Date: 3 August 1981 22:34-EDT From: Alan Bawden Subject: The Archive To: CUBE-LOVERS at MIT-MC Those of you who look through the archives of old Cube-Lovers mail will notice that I have split off a new section of the archive. The mail now lives in: MC:ALAN;CUBE MAIL0 ;oldest mail in foward order MC:ALAN;CUBE MAIL1 ;next oldest mail in foward order MC:ALAN;CUBE MAIL2 ;more of same MC:ALAN;CUBE MAIL ;recent mail in reverse order  Date: 4 August 1981 01:21-EDT From: Chris C. Worrell Subject: Poll To: CUBE-LOVERS at MIT-MC In line with the recent cubeing contest in Boston, I suggest a poll. Such questions as, will be included: 1. age 2. average solving time 3 occupation (if student, undergrad or grad and major) 4. solving method 5. how long it took you physically working (playing) with the cube (or some facimile, like a program) to solve the cube the first time, and reliably 6. how long have you been working with the cube (say since Jan. '81) 7. max #of qtw. in method, avg. # of qtw. in method I would not suggest anybody send their answers to the list, because 1. it would generate a lot of semi-useless garbage in the list,and 2. you would lose anonymity (which is one of the aspects of a poll) If anyone can suggest how this could be implemented please pass it on. Chris Worrell  Date: 4 August 1981 01:58-EDT From: Chris C. Worrell Subject: Identities... To: CUBE-LOVERS at MIT-MC A few corrections to my list: I16-13 has 18 qtw not 16 I18-4 has 20 qtw not 18 and has supergroup effect I18-5 is not distinct from I18-6 so may be taken off the list Additions: I16-15 U'F'UBU'FUB'URU'L'UR'U'L courtesy of Jerry Agin I16-16 FLF'RFL'F'R'B'LBR'B'L'BR I16-17 LD'R'DL'D'RUR'DLD'RDL'U' I18-9 SAME AS I16-13 I18-10 RDLD'R'DU'L'D'LUD'R'DL'D'RD Concerning Sources: I found none of these by exhaustive search so I believe that this is far from a complete list esp. in 18 region. I used several methods to derive these identities 1. from two processes which have same effect , I concatanate the inverse of the second to the first, and remove noops. 2. from 1 process which affects only the d face or not the U face, I twist both the D and U faces, run the reverse process (maybe with an orientation change) then untwist the D and U faces. 3. examine old identities for similarities shift/rotate/invert/reflect appropriatly concatanate and remove noops. Chris Worrell  Date: 4 Aug 1981 1123-PDT From: Alan R. Katz Subject: 10 sided "cube" To: cube-lovers at MIT-MC cc: katz at USC-ISIF I have a 10 sided "cube", which is made by "Wonderful Puzzler" (they also make crappy cheapo regular cubes). The guts are essentially the same as the regular cube, but the corners are cut off. If you look down from the top, you see an octogon. The edges are all the same as a regular cube, but since the corners are cut, they are one color (thus the 10 colors). For example, there are red, blue, and orange faces. On an ordinary cube you would have a red-blue-orange corner cubie, but on this in its place is a pink face. To make this clearer, here is the coloring of the thing: red * light-blue**gold**yellow**blue**pink**orange**violet**green * white (red on top, white on bottom, looking at the blue face, back face is green, right face is orange, left face is gold). The interesting thing about this is that unlike the ordinary cube, every cube does not have a place, you dont know that the pink corner goes between the blue and orange faces (in an ordinary cube it is the red-blue-orange corner so you know where it goes). To solve it, you just put the corners in some order, solve it using the usual transformations, and then if you get a "parity error" you must go back to the top layer but switch two of the corners and solve it again. Thus in general you have to solve it almost 2 times! (almost because you dont have to redo the top layer or half of the second layer (this assumes you solve top down, which I do.)). What I mean by parity error is that if the corners are switched you can get a configuration that in an ordinary cube would tell you the cube is put together wrong. For example, you can be solving it and get to a point where an odd number of edges must be fliped. There may be a transformation to flip an odd number of edges with this cube, but I have not found it. Anyway its more interesting to solve and it changes it shape in general with each transformation. (unlike the cube which stays a cube; this is a octagonal prism). Alan -------  Date: 5 Aug 1981 0948-PDT From: ISAACS at SRI-KL Subject: 10-sided cube To: cube-lovers at MIT-MC The 10-sided cube was discussed a couple of months ago. The main result was that the easiest way to solve it was "side across" - that is, don't start from the octagonal face, but from one of the "sides". Then the last layer should be solvable except for a possible edge-flip. Note that there are two new difficulties with this cube: the one mentioned, and the single edge flip. SPOILER! This is, of course, an optical illusion, brought about by the fact that the second edge, which of course is also flipped, is a uni-colored edge, and you can't see it. --- Stan Isaacs -------  Date: 7 Aug 1981 1005-PDT From: ISAACS at SRI-KL Subject: BARREL AVAILABLE To: CUBE-LOVERS at MIT-MC The "Magic Barrel" or "Ten Billion" or whatever its name is, that was talked about in this digest several months ago, seems to have finally gotten on the American market. In the Bay Area, at the Stanford shopping center, they have them in stock at Macy's, and will soon have them at Games and Things. Games and Things also has the smallest cubes I've seen - about 3/4 inch on a side. That makes at least 4 different small sizes, from 1 1/2 inch down. --- Stan -------  Date: 10 Aug 1981 0841-PDT From: Tom Davis Subject: Barrel Puzzle instructions To: cube-lovers at MIT-MC I just purchased one of the "Wonderful Barrel" puzzles mentioned in Stan Isaac's note, and I found the enclosed set of instructions to be a classic. I figured I'd pass them on, in case some of the members of this list don't buy the puzzle itself. The first couple of paragraphs are not so great, but it gets better toward the end. @begin(verbatim) "Wonderful Barrel" is a kind of mental game plate also can be regarded as an indoor ornament. The red, orange, yellow, green, blue, black color balls which is inner in barrel can create more than 10 billion combination to reach a solution and finally be aligned on 5 straight lines, yet each straight line are with 4 same color balls. * PLAY RULE * Original alignment circumstances is as follow: there are three black balls are set on SHELTER but each black ball is above a shore of PLUNGER; the 5 straight lines are up the SHELTER with different color from one another, each line contains 4 same color balls. The 5 straight lines are aligned in clock-wise with blue, green, yellow, orange, red color. As you know, the original alignment is orderly, now you can circumvolve DRUM, then operate plunger up and down for taking the ball out from SHELTER. While you put the ball into SHELTER again, the order of "each straight line with 4 same color ball" has been broken to confused state. Now, a challenge is ahead of you, you often successful in this line but irregularly in that line, in a word, it always can not come to a satisfactory arrangement of the whole lines fluently. At the beginning, it is not easy to arrange the 5 straight lines return to initial order in short time, so please try to solve this puzzle from one line first, then completion of two, three, four even to whole lines are returned to initial order finally. At last, maybe you can analyse the mystery out, but maybe you can not solve it today; however, how about tomorrow? one month later? one year later? or ten years later you may still continued to challenge to it singlely and unwilling. @end(verbatim) P.S. I @i(tried) to proofread this, but as you can see, it was difficult. -------  Date: 14 August 1981 0111-EDT (Friday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Results of an exhaustive search to six quarter-twists Message-Id: <14Aug81 011137 DH51@CMU-10A> The first answer is that there are exactly 878,880 cube positions at a distance of 6 quarter-twists from solved, and so 983,926 positions at 6qtw or less. These figures reflect a decrease of 744 from the previously known upper bounds. It turns out that the twelve-qtw identities reported by Chris C. Worrell are complete, in a sense. The only reservation here is that a fifth rule must be added to his list of the ways in which ``a generator generates other identities.'' This rule is substitution with shorter identities, and it's not too surprising that it was left out, since the only shorter identities are the ``trivial'' ones like XXXX=XYX'Y'=I, where X and Y are opposite faces. In the case of the twelve-qtw identities, this means that identities of the form aXXb and aX'X'b generate each other. The structure of the 12-qtw identities is clearer if we write them in a transformed way: I12-1 FR' F'R UF' U'F RU' R'U I12-2 FR' F'R UF' F'L FL' U'F I12-3 FR' F'R UF' UL' U'L FU' The fifth rule is necessary so that I12-2 may generate the identities I12-2a FR' F'R UF FL FL' U'F and I12-2b F'R' F'R UF FL FL' U'F'. To see that this rule is necessary, it need only be observed that inversion, rotation, reflection, and shifting all preserve the number of clockwise/counterclockwise sign changes between cyclically adjacent elements. In what sense are the ``trivial'' identities trivial? I have come to believe that they are trivial only because they are short and simple enough that they are well-understood. The only identities for which I can find any theoretical reasons for calling trivial are the identities of the form XX'=I. In spite of the simplicity of the ``trivial'' identities, their occurrence is one of the major reasons that Alan Bawden and I were unable to show earlier that I12-1-3 generated all identities of length 12. I fear that the combination of 4-qtw and 12-qtw identities may turn out to be a major headache when dealing with identities of length 14 and 16.  