve the 3x3x2 version if and when I ever see it. I derived the number of orbits and the reasons behind them, but will not describe them here because the 3x3x2 is a novel idea and I don't want to ruin all the fun. Also I came up with the idea of a 2x2x3 version which basically operates on the same principles as the 3x3x2 , but it looks different. solving the 3x3x2 can give a slight amount of insight about Thistlewaite's algorithim ,described by McKeeman on 12 Aug. @15:10 PDT. 3> On July 15 @1413 EDT, ALAN asked if anyone had learned to solve the extended cube problem independently. I had known the faces could assume different orientations when the cube was solved but hadn't done any work on this and didn't know what the possible transforms might be. In a later note cmb described what his transforms accomplished and from that idea I recently derived similar (if not equal) transforms.With a little foresight I find that only one of these transforms might be needed. This evening a friend and I came up with a modification to the cube construction that would facilitate the extended problem. this is simply affixing another cubie to each face cubie and coloring appropriatly. 4> As suggested by CSD.VANDERSCHEL on 9 Aug. @1610 pdt has anyone given any thoughts to 3-faced cubing (2 types) or 4-faced cubing (again 2 types). As far as I can tell 5-faced cubing is not profitable. Also does CSD.VANDERSCHEL know a good way (at this time) to show why only 1/6 of the 6! possible permutations of the corners are possible? There is also an extended problem for the 2-faced problem involving face orientations. 5>An addendum to WOOD's message of 23 JUL. @5:23pm regarding repetitive sequences to get to the identity. As it turns out, by my calculations the number 1260 is sufficient even including the prblem where the faces are permitted to move. The only subcycles of the faces themselves have lengths 4,3,2,and of course 1. These subccles may immediatly be generated by I (4 times), I^2 (2 times),^2 (also 2 times) and finally (3 times). (I do not use IJK notation on these last two because I have noticed a little disagreement on exactly what these mean.) These subcycles make no difference in the total number because subcycles of lengths 4,3,and 2 have already been taken into consideration in the computation of 1260. I have no idea whether th e face cubies might change their orientations in a sequence of length 1260,when the faces are allowed to move, but I know it does nothe number above 1260 when the faces are not allowed to move. 6> Singmaster's solution from his version 5, as reported by McLure on 16 Aug. @1053 PDT sounds almost exactly like my solution except that I keep the first face completed in the up poition at all times. He reported that his method takes less than 200 (of his) moves. My transformations yield a solution in a maximum of 190 quarter-twists. My actual solving length (from when I bothered counting) averages about 125 qtws , and my time (when I bother) is almost consistantly 2.5 minutes, occasionaly under 2, with worst case about 3 min. 10sec when I messed up. Usually my fast times do not use all of my algorithms techniques , because they are new to me and I don't know them by heart. Breakdown of moves: 1. Top edges in proper position and orientation 20 (3+6+6+5) 2. " corners " " " " " 36 (9+9+9+9) 3> 3 middle edges " " " " " 45 (15+15+15) 4. 4th middle edge in proper position and orientation and bottom edges in proper orientation 23 5. bottom edges in proper position 18 6.bottom corners in proper position 20 7. " " " " " 28 Total 190 Note: in step 4 proper orientation means that say if the down face is white , after step 4 all bottom edges will be showing white on their down sides. This algorhitim has not been optimized much and uses lookahead only in step 4. Step 2 gives the worst case for any corner, but if only worst cases are present then they cancel each other out somewhat. Replys, questions, and comments are welcome. Chris  DMM@MIT-ML 08/30/80 18:41:24 Re: DAL@ucla-security's msg of 27-Aug To: cube-lovers at MIT-MC Genereally, I have found themacros supplied by DAL to be quite useful, but t|ere are a few bugs i've noticed. First of all, oC8&C9, I believe DAL has the cw & ccw notations reversed. Regarding C7: What is M? I have not seen it referenced anywhere. and last, in maco EB: the last move is f, not F. (I`guess it was the 5 beers and 2 Drambuies.)  