Date: 14 August 1981 03:12-EDT From: Chris C. Worrell Subject: CUBE-POLL@MIT-MC To: CUBE-LOVERS at MIT-MC The first CUBE-LOVERS poll has now begun. Send your answers to: CUBE-POLL@MIT-MC The questions: 1. Occupation (if student, undergrad or grad and major) 2. Age 3. Average solving time 4. How long it took you physically working with the cube in order to solve it reliably. 5. How long have you been working with the cube (say since Jan. '81) 6. How many cubes do you own, and how many have died on you. 7. Solving methods: A. Time-efficient, describe method in general. B. Move-efficient, describe in more detail, including number of qtws for each stage. Include a maximum number of qtws for this method. 8. Transformations you have found or know of. Include a description of what the transform does. Include transforms you use in your method, such as top to middle slice edge movers, and transforms which affect only one face. The results of this question will be compiled into a dictionary of transfors and may also aid people in their investigations of the cube. 9. Any specific intrests you have in the cube, such as applications to group theory, or investigation of identities or whatever. 10. Any other intrests you might have, relating or not relating to puzzles. Chris Worrell (ZILCH@MIT-MC)  Date: 17 August 1981 11:34-EDT From: Allan C. Wechsler Subject: 12q relations. To: CUBE-LOVERS at MIT-AI I12-1 FR' F'R UF' U'F RU' R'U I12-2 FR' F'R UF' F'L FL' U'F I12-3 FR' F'R UF' UL' U'L FU' I can't believe I'm the first person to notice this: Suppose we only know I12-1 and I12-2. Then we have I12-1' U'RUR'F'U (FR'F'RUF')' (I12-1')(I12-2) U'RUR'F'U (FR'F'RUF')' (FR'F'RUF') F'LFL'U'F Reduce: U'RUR'F'U F'LFL'U'F Conjugating by (U'RUR'F'U)', we get F'LFL' U'FU'R UR'F'U But this is just the RL mirror image of FR'F'L UF'UL' U'LFU' This is exactly I12-3. So there are really only two independent 12q identities, and the third can be deduced from them.  Date: 19 Aug 1981 09:48 PDT From: Lynn.ES at PARC-MAXC Subject: Re: 10 sided "cube" In-reply-to: KATZ's message of 4 Aug 1981 1123-PDT To: Cube-Lovers at MIT-MC cc: Lynn.es@PARC I last week picked up one of the same octagonal cubes (Wonderful Puzzler brand) as Alan R. Katz in his message. I might point out that it shares the same workmanship and twistability (lack thereof) which Katz attributes to their brand of standard cube. After some experimenting, I found that the "parity error" involved is always a pair of edges in reverse locations on the bottom. My solution algorithm is top, equator, bottom [corner locations, edge locations, edge orientations, corner orientations]. On an ordinary cube, one pair of edges reversed never happens when you go to position edges. I suspect that if your algorithm does edge locations before corners, then two corners in wrong locations would be the parity error. Incidentally, the parity of Katz's cube is the reverse of mine, though the center cubie colors are in the same relation to each other. Edge orientation (I assume that is what Katz meant by "edges must be fliped") parity errors are irrelevant to getting the cut-edge-colors rightly paritied. Edge orientation parity errors happen because one of the cut-edges may have been oriented wrongly. Or more precisely, an odd number of them reversed from the way they were virginally (but they look right either direction). This is easily cured by any of the "flip a pair of edges" macros. The change in cube shape is actually helpful in solving, except for being difficult to grab sometimes. Corners that need orientation really stand out. The Greenwich meridian edges (assuming you solve it from the top down with the octagonal faces left and right), being different shape, are instantly located. The best part of the Wonderful Puzzler is the instructions, which I quote here: THE CHALLENGE: "ORIGINAL PUZZLER" presents not only a unique challenge but offers the possibility of countless hours of relaxation. Your mental ingenuity may be tested for a few hours -- many days -- several weeks -- or even a period of much longer duration. If you can determine the key to unlock the knack for solving the PUZZLER, the final trimph can be the psychological turning point in your life. Mathematicians may be tested to the limit and cry over this one -- and you may, too! You will gain a measure of satisfaction when you align one plane. You will be delighted with the completion of two. You will be elated with the completion of three or four! The completion of the fifth plane will quicken your pulse!! -- and you will have scaled the peak once the last unit of the sixth plane falls into place!!!! PREPARATION & CAUTION: *Spin the PUZZLER several times, as indicated on the cover, until all color units are randomly distributed on each of six planes. *Do NOT remove any color unit in the process of play. *Initially, activate the random distribution GENTLY. With little use, this can be accomplished easily and smoothly. Patience and persistence will beat the Puzzler! Good Luck! Challenge Can you contend with more than 18'000'000'000 combination to reach a solution? *end of quote* I think the errors remaining above are all theirs. It is full of little gems: turning point in your life, solving five planes (on the standard cube, which I believe has the same instructions), 18 billion (better than Ideal's guess). The GENTLY caution is valid; a neighbor kid blew mine apart, including a center cubie, by ignoring this.  Date: 20 August 1981 02:46-EDT From: Alan Bawden Subject: Resending to the right place so that I can digest it later... To: CUBE-LOVERS at MIT-MC Date: 19 August 1981 23:22-EDT From: Eric L. Flanzbaum Subject: A small version if the cube ... To: CUBE-LOVERS-REQUEST at MIT-MC cc: ELF at MIT-MC I don't know if this has been mentioned yet, but why not mention it again? I just saw Rubik's cube being sold on a keychain. It is about 1 1/2 inches and works/looks like the real thing. They sell for about $2.50 which is considerably less than the standard version. I live out on the west coast, so I don't know if they sell it back east. Happy solving, -Eric.  Date: 22 Aug 1981 1404-EDT From: ROBG at MIT-DMS (Rob F. Griffiths) To: Cube-Lovers at MIT-MC Subject: Re: ELF's Small cube Message-id: <[MIT-DMS].207641> I have seen that one, and also one that is absolutely tiny. It is the size of one of the cubies on the full sized Rubik's, and is fully operational. They are really quite flexible and well made, I don't know who manufactures them, but I will try to find out. -Rob.  Date: 22 August 1981 18:46-EDT From: Chris C. Worrell Subject: CUBE-POLL``MIT-MC To: CUBE-LOVERS at MIT-MC SO FAR I HAVE RECEIVE ONLY ABOUT 8 REPLIES, THIS IS HARDLY ENOUGH TO DO STUDIES ON. PLEASE SEND MORE REPLIES. IF YOU NO LONGER HAVE THE QUESTIONNAIRE SEND ME MAIL AND I WILL FORWARD IT TO YOU.  Date: 24 Aug 1981 1032-PDT From: ISAACS at SRI-KL Subject: recoloring the 10-sided cube To: cube-lovers at MIT-MC A nice way to recolor the 10 sided cube is to give a side a quarter twist, so it looks sort of like a baseball, and then make the (fairly) obvious 4 groups of 9 faces each in four colors, and two in-between stripes of 3 facies each. Each of the 9-facie groups will have 2 triangular facies, and 3 slant rectangular facies, and 4 squares. I think its a fairly simple variation to solve, but I just made it last night and have not worked with it much yet. By the way, I just got my first magic tetrahedron. This one came from Japan, but says made in Hongkong by "World-wide"(?) copyright by Meffert. Anyone know who he is? It seems very similar to the one invented by Kristen Meier. -------  Date: 25 Aug 1981 1016-PDT From: ISAACS at SRI-KL Subject: BAY AREA CUBE CONTEST To: cube-lovers at MIT-MC Date: 24 Aug 1981 1612-PDT From: ISAACS at SRI-KL Subject: CUBE CONTEST-BAY AREA To: cube-lovers at MIT-MC There will be a Rubics Cube contest at Games and Things, at the Stanford Shopping Center this Saturday, Aug. 29, from 10:00 am to 4:00 pm. Speed contests, money prizes, a cube display (by me), etc. Contestants are supposed to register at GAMES AND THINGS before 10:00 on Saturday. Address is 128 Stanford Shopping Center, Palo Alto, Ca., (415) 328-4331. --- Stan Isaacs ------- -------  Date: 25 Aug 1981 1019-PDT From: ISAACS at SRI-KL Subject: MORE TERMINOLOGY To: CUBE-LOVERS at MIT-MC More possible terms (from a new cube-solver who was a biology major): Dorsal/Ventral for front/back, Port/Starboard for right/left (left/right?). He doesn't have a consistant term for up/down. -------  Date: 25 August 1981 19:04-EDT From: Alan Bawden Subject: Speed cubing To: CUBE-LOVERS at MIT-MC Does anyone have any idea what the world record for speed cube solving really is? The only times I can find are: 1) In the Scientific American article Hofstadter mentions an Englishman named Nicholas Hammond who averages "down to close to 30 seconds". Anyone have any more information on this? Like where Hofstadter found out about this guy? 2) In the reports about the "Regional Cubing Championship" held here in July the best time listed is 48.31 seconds, held by a 10 year old named Jonathan Cheyer. (See PDL's message to Cube-Lovers dated 27 Jul 1981.) 3) I seem to remember reports that there were Hungarians who averaged around 50 seconds. I thought I had read this in Singmaster, but I can't seem to find it there. 4) The best time anyone will admit to on this list (as determined by scanning the replys to ZILCH's poll) is 2 minutes. This time is claimed by both Richard Pavelle (RP@MIT-MC) and Alan Katz (KATZ@ISIF). Also Stan Isaacs (ISAACS@SRI-KL) claims that his two children take about 1 1/4 minutes using essentially his methods. 5) Finally, Dan Pehoushek (JDP@SU-AI) tells me that a friend of his frequently breaks the 30 second barrier. I should have thought to ask for his name. Anybody know of any more good speed cubists?  Date: 25 Aug 1981 2116-EDT From: ELF at MIT-DMS (Eric L. Flanzbaum) To: Cube-Lovers at MIT-MC Subject: Speed Solving Message-id: <[MIT-DMS].208572> Hi Cube fans, I have a couple of friends at school (oh about 4 or 5) who consistently solve the cube (actually, they have races/contests) at a time of under 30-40 seconds. I don't know if this is really unusually fast, but as from ALAN's previous message, it looks like they all rank in there. By the way, these people are entering the 9th and 10th grade in the fall. Happy Solving, -Eric L. Flanzbaum ELF at MIT-AI  Date: 27 August 1981 18:39-EDT From: Dennis L. Doughty Subject: Speed cubing To: CUBE-LOVERS at MIT-MC cc: DUFTY at MIT-MC My fastest time for the cube is 1 minute 17 seconds (everything worked out correctly). My average time is 1:40-1:45. --Dennis p.s. i'll answer the poll when I get the time.  Date: 28 August 1981 12:59-EDT From: Robert H. Berman To: CUBE-LOVERS at MIT-MC You may be interested to see a cartoon about the cube on page 36 on the August 31 issues of the New Yorker. Yes, the New Yorker. --rhb  Date: 29 August 1981 07:54-EDT From: Thomas L. Davenport Subject: Cube Song To: CUBE-LOVERS at MIT-MC This past week PBS broadcast the latest "Mark Russell Comedy Special" and in it he did a funny song about the cube. He even had one with him on stage. -Tom-  Date: 29 Aug 1981 1906-EDT From: ROBG at MIT-DMS (Rob F. Griffiths) To: cube-lovers at MIT-MC Subject: English Whiz Kid Message-id: <[MIT-DMS].208873> From TIME: August 31,1981: --------------------------- Along with diet books, cat books, and advisories on how to make a profit from the coming apocalypse, there is a growing shelf concerned solely with mastering that infuriating, six sided, 27-part boggler with 42.3 quintillion possible combinations known as Rubik's Cube.. The latest entry: ''You Can Do The Cube'' (Penguin, $ 1.95) by Patrick Bossert, 13, a London schoolboy who discovered the cube only this spring during a family ski vacation in Switzerland. Within five days he had mastered the monster, and later began selling his schoolmates a four-page, mimeographed tip sheet for 45 cents. An alert editor at Penguin saw a copy and persuaded the prodigy to turn pro. The 112 page result contains 3 dozen 'tricks' for solving the cube (using logic rather than math), as well as a chapter on 'Cube Maintenance' (to loosen a stiff cube, ''put a blob of Vaseline on the mechanism''). With 250,000 copies of the cubist's book in print, a Penguin executive marvels: ''It's the biggest, runaway, immediate success we have had since we published 'Lady Chatterley's Lover' in paperback.'' --------------------------- -Rob.  Date: 1 Sep 1981 1442-EDT From: Bob Clements Sender: CLEMENTS at BBNA Subject: [Bob Clements : Rubik's cube sale] To: Cube-lovers at MC I didn't know of the cube-lovers list, but it was suggested to me that I re-send this msg to cube-lovers. If this sale has already been mentioned on this list, sorry for the repeat. /Rcc --------------- Mail-from: MIT-AI Received-Date: 1-Sep-81 1343-EDT Date: 1 Sep 1981 1221-EDT From: Bob Clements Sender: CLEMENTS at BBNA Subject: Rubik's cube sale To: Info-Micro at ai I don't want to start a whole Rubik's discussion, but for those in the Boston area who need to replace their worn Cubes, or get their first one, the Caldor chain has them on special for $4.66 thru Saturday. /Rcc ------- --------------- -------  Date: 3 September 1981 15:40-EDT From: Dennis L. Doughty Subject: Practical use of the Rubik's cube/Speed cubing To: CUBE-LOVERS at MIT-MC cc: DUFTY at MIT-MC This rush week, my fraternity extended a bid to a freshman by the name of Larry Singer who generally solves the cube in 1:13 or thereabouts (my average time is 1:40). All day Monday, our bids were playing such games as "Pledge Pong" or "Pledge Pool." The idea here is that the bid challenges an active to a game of pool or ping-pong, and if he loses, he pledges. Well, Monday night, Larry challenged me to "Pledge Cube." One of the brothers uniformly scrambled two cubes, and we were to compete in a head-to-head competition. Well, we both were under considerable pressure, naturally, and we both made several mistakes while solving the cube. I won, but my winning time was 2:09. Larry and his close friend then both pledged. So now, no one in the house can tell me that there's no practical use for the Rubik's cube. --Dennis  Date: 5 September 1981 1446-EDT (Saturday) From: Bob.Walker at CMU-10A To: cube-lovers at mit-mc Subject: Snake: **SPOILER** Message-Id: <05Sep81 144605 BW80@CMU-10A> I am new to the list, and in reading the archives, I found that the transform for my favorite pattern was listed incorrectly. Specifically, I refer to the Don Woods' message of 6 January which listed transforms for the Snake, Worm, and Baseball (I think). Anyway, the transform listed for the Snake was incorrect. What is listed IS a pretty pattern, merely the wrong one. The blurb about how to "hack" your way to the Snake, however, is correct. The proper transform to achieve the snake (from Singmaster) is: * * * S P O I L E R * * * Snake: B R L' D' R R D R' L B' R R + U B B U' D R R D'  Date: 5 September 1981 01:08 edt From: Greenberg.Symbolics at MIT-Multics Subject: TV special Friday night at 8 NBC Magazine will cover Rubik's cube. The coverage will doubtless include footage of the July 25 Boston Area Cubeathon, at which many MIT-Area cubists were present.  Date: 8 Sep 1981 0942-PDT From: ISAACS at SRI-KL Subject: misc To: cube-lovers at MIT-MC Some short notes on various cube stuff - will try to expand on some of it later: 1. The Stanford Shopping Center/Games and Things cube contest of a couple of saturdays ago: was won by Paul cunningham, 16 yrs old. There were about 40 entries; there were something like 8 rounds to get to a winner, double elimination, each pair fighting it out with the best average of 3 cubes, all cubes in a round scrambled exactly the same way. Best average-of-three time was 56 seconds. Best time was 41 seconds. David Tabuchi, the Games & Things speedster, has a best average-of-ten speed of 43 seconds! Brian Robinson, with whom he works on the cube, has a best average-of-ten of 41 seconds. Davids fastest time was 24.98 seconds!!! 2. Cubes are multiplying like hotcakes. Not only changes in labeling, but also changes in shape. I have seen or heard reports of about 8 shape variations, and multiple size variations, from about 19 mm to about 60 mm. And someone told me of a 12mm or so version. The corners have been cut off in 3 different ways (The nicest cuts them halfway through the neighboring edge, and uses 14 colors). The magic Tetrahedron is readily available now. There is a build-it-yourself cube kit. There are still reports of the elusive 4x4x4 cube, but no actual sightings as far as I know. By the way, I collect puzzles, and am trying to find many of these cube variations. If anyone knows where I can actually buy some, or would get me some, please let me know. I will be happy to pay or trade for them. 3. The snake is not a cube, but it is a toy/puzzle/art object that should a appeal to cube-lovers. It also comes in a variety of colors and sizes. 4. Also lots of new books and such. Such as: a. Don Kolve, of Kirkland, Wash. He solves Top-middle-bottom(position corners,twist corners, flip edges, position edges). b. L.E.Hordern, of England. He does: bottom-middle-top ( position corners, orient corners, orient edges, position edges). c. Bridget Last, of Downham, England. She solves: Define face colors ("The easiest way of deciding which face is to be which colour is to define the centre faces as being correct.") Then: position all corners, orient all corners, position (with orientation) all edges. d. Bob Easter, a friend in San Francisco, uses just one move to do everything. The move is F R' F' R (an old friend). First he "walks" the edge cubes to their proper corners, then rotates them into proper order around the corner, then flips them 2 at a time to get them aligned. Then he does similarly with the corners, using the same move, but done 3 times to leave the edges alone. Lots of quarter twists, but little memory. Perhaps the ultimate solution for people with strong wrists. e. Patrick Bussart, Puffin books (a division of Penguin Books). Patrick has been in the news since he is only 12 or 13 years old; his solution is very popular in England. He does top corners, top edges, bottom corners, position rest of the edges, rotate rest of the edges. 