Date: 3 September 1980 2108-EDT From: James.Saxe at CMU-10A (C410JS30) Subject: Orbit classification revisited To: cube-lovers at MIT-MC Message-ID: <03Sep80 210846 JS30@CMU-10A> Hi, folks. Having read Vanderschel's msg of Aug. 6, it appears to me that the explanations of the orientation parities are unnecessarily complex, though the material on permutation parities is presented in a way that should be immediately convincing to anyone familiar with the notion of even and odd permutations from elementary group theory (and anyone who isn't should be!). Here's my attempt at a more elegant demonstration. In what follows, I will use the term "facelet" to denote any (visible) face of a cubie. Thus, each edge cubie has two facelets and each corner cubie has three facelets. I will address Edge Orientation Parity (EOP) first. Consider the following diagram: +-------+ | 1 | |0 U 0| | 1 | +-------+-------+-------+-------+ | 1 | 0 | 1 | 0 | |0 L 0|1 F 1|0 R 0|1 B 1| | 1 | 0 | 1 | 0 | +-------+-------+-------+-------+ | 1 | |0 D 0| | 1 | +-------+ The numbers label absolute positions, not facelets, and therefore remain in the same configuration when the cube is manipulated. Imagine that a mark is placed on each facelet that occupies a position labeled "0" when the cube is in the solved configuration. Thus, each edge cubie will have one marked and one unmarked facelet. (Unlike the numbers, the marks are attached to the facelets and will move as the cube is manipulated). The parity of an edge cubie in an arbitrary configuration is defined as the number labelling the position occupied by its marked face, and the EOP is defined as the sum of the parities of all edge cubies modulo 2. A quarter turn of any face reverses the parity of 4 edge cubies, thus leaving the EOP fixed. By induction, no sequence of manipulations starting from the solved configuration can produce a configuration EOP = 1. So much for EOP. [Just for grins, here's a cute way to determine the parity of an edge cubie without consulting (or reconstructing) the diagram: Assign each of the cubie's two facelets a number in {0,1,2} according as it is oriented parallel to its home position ("Self" -> 0), parallel to the other facelet's home position ("Other" -> 2) or perpendicular to both home positions ("Neither" -> 1); add these two numbers and reduce modulo 3 (yes, three!) giving the parity of the cubie. The sum can never be 2 because that would imply that the two facelets were parallel to each other. Here's an addition table with double-ended arrows showing the possible transitions: (S) 0 (N) 1 (O) 2 +-------+-------+-------+ | | |no | (S) 0 | 0 <-+-> 1 | + | | ^ | ^ | way| +---+---+---+---+-------+ | v |no | | | (N) 1 | 1 <-+---+---+-> 0 | | | |way| ^ | +-------+---+---+---+---+ |no | v | v | (O) 2 | + | 0 <-+-> 1 | | way| | | +-------+-------+-------+ ] Now for Corner Orientation Parity (COP). Consider the diagram below: +-------+ |0 0| | U | |0 0| +-------+-------+-------+-------+ |2 1|2 1|2 1|2 1| | L | F | R | B | |1 2|1 2|1 2|1 2| +-------+-------+-------+-------+ |0 0| | D | |0 0| +-------+ Once again, imagine that we mark all (corner) facelets which occupy positions labeled "0" when the cube is in the solved configuration. The parity of a corner cubie in an arbitrary configuration is defined as the number which labels the position of the cubie's marked face. The COP is the sum of all the parities of all corner cubies modulo 3. Inspection of the diagram will reveal that the twists U, u, D, and d leave the parities of all corner cubies unchanged. Any of the other possible quarter twists will increment (modulo 3) the parities of two corner cubies and decrement the parities of two others, thereby leaving the COP unchanged. [As Vanderschel pointed out, one way to compute the parity of a corner cubie by looking at it is to note the number of clockwise (as viewed from outside the cube) 120 degree twists of the cubie that it would take to bring the marked facelet parallel to its home position. Note that the parity of a corner cubie, unlike that of an edge cubie, depends on the selection of a particular pair of opposing colors for the marked facelets. While this lack of symmetry may be considered unfortunate, it is an inevitable result of the fact that four is not divisible by three. It is easy to show that the COP as a whole is independent of the choice of distinguished faces.] Vanderschel also mentions the extended problem, but does not quite make it clear that the FOP changes every time a qtw is done. This constrains all three of {FOP, EPP, CPP} to be the same, so that only two of the eight plausible states of the vector are actually achievable by twisting. Jim  Date: 3 SEP 1980 2123-EDT From: DCP at MIT-MC (David C. Plummer) Subject: New problem To: CUBE-HACKERS at MIT-MC This was inspired by Tanya Sienko (if that means anything!!) It is possible for each of the six faces to have a capital "T" on them. That is, each face looks like X X X O X O O X O QUESTIONS: 1) How many classes are there? 2) How many members are in each class? 3) How easy is it to get to them from solved? My answers: [in a few days, I don't want to spoil the fun] Good luck, and happy hacking. - DCP  Date: 3 September 1980 2149-EDT From: James.Saxe at CMU-10A (C410JS30) Subject: Re: Lexicon To: Woods at Parc-Maxc CC: Cube-Lovers at MIT-MC, James.Saxe at CMU-10A Message-ID: <03Sep80 214930 JS30@CMU-10A> My suggested term for the fear that someone will randomize one's cube is "nobility," since that is what sufferers of this phobia typically have to solve their cubes. Jim  Date: 3 September 1980 2328-EDT (Wednesday) From: Dan Hoey at CMU-10A To: Cube-Lovers (and Hackers) at MIT-MC Subject: Addictiveness/Disassembly/Taxonomy/Lubrication/Spoilers Message-Id: <03Sep80 232808 DH51@CMU-10A> Hello. I saw the notice announcing the formation of this list a couple months ago, and decided that it was one of those things I could forgo -- until I got my hands on a physical cube. It was an immediate necessity to own one. I bought an Ideal brand cube, which, I understand, is of the species C. Americanus in spite of its "Made in Hungary" label. I had owned the cube less than ten minutes before a facie cover fell off, without the aid of chemical additives. This was not very destructive; just about any gummy material (I used gluestick) suffices to hold it on. However, the screw head revealed by this unusual transformation leads to a new method for disassembly. Unscrewing does not stress the cube as does prying, and probably avoids the deleterious side-effects observed by Greenberg (16 Aug 1453). This method is not without its hazards, however. It is EXTREMELY easy to strip the threads on the plastic X that holds the cube together. I have paper shims in the two threads I stripped and they seem to suffice. Still, it is probably better just to loosen the screw until the cube comes apart with gentle prying. There is at least one good reason for taking a screwdriver to the cube. Mine had been assembled with several of the screws so tight that the springs were completely compressed. Due to mfg inaccuracies in the cubies, this made the cube difficult to twist. By prying each of the facies off with a fingernail I was able to correct the tension. Jim Saxe's cube, putatively of the same species but purchased in a different store at 80% the price, has facies which seem impervious to this prying even after disassembly a la Greenberg (17 Jul 2118). Jim and I exhausted our fingernails to no avail, and careful prying with a knife was unsuccessful. Additionally, this cube has a strong tendency to jam, due either to its uncorrectable looseness or to its edge cubies, which have oversized tongues with extremely sharp edges. The differences lead us to believe that our cubes may belong to different species in spite of their outward similarity. I am amazed that anyone would put molybdenum disulfide on their cube. Isn't that stuff poisonous? Graphite works well but is messy if you overdo it. Silicone lubricant was mentioned by Zimmerman (25 Aug 0907) -- has anyone any experience with this? Merrick Furst recommends soap. For anyone who cubes in public, the only word for LACK of fear that someone will F your cube is Cubemeistership. I made a mistake in taking the cube to one session of a recent conference. The sequence 4(Borrow)4(Borrow') appeared to have an entropic effect on the cube and a negative effect on the transfer of information. SPOILER WARNING: I have one transform which I haven't seen here, and which I find useful: an 8-qtw move to permute three corners of a