5. I have been re-labeling cubes to make new puzzles, and would appreciate suggestions. I have made tactile cubes of various types (that is, with various materials). I'm trying to make one with 5 or 6 grades of sandpaper, but find my touch cannot distinguish between the two middle grades. I have made a magic square cube (each face is a 3x3 magic square; the problem is to decide what the relative orientations and forms of the squares should be; any suggestions?) and a magic cube cube (the center 14 cube is, of course, invisible). And 2 word cubes - one has word squares (there are three 3 letter words in each direction), and the other, six 3 word sentences (FIX THE BOX, YES YOU CAN, FUN FOR ALL, etc. I haven't had these long enough to know if they are "solvable" without trial and error; I think that the magic square cube, especially, is difficult unless you have the order in advance. So far, I have made 2 of them, the first had one square marked, which makes it fairly easy (the other 5 squares are forced by the first); in the second, I made sure each corner was unique, and the edges as different as possible (2 have to be the same); but I haven't had time to try it yet. Enough - this message is too long. -------  Date: 10 Sep 1981 1447-PDT From: ISAACS at SRI-KL Subject: cube query To: cube-lovers at MIT-MC The 2x2x2 cube is solved in the corner sub-group, ignoring the (non-existent) edges. The so-called "Dinman Style" cube (probably meant to be "diamond") has the corners cut off and everything stretched to make a somewhat distorted rhombicuboctahedron (the six center facies are square; the corners are now triangles, and the old edges are rectangles). Solving this involves only positioning moves - all orientation (twisting) is invisible. Thus these two cubes involve two "pure" subgroups. Can anyone design (by either cutting, recoloring, or even inventing new mechanisms) cubes or pseudo-cubes which only involve edge-type moves, or which only involve twisting, with positioning ignored? --- Stan Isaacs -------  Date: 11 Sep 1981 0917-PDT From: ISAACS at SRI-KL Subject: half query answer To: cube-lovers at MIT-MC First of all, I was inaccurate about the "Dinman" style "cube" - it doesn't only eliminate twisting, but also some positioning (ie, after the top and middle are solved, the bottom is automatically solved). Perhaps by the judicious adding of numbers to the facies, the pure position cube can be made. Also, the answer to the edge-only subgroup is easy - just remove all the labels from the corners, so they are all identically monochromatic. Is there a more elegant solution? --- Stan -------  GENTRY@MIT-AI 09/11/81 20:43:40 To: CUBE-LOVERS at MIT-MC It appears that due to time restrictions, the segment about the RUBIK's cube on NBC Magazine has been postponed until next week. Check the listings for your area to see when it will be televised.  Date: 15 Sep 1981 1553-PDT From: ISAACS at SRI-KL Subject: lower bounds To: Hoey at CMU-10A cc: cube-lovers at MIT-MC [This message is being sent to Dan Hoey, and refers to his message of 9-Jan-81, subject: The Supergroup -- Part 2: at least 25 qtw and why] Appended to this message is a longish message I recieved, which has some good ideas to use. In particular, what about using your technique on a 2x2x2 cube, or an (idealized) edge-only cube? And then comparing it with his clculations for the 2x2x2. I'm not sure without a 2x2x2 in front of me, but I think there are only 2 distinct 1 qtw per set of opposite faces, and only one 2qtw move. And that the period is only 2. Is that true? However, there should be more low-number-of-twists identities. I'm distrustful of the actual calculations in the message below, because I don't see the 9 new configurations after only 1 twist. I think there are only 6. Or am I missing something? Also, Dan or someone else on the cube-lovers network: how about compiling all the messages about lower bounds and identities (after a while) into one file we can ftp and look at all together. 11-Sep-81 12:26:52-PDT,6785;000000000001 Mail-from: ARPAnet host BERKELEY rcvd at 11-Sep-81 1223-PDT Date: 11 Sep 1981 11:43:07-PDT From: ARPAVAX.sjk at Berkeley To: isaacs@sri-kl Subject: in case you haven't seen this ... Article 16: >From csvax:mhtsa!harpo!chico!esquire!psl Wed Sep 9 17:16:32 1981 Subject: Rubik's Cube Newsgroups: net.games Want to knoe how far away you can get from the solution on a Rubik's Cube? A Simple Lower Bound As everybody knows, the number of discrete configurations of the 3x3x3 Rubik's Cube is: (8! * 12! * 3^8 * 2^12) / 12 = 4x10^19 = 43,252,003,274,489,856,000 One approach to a lower bound is to calculate the maximum possible number of configurations you can reach with a particular number of moves and then see how many moves you would have to make to reach the number above. With no moves at all you get 1, the starting position. The first move gets you 18, (any one of six faces turned one of three ways). The next move gets you 18*15, (no point in turning the same face twice in a row), for a total of 1+18+270 configurations reached after two moves. A table of these values looks like: ---------possible configurations--------- moves new % max total 0 1 0.0% 1 1 18 0.0% 19 2 270 0.0% 289 3 4050 0.0% 4339 4 60750 0.0% 65089 5 911250 0.0% 976339 6 13668750 0.0% 14645089 7 205031250 0.0% 219676339 8 3075468750 0.0% 3295145089 9 46132031250 0.0% 49427176339 10 691980468750 0.0% 741407645089 Notice that 11 10379707031250 0.0% 11121114676339 not until 17 12 155695605468750 0.0% 166816720145089 moves has the 13 2335434082031250 0.0% 2502250802176339 total number 14 35031511230468750 0.1% 37533762032645089 of possible 15 525472668457031250 1.2% 563006430489676339 configurations 16 7882090026855468750 18.2% 8445096457345145089 exceeded the 17 118231350402832031250 273.4% 126676446860177176339 maximum. So there is no possible way to reach some configurations in fewer than 17 moves. However, this analysis has assumed that each configuration generated was a NEW one, but there are MANY cases where this will not be so. A simple example is turning one face 180 degrees, the opposite face 180 degrees, and then repeating those two moves -- four moves that get us to an old, familiar configuration. If we factor out the sequences that involve these opposite face identities the minimum number of moves becomes 18. Needless to say there are still lots of useless move sequences, but detecting them becomes a lot trickier. A Rumored Upper Bound Rumor has it that a computer program exists, (attributed to Thistlethwaite), that provably will solve any Cube configuration in at most 41 moves. Narrowing it Down So the answer is somewhere between 18 and 41. How do you get further? One way is to write a computer program that tries every sequence of moves until it has generated every possible configuration at least once. That sounds easy, and it is, but such a program would take a \\\L O N G/// time to run. However, if we limit the problem a little by considering a Cube that is two squares on a side (2x2x2), we have a chance of learning something. 2x2x2 Cube By the same considerations stated above we can get a lower bound for the 2x2x2 Cube. There are 7! * 3^6 = 3,674,160 configurations and, since we can limit ourselves to moving only three "orthogonal" sides of the 2x2x2 cube, on the n-th move you could reach 9 * 6^(n-1) new configurations thus we find that with 8 moves you could reach at most 3,023,307 and with 9 you could reach at most 18,139,851. (Note that this doesn't have the problem with opposite side moves that the 3x3x3 cube has.) Because the 2x2x2 cube is relatively simple we can actually run a program to try all the possible move sequences and compare our bound with fact. Listed below are the findings ------new configurations------- total configurations moves -----possible---- ---actual--- ---possible --actual number % number % number number 0 1 0.0% 1 0.0% 1 1 1 9 0.0% 9 0.0% 9 10 2 54 0.0% 54 0.0% 63 64 3 324 0.0% 321 0.0% 387 385 4 1944 0.0% 1847 0.0% 2331 2232 5 11664 0.3% 9992 0.3% 13995 12224 6 69984 1.9% 50136 1.4% 83979 62360 7 419904 11.4% 227536 6.2% 503883 289896 8 2519424 68.6% 870072 23.7% 3023307 1159968 9 15116544 411.4% 1887748 51.4% 18139851 3047716 10 90699264 2468.6% 623800 17.0% 108839115 3671516 11 544195584 14811.4% 2644 0.071% 653034699 3674160 Interestingly enough there are 2,644 configurations that require eleven moves to reach a solution; this is less than one tenth of one percent of the total configurations! It's also interesting that it's better than a 50-50 bet that a randomly ordered 2x2x2 cube can be solved in exactly nine moves, (it's not clear how to turn this into a profitable bar bet, however). Noticing that there are only 321 new configurations after three moves instead of 324 leads us to guess that there are six non-trivial sequences of six moves that end with the original configuration, (why?). These results came from a C program running on a VAX 11/780 and even though the 2x2x2 cube is simple compared to the 3x3x3 it took a lot of time. The figures for 11 moves took over 51 hours of cpu time. If you'd like to make a 2x2x2 cube with which to experiment you can simply take all the little labels off a 3x3x3 cube except the ones on the corners and then ignore the unlabeled cubes. Here's one sequence that gets you to one of the 2,644 configurations: f r f r f d2 f d- f d2 r2 f = rotate front face 90 degrees r = rotate right face 90 degrees d2 = rotate "down" face 180 degrees d- = rotate "down" face 270 degrees So Where's That Leave Us? I just thought of a dandy way to get the answer for the 3x3x3 cube, but the margins on this news item are a little too small for me to include it ... -------  Date: 15 September 1981 21:54-EDT From: Alan Bawden Subject: Editor's note to the last message. To: CUBE-LOVERS at MIT-MC I will look into collecting all of the relevant messages on God's number into one place. If you want to poke around in the archives yourself (please be carefull, and don't delete them again) I will remind you all that old cube-lovers mail is archived in the following places: MC:ALAN;CUBE MAIL0 ;oldest mail in foward order MC:ALAN;CUBE MAIL1 ;next oldest mail in foward order MC:ALAN;CUBE MAIL2 ;more of same MC:ALAN;CUBE MAIL ;recent mail in reverse order (I someone else wants to attempt the compilation, there is a better chance it will get done. Let me know and I will be happy to lend a hand.) Some of the seeming inconsistencies in the message included by Isaacs in his message are a result of the usual half versus quarter twist screw. The reason the writer can see 9 configurations after a single twist is because he has a different definition of a "single twist". I also am not sure, but I also think that the counting argument given here suffers from the some confusion Singmaster had when he computed a lower bound of 17 htw. I think, in fact, that a lower bound of 19 htw results if the argument is executed correctly (Singmaster corrected himself about this by the fourth edition, I think). Someone with a copy of Singmaster handy should look this up. The 41 move count for Thistlethwaite's algorithm is probably a half twist count given that it was reported by Singmaster.  