face. Specifically, for {fdl flu fur -> ufl rfu lfd} do . Mnemonically, you move a socket back and forth between flu/fdl with the f/F transforms, alternating with moving one of the three pointies (cornies?) to be permuted into the socket. Why it works is a mystery to me, but it's useful. Another, which should be obvious but improves Landauer's (27 Aug 0128) C7, is the cw monotwist {flu -> luf in u-face} =. Then {flu fur -> luf rfu} is = taking 16qtw instead of 60 (assuming =, =). All for now. -- Dan  Date: 4 SEP 1980 0852-EDT From: JURGEN at MIT-MC (Jon David Callas) Subject: Lexicon... To: woods at PARC-MAXC CC: CUBE-LOVERS at MIT-MC Worse yet is the fear that someone will put your cube in an unsolvable orbit...... Happy phobias, Jurgen at mit-mc  Date: 6 Sep 1980 0021-PDT (Saturday) From: Dal at UCLA-SECURITY (Doug Landauer) Subject: errata To: cube-lovers at mit-mc Attn: DMM@mit-ml: No, unless my message got garbled somewhere in the midwest, it was correct about c8 and c9, the three corner-cubie movers. I.e., c8 ( <3(FrfR)3(RUru)> ) cycles them cw, and c9 ( <3(FrfR)3(fuFU)> ) ccw. (Much shorter ways to do similar things exist, e.g. to move around three front corners (left two and fur) cw). In these descreptions, the `cw' and `ccw' do not refer to individual cubies but rather to the triangle of three cubies involved, that's why I call it "cycle" == move cubies around, rather than "twist" == re-orient a corner cubie in place, or "flip" == do the same for an edge (side) cubie. You're right about EB. Attn: Dan Hoey at CMU-10A Regarding C7: When I first made this list up, I used m,n and o to represent i,j and k because it was unclear which of i,j and k was which. m is the same as i, and M == I. My cube-turning moves are: I,J,K=front-top-right clockwise move cube's front to top or top to front move top to right side or right side to top move right side to front or front to right side thank you... ... dalgorf -------  Date: 8 SEP 1980 0853-EDT From: ZIM at MIT-MC (Mark Zimmermann) Subject: Singmaster 5th; 2x3x3 "domino" To: CUBE-HACKERS at MIT-MC I received the 5th edition of Singmaster's booklet a few days ago, from Logical Games, Inc.--presumably they have some $4 copies left (they sent mine 1st class mail)....Bela Szalai (LGI) loaned me a 2x3x3 to play with. Conceptually, of course, it's the same as the 3x3x3 with moves restricted to ...physically, however, it was MUCH less aesthetic to play with... turned poorly, and looked ugly. Bela had a couple of them, both of which were rather more "delicate" than the 3x3x3 (in that they tended to get jammed or come apart). -- Mark  Date: 10 Sep 1980 1500-PDT From: Alan R. Katz Subject: randomizing To: cube-lovers at MIT-MC Here's a question for all of you. How do you maximally randomize a cube? In other words, suppose you were going to have a cube solving compitition and wanted to make it as hard as possible to solve a cube, what precautions would you take? One thing I would do would be to make sure the corners are not correct in relation to each other. Anyone have other ideas?? Alan (Katz@isif) -------  Date: 10 September 1980 18:12 edt From: Greenberg at MIT-Multics (Bernard S. Greenberg) Subject: Re: randomizing Sender: Greenberg.Multics at MIT-Multics To: Alan R. Katz cc: cube-lovers at MIT-MC In-Reply-To: Message of 10 September 1980 18:00 edt from Alan R. Katz My personal theory on making a cube as hard to solve as possible is as follows: Put the cube in some very complex pretty, regular, pattern, such as the Plummer's Cross with centers trebly rotated, or the Pons Asinorum on that which looks like the simple Pons: the challenged cubemeister will INVARIABLY say, "Oh, this, this is just one of those and solved like this" and go through a long hairy procedure, hopefully the wrong one, or make an error, or try hard at understanding it, and so forth. In short, s/he will take MORE time trying to undo what s/he perceives as a "simple" hairy hack than a straightforward application of solve-from-random technology.  