Date: 16 September 1981 0003-EDT (Wednesday) From: Dan Hoey at CMU-10A To: ISAACS at SRI-KL, Cube-Lovers at MIT-MC Subject: Re: lower bounds In-Reply-To: Stan Isaacs's message of 15 Sep 81 17:53-EST and Alan Bawden's message of 15 Sep 81 20:55-EST Message-Id: <16Sep81 000353 DH51@CMU-10A> Hi. I'm really pressed for time, but I'll drop a couple of comments. Alan pretty well said it--there are half-twisters and there are quarter-twisters and the included message is one of the former. I strongly favor the latter, since then all the moves are equivalent, (M-conjugate, to you archive-readers). But Singmaster's book, though in the other camp, is too good to ignore. To extend the argument I gave on 9 January to the case where quarter-twists and half-twists are counted equally (we call such a move a `htw' whether it is quarter or half) let PH[n] be the number of (3x3x3-cube) positions at exactly n htw from SOLVED. Then PH[0] = 1 PH[1] <= 6*3*PH[0] PH[2] <= 6*2*PH[1] + 9*3*PH[0] PH[n] <= 6*2*PH[n-1] + 9*2*PH[n-2] for n > 2. Solving yields the following upper bounds: htw new total htw new total 0 1 1 10 2.447*10^11 2.646*10^11 1 18 19 11 3.267*10^12 3.531*10^12 2 243 262 12 4.360*10^13 4.713*10^13 3 3240 3502 13 5.820*10^14 6.292*10^14 4 43254 46756 14 7.769*10^15 8.398*10^15 5 577368 624124 15 1.037*10^17 1.121*10^17 6 7706988 8331112 16 1.385*10^18 1.497*10^18 7 102876480 111207592 17 1.848*10^19 1.998*10^19 8 1373243544 1484451136 18 2.467*10^20 2.667*10^20 9 18330699168 19815150304 At least 18 htw are required to reach all the 4.325*10^19 positions of the cube. This is the same argument that was used in Singmaster's fifth edition, p. 34, and is the best I know. Lest ye be tempted to pull the trick I did in the January message, remember that half-twists are even permutations, so there is no assurance that half the positions are an odd distance from SOLVED. This is illustrated in the 2x2x2 case, where more than half of the positions are at a particular odd distance. And yes, all of Thistlethwaite's analysis seems to use the half-twist metric. I am quite surprised, however, to hear the rumor of 41 htw. As of Singmaster's fifth edition, the figure was 52 htw ``... but he hopes to get it down to 50 with a bit more computing and he believes it may be reducible to 45 with a lot of searching.'' If anyone has harder information on the situation, I would like to hear it. Well, back to real work. I saw a Rubikized tetrahedron in a shop window earlier this evening; I'm not sure whether I'm relieved or infuriated that the store was closed for the day.  Date: 21 Sep 1981 08:49 PDT From: Eldridge.ES at PARC-MAXC Subject: An incomplete solution To: Cube-Lovers@MC cc: XeroxCubeLovers^.pa Reply-To: Eldridge For months now I have been living in blissful ignorance thinking that I too could solve "the cube". To my horror and dismay I have found that my solution is not complete. There are some cubes on the market that have pictures of fruit on the faces rather than solid colors. I wondered if the solution I use would get the pictures on the faces all lined up in the proper direction. I found that it didn't! The problem is that some of the center cubies do not line up in the same direction as all the other cubies on the face. I am currently working on finding some macros that rotate the center cubies without affecting the rest of the cube. I would suggest that you might find one of these fruit cubes and try it. Or you can do as I did and mark the faces of an original cube so that you can tell the orientation of the cubies. Good Luck! George  Date: 21 September 1981 15:50-EDT From: Richard Pavelle Subject: 4x4x4 To: CUBE-LOVERS at MIT-MC A person in sales at Ideal Toy Corp told me today that they will begin selling the elusive 4x4x4 next May or June. The die casts are being made now. They will be showing it at Toyfair (a trade show?).  Date: 22 Sep 1981 0552-EDT From: ZILCH at MIT-DMS (Chris C. Worrell ) To: CUBE-LOVERS at MIT-MC Subject: Article on Erno Rubik Message-id: <[MIT-DMS].210992> Yesterday in a store I noticed that this week's PEOPLE magazine has an article on "Erno Rubik: The Inventor of that !#&%$*@ Cube" (this title is approximatly right) The article is on page 30 (40?) and runs for a total of 4 pages, however I didn't read it so I don't know what it's content is.  Date: 23 September 1981 0005-EDT (Wednesday) From: Guy.Steele at CMU-10A To: cube-lovers at MIT-MC Subject: The Cube in the Comics Message-Id: <23Sep81 000559 GS70@CMU-10A> The "Ferd'nand" comic strip for 9/22/81 features the cube. (For those who don't see this strip: it's pure mime, with no word balloons. Panel 1: Ferd'nand sits in an armchair, struggling with a (scrambled) cube. Panel 2: In disgust he throws it out of a window. Panel 3: The cube sits on the lawn. Ferd'nand turns away, but still is glaring over his shoulder at the cube. Panel 4: Fred'nand is sitting on the grass where the cube had landed, struggling away again. Know the feeling?) --Guy  Date: 23 Sep 1981 1554-PDT From: HOROWITZ at USC-ISIF Subject: question for cube-lovers To: cube-lovers at MIT-AI Minh Tai, a high school senior in the Los Angeles area who can do the cube in average 50 seconds proposes the following problem: Is it possible to make every face have either a Z or an S (two colors, one forming the Z pentomino)? He signs himself with a quick routine for six T's as follows: (U2 R' L)2 (B2 D2)3 ------!----!------- ! x ! x ! ! ! ! ------!----!------- ! ! x ! ! ! ! ------!----!------ ! ! ! x x ! ------!-----!------ -------  Date: 23 September 1981 2236-EDT (Wednesday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Re: question for cube-lovers (S and T patterns) CC: HOROWITZ at USC-ISIF In-Reply-To: Alan Bawden's message of 23 Sep 81 19:31-EST Message-Id: <23Sep81 223642 DH51@CMU-10A> It is impossible to put Z patterns on all six faces of the cube, just as it is impossible to extend the Laughter (or Zig-Zag) pattern to six faces. The problem is with the corners. If every face is to have a pair of opposite corners that agree with the face center and a pair of opposite corners that agree with each other but not with the center, then the only constructible pattern is like (i.e. M-conjugate to) the following: F - U - U - U - F D - L F - R U - R B - L - L - - F - - R - - B - L - D R - F R - U L - B B - D - D - D - B But this pattern has incorrect corner orientation, and so is not achievable. The T patterns were introduced to this list by David C. Plummer [3 September 1980 2123-EDT], who assigns credit to Tanya Sienko for the idea. Jim Saxe and I [27 January 1981 0102-EST] gave a process for Tanya's T, the pattern that Minh Tai uses to sign with. Our process, (FF UU)^3 (UU LR')^2, is four quarter-twists shorter than Tai's ``quick routine'' because of the cancellation in the middle. The other T pattern, Plummer's T, can be achieved in 28 qtw: FF UD' F'B' RR F'B U'D RL FF RL' UD' RL FF R'L U'D'.  Date: 24 Sep 1981 0835-PDT From: Chris Van Wyk Subject: Soviet view of the cube To: cube-lovers at MIT-MC September 1981 Atlas World Press Review reports Americans may call the Rubik Cube a fad, but to Izvestia's Melor Sturua it is a "new psychosis." He reports [July 12] the interpretations of unidentified sociologists: "Some assert that the Rubik Cube reflects the philosophy of the Reagan Administration--to build and destroy aimlessly in a futile search for a solution to the world situation. Others see Americans' efforts to solve the puzzle as the desire to escape from a disordered life. -------  Date: 25 September 1981 03:11-EDT From: Chris C. Worrell Subject: Poll results To: CUBE-LOVERS at MIT-MC Here are the results of the CUBE-POLL. DATE AVG. WORK WHEN # # REPLY AGE SOLVE TIME GOT OWN DIED OCCUPATION (MINS) ------------------------------------------------------------ 8/28 15 -- -- 4/81 1 0 STUDENT 9/7 16 -- -- 9/81 1 0 STUDENT 9/8 18 3-3:15 1.5-2m 9/80 2 .75 STUDENT 9/24 19 1:40 <10h 7/80 0 0 UG-ENG 8/17 20 4 1w 12/79 1 0 UG-CS/EE 8/29 20 1:50 1w 7/81 2 0 UG 8/25 22 1:30 ? 6/80 7 0 COMP. ANALYST 8/19 22 10 unreliably 7/81 2 0 HACKER 8/15 24 4 2d 1/80 1 0 PROGRAMMER/GRAD-CS 8/17 25 2 40h 6/80 4 0 PROGRAMMER/GRAD-PHYS 8/17 27 4 2-3w late78/79 2 0 HACKER/RECENT G.-CS 8/15 28 10-15 30h since start 2 0 PROGRAMMER 8/17 30 10 ? 5/80 1 0 TEACH/PROGRAMMER 9/11 30 5-10 50-60h 12/80 1 0 TEACHER-CS 8/15 31 2:15 8d 2/80 4 0 HACKER 8/18 31 5 2w 5/80 2 0 COMP. SCIENTIST 8/24 38 2 100h 5/80 4 3 PROFFESSOR 8/31 39 3 3d fall80 1 0 ROBOTICS RESEARCH 8/24 41 3 ? early79 24 0 COMP. ANALYST (the 24 cubes owned is mostly non-standard ones) DEAD CUBES: Many people replied that none of their cubes have died on them though some admit that some of their cubes are in pretty bad shape, or they have gotten additional cubes to keep their old ones from dying. One person reported that he has a cube which still works, even though it got left in a hot car, and got partially melted so is now some what surrealistically distorted. (I wonder what the temperature was) METHODS (INDEXED BY AGE(DATE-REPLIED)): 18. CORNERS,"MIDDLE-RING",TOP-SIDES,BOTTOM(MAY MESS UP TOP),? 19. TIME-EFFICIENT: U-EDGES,U-CORNERS,MIDDLE-SIDES,D-EDGES-ORIENTED, D-EDGES-POSITIONED,D-CORNERS-POSITIONED, D-CORNERS-ORIENTED MOVE-EFFICIENT U-EDGES 20 U-CORNERS 28 MIDDLE-SIDES 46 (primary target for improvement) D-SIDES 14 D-CORNERS 20 TOTAL=128qtw 20.(8/17) 2 METHODS BOTTOM,MIDDLE,TOP SLICES 4-CORNERS,8-CORNERS,BOTTOM,TOP,MIDDLE 20.(8/29) SOLVE GREEN FACE,OPPOSITE CORNERSIN CORRECT ORIENTATION OPPOSITE CORNERS IN CORRECT POSITIONS,OPPOSITE SIDES IN CORRECT POSITIONS,MIDDLE LAYER ABOUT 150qtw MAX. 22.(8/25) TOP 4 CORNERS,BOTTOM 4 CORNERS PLACED, ..ORIENTED EDGES PUT INTO PLACE IN PAIRS 22.(8/19) TOP(EDGES THEN CORNERS),MIDDLE,BOTTOM 24. MOVE AND TIME EFFICIENT: TOP CORNERS 22 BOTTOM CORNERS POSITION 26 " " ORIENTATION 24 TOP EDGES 40 BOTTOM EDGES 68 CENTERS 8 (initial posn of corners is by convenience) MIDDLE EDGES POSITION 20 " " 24 TOTAL=232qtw 25. TOP,MIDDLE,BOTTOM CORNER ORIENTATION,BOTTOM EDGE ORIENTATION,BOTTOM CORNER POSITION,BOTTOM EDGE POSITION 27. TOP CORNERS,TOP EDGES,3 MIDDLE EDGES(POSITION,POSSIBLY ORIENTATION),POSITION OTHER CORNERS,ORIENT OTHER CORNERS,POSITION OTHER EDGES,ORIENT OTHER EDGES 28. BOTTOM LAYER,MIDDLE SIDES,ORIENT TOP SIDES,TOP CORNER POSITION, ..ORIENTATION,POSITION TOP SIDES 30.(8/17) SOLVE TWO LAYERS SIMULTANEOUSLY, THEN THIRD 30.(9/11) UPPER FACE EXCEPT ONE EDGE,LOWER CORNERS POSITIONED, ..ORIENTED,LOWER EDGES,LAST UPPER EDGE,MIDDLE POSITIONED, ..ORIENTED 31.(8/15) TOP CORNERS,TOP EDGES,BOTTOM CORNERS ORIENTED, ..POSITIONED,BOTTOM EDGES,MIDDLE POSITIONED, ..ORIENTED 31.(8/18) TOP CORNERS,TOP EDGES,(TURN CUBE OVER),TOP CORNERS POSITIONED, ..ORIENTED,REMAINING EDGES POSITION, ..ORIENTED 38. 2-METHODS 3 CONSECUTIVE LAYERS CORNERS FIRST (FOR SPEED) ONE OF THESE IS 120qtw 39. TIME CORNERS,BOTTOM,MIDDLE,TOP MOVE CORNERS,EDGES IN RANDOM ORDER AVERAGES 75qtw 41. 2 METHODS UPPER EDGES,UPPER CORNERS,(TURN OVER),MIDDLE, ORIENT EDGES,ORIENT CORNERS,PLACE CORNERS,PLACE EDGES UPPER EDGES,3 UPPER CORNERS,3 MIDDLE EDGES LAST UPPER CORNER,LAST MIDDLE EDGE,(TURN OVER) ORIENT EDGES,ORIENT CORNERS,PLACE CORNERS,PLACE EDGES As far as I can tell no two people who reported in detail solve the cube in quite the same way. INTRESTS IN THE CUBE: investigation of identities to calculate GOD's number application to the teaching of group theory training of mind and logic, something to do variations (tetrahedron, 10-sided, interesting faces) what different solving methods are analogy with quark confinement Combinatorial algorithims introducing others to the cube how people learn to solve cubes (AI) pretty patterns, cube graphics(for fun) determining GOD's number Showmanship, flaming about cubes in public cubing to attract or repel people analytic, as opposed to heuristic or exhaustive aids to solve the cube NON-CUBING INTRESTS OF PEOPLE: almost any types of puzzles, science fiction COMPUTERS!!!, ADVENTURE Subject: 4x4x4 TRANSFORMATION To: CUBE-LOVERS at MIT-MC On the normal cube the relative positions of the faces is immutable, however this is not so on the 4x4x4. I have found a trransformation which i beleive will facilitate the switching of any two of the cental blocks of 4 on a face. I am not sure as I did this totally in my head, without the benifit of a 4x4x4. This trandsformation would be extremelly nasty to pull on someone who has learned how to solve the standard cube from someone else, and as such does not know much about cubolgy.  Date: 25 Sep 1981 21:41:35 EDT (Friday) From: Roger Frye Subject: Cube in Political Cartoon To: Cube-Lovers at MIT-MC Cc: frye at BBNP The Boston Globe editorial page for 9/25/81 has a cartoon entitled "Stockman's CUBE". I can't make out the signature on the cartoon. It could be M. H. Beane. It shows much struggling to solve the cube followed by a realization. The character, presumably Stockman, Reagans's economist, then smashes off a corner or two of the cube with a hammer. In the last frame, he has his suit jacket back on (which he had lost while struggling). He holds up a solved cube and says, "Simple... I just applied my economic method... It's everywhere. Last week we argued about it in couples therapy; this week our therapist said she had an argument with her husband because he just got one and wasn't listening to her. I join Dame Ollerenshaw on the casualty list; a few nights ago I sprained my wrist twisting too long, too late, in too cramped a position, and sinning too fast against a jam. Happy Cubing, Roger Frye  Date: 25 Sep 1981 22:03:27 EDT (Friday) From: Roger Frye Subject: spinning not sinning To: Cube-Lovers at MIT-MC Cc: frye at BBNP Spelling corrections for my last message. Add a quote mark after the second ellipsis. Change "sinning" to "spinning". Happy sinning, Roger  Date: 26 September 1981 14:08-EDT From: David C. Plummer Subject: ZILCH's 4x4x4 center block exchange To: CUBE-LOVERS at MIT-MC Just a clarification (and a discussion promoter, perhaps). I can believe it is possible to put the center 2x2 blocks of a 4x4x4 cube into any relative position. But, there is still only one way to solve it. This is because the relative position in the solved state is determined by the corners. I guess this allows for a new class of pretty patterns, especially DOTS: DOTS as we know it in the 3x3x3 (two sets of three; one set clockwise, the other couterclockwise) DOTS (two sets of three; both (counter)clockwise) DOTS (two sets of three; similar to BASEBALL) DOTS (three sets of two; like Christman's Cross) DOTS (three sets of two; pairs opposite) (a local max) DOTS (three sets of two; all pairs adjacent) DOTS (one set of six; similar to the checkboard obtained by Plummer's Cross + dots + pons) etc. Anybody want to take a shot at trying to catalog all the local maxima?  Date: 28 Sep 1981 1722-PDT From: ISAACS at SRI-KL Subject: notation To: cube-lovers at MIT-MC Anybody have a good notation for 4x4x4? How about for the pyramid? Also, anybody have a good (less than 8 moves; QTW or HTW) to move a cubie from the bottom layer to a top corner, without disturbing top or middle, and making bottom facie into top facie (eg DLF --> UFL or suchlike). --- Stan -------  Date: 9 Oct 1981 10:55:04 EDT (Friday) From: Roger Frye Subject: Cube-a-thon To: Cube-Lovers at MIT-MC Cc: Frye at BBNP Does anyone have any more details on the following (as clipped from Boston Globe 10/8/81): JEFF VANASANO, 15, of the Bronx, beat more than 2000 competitors in a Rubik's cube-a-thon in Jackson Township, N.J., by solving a Rubik's Cube in 24.67 sec- onds. Vanasano will go on to the US finals in Los Angeles in November and there will be a world championship next spring. I wonder who sponsored the competition, what form the contest took, who Vanasano is, what techniques he uses. I found the analysis of Kimmo Eriksson's technique by Lars S. Hornfeldt (9 May 1981 08:47-EDT) to be most enlightening, and would love to see similar analyses. -Roger Frye  Date: 11 October 1981 13:26-EDT From: Andrew Tannenbaum Subject: New Jersey Cube-a-thon To: CUBE-LOVERS at MIT-MC The NJ Cube-a-thon was at (Six Flags Over) Great Adventure amusement park, sponsored, I believe, by Ideal. I saw the kid who won on television, zip, zip, zip, presto. He said he was solving 300 cubes a day as practice, and has been improving his speed five seconds a month for the past six months or so. I think there were about 1500 entrants. How about this for a bumper sticker? "Cubists do it in under a minute" Andy Tannenbaum Bell Labs Whippany, NJ  Date: 11 October 1981 17:52 edt From: Greenberg.Symbolics at MIT-Multics Subject: Under a minute To: CUBE-LOVERS at MIT-MC Several people have noted that there must be something sick about a bunch of people whose "do it" bumperstickerisms are all UNCOMPLIMENTARY, viz., "faster", "in under a minute"....  Date: 12 October 1981 0028-EDT (Monday) From: Guy.Steele at CMU-10A To: cube-lovers at MIT-MC Subject: Cubists do it... Message-Id: <12Oct81 002811 GS70@CMU-10A> Okay, then, how about: Cubists do it... ... with both hands. ... in groups. ... symmetrically (!). ... first on top, then on the bottom. ... by swapping. ... with their fingers. ... while twisting their faces. ... rectilinearly. ... by conjugation (tautologous?). ... despite odd orientations. --Quux  Date: 12 Oct 1981 0919-PDT From: HOROWITZ at USC-ISIF Subject: Latin square question To: cube-lovers at MIT-MC Herb Taylor, a research associate at USC in the EE dept. has proved that it is impossible to put a latin square on every face of the 2 x 2 x 2 cube. Can anyone answer the question on page 15 of his book UNSCRAMBLING THE CUBE? It asks whether or not it is possible to put a latin square on every face of the 3 x 3 x 3 cube. Ellis Horowitz p.s. Minh Thai of Los Angeles did ten scrambles with average time 40 seconds -------  Date: 14 Oct 1981 13:39:30 EDT (Wednesday) From: Roger Frye Subject: Latin Square Answer To: Cube-Lovers at MIT-MC Cc: Frye at BBN-UNIX Exhaustive search shows that there are several ways to fill all faces of Rubik's 3^3 with Latin squares, but none lie in the primary orbit. Here are two arrangements in the orbit where one corner is twisted 1/3 turn anticlockwise: UFB UBF BUF FUB FBU BFU LUD RLF RUD LRB LDU RLF RDU LRB DLU LFR DRU RBL ULD LFR URD RBL UDL FRL UDR BLR DUL FRL DUR BLR DBF DFB FDB BDF BFD FBD I did the search with pencil and scissors on quadrille lined paper. The following observations speed the search: 1) The 3*3 Latin square whether reduced or not must be some rotation or relabeling of the following pattern: ABC BCA CAB 2) The diagonal bars of the squares must be arranged as in the pretty pattern called "Laughter" because of the shape of the corner cubies. (See the bars in the patterns above.) 