Date: Wed, 10 Sep 1980 1854-CDT Message-id: <337474464.2@DTI> From: aramini at DTI (Michael Aramini) To: cube-hackers@mit-mc Subject: randomizing There are two types of aproaches to what can be meant by total (or maximal) randomizing. One is to consider states of the cube that are maximally far away from being solved (that is the minimal number of quarter twists needed to solve from a given state is maximallized. the other, which may be more useful for making the cube more difficult to solve, is based on maximallizing the amount of time it would take a person (or a program, for that matter) to solve the cube. Of course, since this amount of time is both dependent on the state of the cube, and how the person (or program) goes about solving it. Thus the set of most randomized positions is dependent on who is solving the cube. the advantage of the first definition is that it seems to be of more theorectical significance (the number of quarter turns needed to solve from this position gives you the diameter of the state graph). The second approach seems more kludgy since it much less well defined since the amount of time it takes to solve the cube if a function of many variable besides simply the initial state. This distinction is much like differing strategies for writing chess playing programs, you could write it assuming the oponent to be a "perfect" player (as if a complete look ahead to the leaves of the tree appraoach were being used by the oponent) or by considering how the oponent is likely to play (have the program try to confuse the oponent by taking advantage of something that the oponent wont recognise). the second method thus will take into account the knowledge-base available to the solver (for example trying to trick the solver into thinking that the cube is in one of the easy to solve classes but really isnt, thus leading the solver down a blind alley) This gives me an idea for a game where oponents are each trying to bring a randomized cube into two different final states (each of which is hopefully equally far away from the initial state, just to make the game fair) by alternately taking turns making quarter turns. of course one must make up some rules so that that the game terminates (ei going around cycles in the state graph an indefinite number of times is disallowed), and these rules might not be very obvious,although they would prob be similar to some of the stalemates rules of chess. -----  Date: 10 September 1980 2303-edt From: Bernard S. Greenberg Subject: Randomizing To: CUBE-LOVERS at MIT-MC I enter the following tidbit into the transcript of this society, apropos to the current discussion: RMS at AI (Richard Stallman) has suggested the following paradigm for cubing tournaments: (Assuming physically standard cubes are used) The two participants acquire (or create) solved cubes. Each one randomizes it his/her favorite way, and hands it to the other to solve. The resources expended by the solver (either qtw or real-time) are added to his/her resource expenditure in solving the other's cube. Minimum resource expenditure wins.  Date: 11 SEP 1980 0016-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: How do you maximally randomize a cube? To: CUBE-HACKERS at MIT-MC I am interested in maximally distant states of the cube. I have often wondered just what a maximally distant state would look like. I also wonder HOW MANY of them there are! Interesting fact (offered without proof (it's not hard)): Assuming we are counting quarter-twists. If I hand you a cube in a maximally distant state, and ask you to solve it in as few twists as possible, you don't have to think at all in order to know what to do first! ANY first twist will bring it closer to home (after that it gets harder). Call a state with this property a "local maximum". Any maximally distant state is also a local maximum. Also, any symmetric state is a local maximum. This doesn't mean that a maximally distant state is symmetric, but it does get you thinking along those lines!  Date: 10 Sep 1980 9:37 pm PDT (Wednesday) From: Woods at PARC-MAXC Subject: Re: How do you maximally randomize a cube? In-reply-to: ALAN's message of 11 SEP 1980 0016-EDT To: ALAN at MIT-MC (Alan Bawden) cc: CUBE-HACKERS at MIT-MC It is not obvious to me that any twist in a maximally distant state brings you closer to the home position. How do you know there aren't two equally distant states that are a qtw apart? There is probably a parity argument that proves this can't be so, but if you count half-twists as single operations I'd be much surprised if you could find a simple proof. -- Don.  