3) When you attempt to place one of the remaining four corner cubies, the corner color propagates to two edges which restricts the other color on those edge cubies to not be that color and also not that color's complement (e.g. U and D). This restriction then propagates to another corner. -Roger Frye  Date: 15 October 1981 10:05-EDT From: Andrew Tannenbaum Subject: maybe we are a little sick. To: CUBE-LOVERS at MIT-MC Date: 11 October 1981 23:08-EDT From: Alan Bawden Date: 11 October 1981 17:52 edt From: Greenberg.Symbolics at MIT-Multics Several people have noted that there must be something sick about a bunch of people whose "do it" bumperstickerisms are all UNCOMPLIMENTARY, viz., "faster", "in under a minute".... ----- It so happens that cubists like to do it faster instead of slower. Or by twisting faces. There are those who allege that anyone who would try and try to solve the dumb puzzle is sick, or that anyone who would solve and scramble a cube tens of thousands of times is sick. Lots of people can't imagine anyone finding joy in hacking. It can't give the satisfaction of the personal communication and conquest of marketing, for instance. This goes for many interests. My mother always told me that I shouldn't play with my cubies. Andy Tannenbaum Bell Labs Whippany, NJ  Date: 17 October 1981 22:54-EDT From: Mark K. Lottor To: CUBE-HACKERS at MIT-AI Where is old mail located?  Date: 19 October 1981 12:23-EDT From: Andrew Tannenbaum Subject: assembling your own To: CUBE-LOVERS at MIT-MC What ever happenned to Bela Szalai of LOGICAL GAMES (mentioned in mid 1980 in this list)? He re-mortgaged his house to produce white cubes, did he succeed? Does anyone make high quality cubes? Is it possible to buy unassembled cubes which you can tweak and lube yourself? I don't want to try to figure out how to get at the screws in my good cubes, I fear I'll destroy them. What is the current edition of Singmasters? Has anyone else written a worthwhile document (aside from the compilation of this digest)? Andy Tannenbaum Bell Labs Whippany, NJ  Date: 20 Oct 1981 10:26:57 EDT (Tuesday) From: Roger Frye Subject: Do It in the Dark To: Cube-Lovers at MIT-MC Cc: frye at BBNP I have constructed a tactile cube by epoxying buttons, snaps, paper clips and Q-tip stems to a store-bought cube. It takes me about 20 minutes to solve with my eyes closed as opposed to about 5 minutes with my eyes open. Maybe Conway cheats in his 5 look solution by feeling colors. I minimize search times by always working faces in a fixed order and by orienting colors (i.e. textures) before moving cubies on the last face. But the ability to see (i.e. feel) the back of the cube doesn't compensate for my as yet poorly developed tactile pattern recognizer. - Roger Frye  Date: 22 October 1981 14:02-EDT From: Alan Bawden Subject: fowarding To: CUBE-LOVERS at MIT-MC Date: 22 Oct 1981 0928-PDT From: ISAACS at SRI-KL Subject: Re: assembling your own To: ALAN at MIT-MC In-Reply-To: Your message of 19-Oct-81 1735-PDT There seem to be at least two high quality cubes, one called "Deluxe" and I think sold by Ideal for about $14.00 around here (S.F.), the other packaged differently (in a half-cardboard box) and sold for about $8.00. Last night I heard that there may be still another version (perhaps identical) for about $4.00. Anyway, they all have the colors on the facies put on with thin plastic squares instead of stickers. The one I have also turns very well. But the plastic seems to be what makes them "deluxe". There is also a "build it yourself" cube kit sold; I've seen it, but don't have one. It comes with the pieces and screws, and each facie has a small hole in the center. Plastic colored faces are snapped in. Maybe it is an early version of the "Deluxe". Documents on the cube: There seem to be more books published on the cube every day. Almost all of them are just solutions, with a little extra information (pretty patterns, anecdotes, other types of cubes, etc.). I know of not other book as interesting mathematically or theoretically as Singmasters. An article by Dame Kathleen Ollerenshaw, called "The Hungarian Magic Cube" in Bulletin, The Institute of Mathematics and its Applications, Vol. 16, aPril 1980, may have interesting information (I haven't seen it), and the book? article? by Conway and others, mentioned in Singmaster, ought to be interesting if/when it comes out. --Stan -------  Date: 29 Oct 1981 0932-PST From: ISAACS at SRI-KL Subject: Re: "Deluxe" cube To: Bob.Walker at CMU-10A, cube-lovers at MIT-MC In-Reply-To: Your message of 22-Oct-81 1953-PDT Last night my friend brought me the "deluxe" cubes. They came from: Ckoach House Cards and Gifts Sunnyvale Town Center 2502 Town Center Sunnyvale, Calif. (address from phone book - it may be inaccurate) (408)736-7244 They had the cubes there for $3.95 per cube. They were made in Korea, I forget the brand name. Coach House was also apperently selling the regular cheap-type cube (Wonderful Puzzler?) for $1.10 apiece. A good price. ---Stan -------  Date: 1 November 1981 19:12-EST From: Andrew Tannenbaum Subject: Trick or Treat To: CUBE-LOVERS at MIT-MC Last night I dispensed treats to two little walking cubes. Both solved. They made my evening, God bless 'em. Also one of those missing link frobs. Andy Tannenbaum Bell Labs Whippany, NJ  Date: 2 Nov 1981 1153-PST From: ISAACS at SRI-KL Subject: more trick or treat To: cube-lovers at MIT-MC In my neighborhood, too, there were at least two Rubik's Cubes at the door. Also solved - did anybody see any walking unsolved cubes? Next someone ought to develop a walking, solvable, cube costume. --- Stan -------  Date: 3 Nov 1981 0942-EST From: ELF at MIT-DMS (Eric L. Flanzbaum) To: Cube-Lovers at MIT-MC Subject: Trick or treat (more of) Message-id: <[MIT-DMS].214352> Of course I, too, saw some Rubik's cubes wandering around. And they were solved. Just yesterday I talked to someone who is going to a costume party next weekend (11/7) as Rubik's cube. Wow, everyones getting into the act. Next thing you know they'll have already made constumes sold in stores!  Date: 3 November 1981 10:41 est From: Greenberg.Symbolics at MIT-Multics Subject: Re: more trick or treat To: Cube-Lovers at MIT-MC In-Reply-To: Message of 2 November 1981 23:40 est from Alan Bawden A party I went to Sat Nite had a walking Rubik's cube, but alas, the wearer was cube-gnorant, and it had multiple center cubies of the same color and cubies with identical faces on two sides, etc.  Date: 3 Nov 1981 1452-EST From: ROBG at MIT-DMS (Rob F. Griffiths) To: cube-lovers at MIT-MC Subject: Walking Talking Cubes Message-id: <[MIT-DMS].214388> Down on Boulder's mall Halloween night, I saw about 4 cubes. All of these were solved, except one. I asked the man why he had designed his cube with only one face complete. His reply was that that was the closest to sloving his cube he had ever come, and he figured looking at a bigger model might help him! Speaking of larger cubes, does anyone know of any cubes that are larger than the standard Ideal Toy size cubes? And where could I obtain one? -Rob.  Date: 3 Nov 1981 1608-PST Sender: OLE at DARCOM-KA Subject: It's bigger, but its round! From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Message-ID: <[DARCOM-KA] 3-Nov-81 16:08:12.OLE> Following Rob's question about bigger cubes I would like to report on "Wonderful Puzzler"'s latest: The Rubik Sphere. Yes, it is here and is about 90mm in diameter with six circular faces. Solving is hence just as for the cube apart from some confusion caused by the fact that there are no edges to hold onto, the thing has no distinct orientation. This also makes twisting a bit tricky and I would think those 24 second champions would have trouble with this one. The ball or sphere has a smaller brother (keyring size) and this one has a different coloring scheme, "stripes" instead of "faces" and is in this respect very similar to the 10 sided drum discussed here earlier. The mechanics of the sphere is very much as you would expect, just an extension of the original cube mechanism,- or if you like: imagine heating your cube up to near melting point and squeezing it into a ball (not recomended!) Happy.. eh, well Sphering! OLE  Date: 4 Nov 1981 1133-PST Sender: OLE at DARCOM-KA Subject: Global Cube From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Message-ID: <[DARCOM-KA] 4-Nov-81 11:33:04.OLE> Having seen the cover of next weeks 'Newsweek' (November 9) I got the idea of continuously coloring the Sphere (see my previous message) with a map of the world, i.e make it into a globe. This would also incorporate the Supergroup and per- haps help people learn geography (if they already know how to solve the cube/sphere...) What will they think of next? OLE  Date: 5 Nov 1981 0954-PST From: ISAACS at SRI-KL Subject: Rubic's Sphere To: CUBE-LOVERS at MIT-MC One of my favorite pretty patterns on the Sphere is the "universe", from "Unscrambling the cubE": every edge and corner flipped/twisted in place; the each adjacent pair of corners twisted towards each other. Forms a nice, symmetric pattern of each face "exploded" outward from its center, which is more visible on the sphere than the cube. ---Stan -------  Date: 8 Nov 1981 1151-EST From: ROBG at MIT-DMS (Rob F. Griffiths) To: Cube-Lovers at MIT-MC Message-id: <[MIT-DMS].214863> I don't know how many of your newspapers carry this strip, but in today's 'Goosemeyer', there is a set of pictures of a general going through many secturity checks and M.P.'s. He then walks into a 'Top Secret' room and says 'Sorry Im late...Whats the situation, men?' One of the others replies 'It looks hopeless sir...Everything we have tried has failed!' The general coutners 'Thers precious time left to come up with the right solution...' The last frame shows them standing around a table looking at a scrambled cube, and the general saying 'Ive got to get the back to my son by three o'clock'!! -Rob.  Date: 3 December 1981 15:19-EST From: Richard Pavelle Subject: MASQUERADE To: CUBE-LOVERS at MIT-MC As you may have noticed, cubism seems to be stagnating. Therefore I suggest that we start discussions on a different sort of puzzle called MASQUERADE! MASQUERADE is a book of about 15 pages beautifully illustrated by the author and artist Kit Williams. The story and pictures contain clues which supposedly will lead the solver to a "treasure" described by Williams as "... a golden hare adorned with precious stones and faience". It is hidden somewhere in Britain. The book is published by Shocken Books in Manhattan and sells for about $10. There is also an ad on Page 35 of Scientific American for December (North American Edition). If any of you have interest in this idea send mail to me, RP@MC, and I will keep accumulate comments and names.  