Date: 11 September 1980 01:04-EDT From: Alan Bawden Subject: Re:"Assuming we are counting quarter-twists." To: Woods at PARC-MAXC cc: CUBE-HACKERS at MIT-MC I said: "Assuming we are counting quarter-twists." I realize that there are people out there who like to count half-twists as a single twist. I don't. I'm not trying to force my way of counting twists on anyone. I always try to be carefull to make this assumption explicit out of courtesy to those who might want to count otherwise. (Sort of like the Axiom of Choice.) In fact I DON'T know if that result is true if you count half-twists as well. I suspect it is, but I don't have a proof. You needn't jump up and down and point every time one of us quarter-twisters reveals himself. This one, at least, is well aware of his assumptions.  Date: 10 Sep 1980 11:09 pm PDT (Wednesday) From: Woods at PARC-MAXC Subject: Re: "Assuming we are counting quarter-twists." In-reply-to: ALAN's message of 11 September 1980 01:04-EDT To: (Alan Bawden and the rest of) CUBE-HACKERS at MIT-MC If you'll reread my message, you'll note that I claimed the parity argument wasn't obvious to me even in the case of a qtw metric; my reference to half-twists was intended along the lines of "this is even less obvious". I'd be interested in seeing your "obvious" proof in the qtw case, if you'd care to send it to me (no need to bother the whole mailing list with it). -- Don.  Date: 11 September 1980 22:06 edt From: Greenberg.Multics at MIT-Multics Subject: Singmaster v.5 To: cube-lovers at MIT-MC (To MIT area cubists only:) I have received "Notes on Rubik's Magic Cube" version 5 from Singmaster, if anybody wants to see it. It is 75 pages long and AWESOME.  Date: 15 Sep 1980 1842-PDT From: Alan R. Katz Subject: number of reachable states To: cube-lovers at MIT-MC I have seen the number 4.3 * 10^19 for the number of reachable states for the cube, can anyone tell me how you calculate it? This may have been answered before in this list, but I couldn't find it. Also, someone mentioned that one can make a checkerboard pattern from the Pons Asinorum by trebly rotating the centers by a simple transformation. Can anyone tell me this transformation? (again I may have missed reading it) Reply to either me or the list. Alan -------  Date: 15 SEP 1980 2156-EDT From: ALAN at MIT-MC (Alan Bawden) Subject: Where to find old mail. To: CUBE-HACKERS at MIT-MC Just a reminder to you all that ALL of the old cube-lovers mail is archived in the file ALAN;CUBE MAIL on MIT-MC.  Date: 16 SEP 1980 0746-EDT From: RP at MIT-MC (Richard Pavelle) Subject: number of reachable states To: KATZ at USC-ISIF CC: RP at MIT-MC, CUBE-LOVERS at MIT-MC Date: 15 Sep 1980 1842-PDT From: Alan R. Katz I have seen the number 4.3 * 10^19 for the number of reachable states for the cube, can anyone tell me how you calculate it? This may have been answered before in this list, but I couldn't find it. The number is (12! * 2^12 * 8! * 3^8)/12. This comes from the following. There are 8 corners and there are 3 positions- hence 8!*3^8. There are 12 edges with 2 positions hence 12!*2^12. Finally, the /12 comes from parity considerations. Only 1/4 of the positions in the flippling of two edges are possible while 1/3 of the toppling of two edges are possible. Also, someone mentioned that one can make a checkerboard pattern from the Pons Asinorum by trebly rotating the centers by a simple transformation. Can anyone tell me this transformation? (again I may have missed reading it) The moving of centers is easy- 4 moves of the center slice while rotating the cube 90 degrees in your hand between moves. With the transformation in hand you can move the centers easily to possible positions.  Date: 16 SEP 1980 0946-EDT From: DCP at MIT-MC (David C. Plummer) Subject: number of reachable states To: KATZ at USC-ISIF CC: CUBE-HACKERS at MIT-MC Date: 15 Sep 1980 1842-PDT From: Alan R. Katz I have seen the number 4.3 * 10^19 for the number of reachable states for the cube, can anyone tell me how you calculate it? This may have been answered before in this list, but I couldn't find it. Also, someone mentioned that one can make a checkerboard pattern from the Pons Asinorum by trebly rotating the centers by a simple transformation. Can anyone tell me this transformation? (again I may have missed reading it) Reply to either me or the list. Alan ------- Consider the corners. There are 8 of them, and they can go anyplace. This leads to 8 factorial permutations. Each corner can take on three orientations, so this is another factor of 3^8. But the corners have three possible states (trarity [three way parity]) so divide by 3. Now do the same with the edges. 12 edges gives 12 factorial arrangements, times 2^12 oreintations. But the edges have two parities involved, so divide by four (thus giving rise to the 12 states of the cube, one of which has the solved configuration as a member). So if you evaluate 8 12 8!*3 *12!*2 ----------- 3*4 you will get 4.3 * 10^19.  DPC@MIT-AI 09/17/80 17:02:00 Re: cube mode on lisp machines To: cube-lovers at MIT-MC cube mode has been fixed (at least temporarily) to work in the new window system on the lisp machines. to invoke it: (load "bsg;cubpkg") (cube) it is self explanitory once you get in. have fun. -dpc  Date: Wed, 17 Sep 1980 2047-CDT Message-id: <338086051.13@DTI> From: aramini at DTI (Michael Aramini) To: cube-hackers@mc Cc: boken@mc Subject: cubing hazordous to your health? I saw someone cubing near the digital computer lab at the Univ. of Ill. today. Considering how much traffic there is on springfeild av (the street in front of said bldg.) i though it might be a good idea to be crossing the street while intently solving a cube (i didnt see the person try crossing the street, but he was at least walking without looking where he was going). -----  Date: 20 September 1980 2144-edt From: Bernard S. Greenberg Subject: Cubesys/LISPM fixed To: CUBE-LOVERS at MIT-MC :cube's Lisp Machine instantiation has now been fixed to work in color on Color LISPM's again. To invoke, (load "bsg;cubpkg"), wait till it loads, and (cube). Should be self-documenting.  Date: 23 SEP 1980 0851-EDT From: ZIM at MIT-MC (Mark Zimmermann) Subject: report on DC area cube contest To: CUBE-HACKERS at MIT-MC Hecht's (a local dept. store chain) offerred $100 gift certificate to any one who could get the cube done in 5 minutes (and T-shirts promoting the cube to anyone getting 1 face right in that time), for a 4-hour period last Saturday...I was the 7th to get the cube done (there were a bunch of mathematicians from the University of Maryland there before me)...the store took everyone's name & address & social security numbers...haven't heard anything from them yet, though...there may have been more solvers after I left. --Mark  Date: 25 SEP 1980 0859-EDT From: ZIM at MIT-MC (Mark Zimmermann) Subject: search-from-both-ends ultimate cube algorithm To: CUBE-HACKERS at MIT-MC There are some standard algorithms involving time-memory tradeoffs for solving problems like Knapsack in N^(1/2) steps instead of N...Hellman had a paper in some IEEE journal about an application to breaking cryptosystems. The same could be applied to solving the cube "exhaustively", given several hundred billion memory locations storage and a willingness to go through a similar number of table-lookups. Just make a table of all the positions that can be reached within 10 meoves or less (with the route to that position recorded too!)...then in order to solve an arbitrary cube set-up, begin working out from that set-up, looking for a position included in the table. If Singmaster is right and any position can be reached in 20 moves or less, one will always succeed within 10 moves from the arbitrary start.... The several hundred billion entries in the table are a bit too large for my home computer, but perhaps a smaller sub-group of the cube (slice, or anti-slice, or such) could be done this way in a reasonable amount of time.... Hellman seems to think that solving the DES (which has 2^56 keys, I think) is not impractical, given enough money and a few years. How about a competition to find the shortest ways to get to the Crux Christmanni & Plummeri(?)...a simpleminded corner-shifting I tried took 84 moves, I think, to get whichever one has two cycles of 3 colors...that leaves lots of room for improvement! --Mark  Date: 23 Oct 1980 2155-PDT From: KATZ at USC-ISIF Subject: whats happening? To: cube-lovers at MC cc: katz There hasn't been much activity on this list for long time, whats happening? Maybe we are all cubed out?? Anyway, heres a question for the LA people. Does anyone know where I can get a Hungarian version of Rupic's cube (NOT the ideal version) in the L A area?? Alan -------