Date: 4 Dec 1981 1459-PST Sender: OLE at DARCOM-KA Subject: Diversion From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Message-ID: <[DARCOM-KA] 4-Dec-81 14:59:07.OLE> Folks, Masquerade might be a good continuation for cube-lovers, I have not seen it so I can't really say. However, I am having sleepless nights about another of our puzzle friends the Ten Billion Barrell described herein by David Plummer a few months back. No-one ever seemed to come up with a good notation nor any hints on how to solve this toy, an as far as I can tell no good books are out which offer any help either. So, could we perhaps have a discussion on the barrell or would the masters of this time stealer come forward and advice me. I am much to young to be getting grey hairs, but found a few this morning! OLE  Date: 4 Dec 1981 1802-PST From: ISAACS at SRI-KL Subject: Globe Cube To: cube-lovers at MIT-MC A friend of mine brought me a Spherical "Rubiks cube" with a world map on it from Hawaii. Gotten in a clothing store (Casablanca?) on Maui, not a puzzle place. She also got me some Hawaiian cubes - each face with one of the islands. There's a great proliferation of Cube (and related items) books. "Not Another Cube Book", a 'humorous' book (not very good). A new book from Nourse which has solutions to the barrel, the tetrahedron, the Missing Link, and the Snake. A cube smasher. And many more. I've heard there is or will be a book on what to do with a smashed cube. I've been thinking of trying to cash in on this craziness and write my own. Q: What's Red and Orange and Green and Blue and White and Yellow and lives at the bottom of the sea? Answer: Moby Cube! Oh well. Maybe I'll stick to computers and puzzles. --- Stan -------  EH@MIT-AI 12/05/81 19:20:20 Re: Tiny cubes To: cube-lovers at MIT-MC Does Ideal Toy Corp. make those tiny cubes? (about a inch square) Beware of stuff made in Tiawan as they break QUICKLY..as what happened to me with a 3/4 inch cube..and I heard from my friend that his tiawan-made cube (standard size 2 1/2 inch) broke in a few days.... The only RUbic cube is the Ideal Toy corp cube and it's great! Edward  Date: 5 Dec 1981 1654-PST Sender: OLE at DARCOM-KA Subject: Rubikmania From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Message-ID: <[DARCOM-KA] 5-Dec-81 16:54:23.OLE> Check out TIME magazine, December 7th page 35, for an update on the cube and its many spin-offs and solution guides. The article mentions the Cube Smasher, "...guaranteed to pound the puzzle to bits", as well as the "Not Another.." book that Stan was talking about. Most interesting is perhaps the figure 10 MILLION (!) which is how many cubes have been sold Worldwide so far. This is presumably the official Ideal figure, so one wonders how many have been sold all together, Wonderful Puzzler seem to have done very well, at least here in the UK. If you are interested in computers, and presumably you are since you're on this list,- then check out page 33 of the same magazine for a very interesting article on Texas Instruments. OLE  Date: 6 December 1981 03:14-EST From: Eric L. Flanzbaum Subject: Sticky cubes ... To: CUBE-LOVERS at MIT-MC cc: ELF at MIT-MC Today, I just purchased a Rubik's cube (The real thing, not a fake!). I would like to know if there is any way to make the cube move more freely and loosely w/out having to wait for it to loosen up itself by using it. Any suggestions? --- Eric ---  EH@MIT-AI 12/06/81 16:34:03 Re: Sticky Cubes... To: Cube-Lovers at MIT-MC CC: ELF at MIT-AI From: "You can do THE CUBE" by Patrick Bossert / published by Puffin. To loosen up the cube if it's sticky to turn, turn the top slice 45 degs and insert a screwdriver under one of the edge pieces (the one in the middle) .... (1) [] [] [] <- turn this slice 45 deg [] [] [] [] [] [] (2) [] [] [] ^ insert a screwdriver blade under that part and lift up until it pops out. (3) Then you can remove the two corner pices. Looking inside, you can see the + shape of the cube mechanism. Put a BLOB of vaseline inside. (4) put the corner pieces back and the middle piece too. MAKE SURE you get them in exactly as they were before you opened them up. Otherwise,you WILL NOT be able to solve the cube!! Hope this helps, Edward ps: PLEASE feel free to write to me if you're confused as this is the best I can do without graphics.... (use graphics to show but you can't here)  Date: 7 December 1981 1911-EST (Monday) From: Dan Hoey at CMU-10A To: Cube-Lovers at mit-mc Subject: Brute Force Report: The fourteen-qtw identities Message-Id: <07Dec81 191129 DH51@CMU-10A> Several messages in August of this year [mail to Hoey@CMU-10A for copies] concerned short identities of the cube, i. e. processes which return the cube to solved. Later in that month I assisted David Plummer in a brute force attack on the problem. We had plans to investigate all the positions up to eight qtw, but unfortunately became busy on other projects. I have finally come up with enough time to analyze and report the data from the seven-qtw search. There are 8,221,632 cube positions at a distance of seven quarter-twists from solved, and 9,205,558 positions at seven or fewer qtw. By recording cases of different seven-qtw processes yielding the same position, a complete list of fourteen-qtw identities is obtained. The task is then to reduce the list to exclude multiple instances of equivalent identities. We call two identities equivalent when one can be obtained from the other by some combination of the following operations: - uniformly relabeling the twists according to a rotation or reflection of the cube, - cyclically permuting the twists, - reversing the order of the twists and inverting each one, and - substituting a sequence x for a sequence y, where xy' is a shorter identity. The first three criteria are easily implemented on a computer. The fourth can be performed for the shortest identities, those equivalent to F^4 and FBF'B', but I know of no algorithm to detect all cases of equivalence due to substitution of the longer identities. My strategy was to reduce the (several thousand) identities by computer for the simple kinds of equivalence, and then to look by hand for substitution equivalence between the fourteen identities then remaining. I found three equivalences, listed at the end of this note, but the possibility remains that some of the following identities are equivalent. The list is, however, complete (modulo bugs and cosmic rays). Identities equivalent to the first six on this list were independently discovered by Chris C. Worrell; I follow his numbering for them. Identities I14-5 through I14-7 do not hold in the Supergroup, because they twist face centers as noted in the brackets. I14-1 BF' UB'U'F UL' BU'B'U LU' I14-2 B UBL' B'D'BD LB'U' L'B'L I14-3 BB U BB UD' RR U' RR U'D I14-4 BUB'U' L'FL UBU'B' L'F'L I14-5 (BB UD B U'D')^2 [Supergroup BB] I14-6 BF' U B'F LR' UD' F' U'D L'R [Supergroup UF'] I14-7 BF U B'F' LR' UD F' U'D' L'R [Supergroup UF'] I14-8 BF' UFRU'R'B'U'B'RBUR' I14-9 BF' UFRU'B' UD' F'U'R'FD I14-10 (BUBU'L'B') R (BLUB'U'B') R' I14-11 (BUBU'L'B') D'R'B' DLD'RD The twelve-qtw identity I12-2 = (BUBU'L'B') (B'D'B'DLB) can be substituted into identities I14-10 and I14-11 to yield: I14-10a (B'L'D'BDB) R (BLUB'U'B') R' I14-10b (BUBU'L'B') R (B'D'B'DLB) R' I14-10c (B'L'D'BDB) R (B'D'B'DLB) R' I14-11a (B'L'D'BDB) D'R'B' DLD'RD Identity I14-10c can be obtained from I14-10 [by shifting seven places and reflecting the cube through the UD plane] but I14-10, I14-10a, and I14-10b are mutually inequivalent when twelve-qtw identities are ignored. The same holds for I14-11 and I14-11a. Strangely enough, I14-11a can also be transformed to I14-11 by substituting with the identity (BDBD'R'B') (B'U'B'URB), which is equivalent to I12-2.  Date: 8 Dec 1981 12:45 PST From: Hoffman at PARC-MAXC Subject: Re: Diversion In-reply-to: Ole's message of 4 December 1981 To: Cube-Lovers at MIT-MC About the "Wonderful Barrell". I, too, had been searching for a good notation and some ideas about how to solve it. I finally found it! James Nourse, author of 'The Simple Solution to Rubik's Cube' has a new book out entitled 'The Simple Solution to Cubic Puzzles' (or close to that title). It goes for just $2, and I picked mine up at a Waldenbooks. It includes a complete solution for the barrell, as well as the pyramid, the picture cubes, the Missing Link, and some items about the Rubik Snake. A good deal, but BEWARE of some most aggravating typographical errors. (I know of two in the barrell solution.) -- Rodney Hoffman  Date: 8 Dec 1981 14:30 PST From: Lynn.ES at PARC-MAXC Subject: Re: Diversion In-reply-to: Ole's message of 4 Dec 1981 1459-PST To: Cube-Lovers at MIT-MC cc: Lynn.es I just got my first look at a borrowed copy of Nourse's new "solve everything" book, specifically to look at the barrel puzzle solution he gives. His notation is: The two rotating pieces are "rings". They rotate through 5 "notches". "Layers" (horizontal slices containing 5 balls each) are numbered 1 thru 4 from bottom to top, plus T for the 3 balls immediately under "plungers" (when they are up). No notation is used for positions within the puzzle, except this numbering of the layers. Positions are given by word descriptions or drawings. Moves: Up-arrow move the plungers up Down-arrow move the plungers down Two boxes, one above the other, which represent movement of the two rings. A "T" above, or inverted "T" below the boxes shows the current plunger position. Each box may contain: blank don't move this layer at all Left-arrow rotate the ring left (clockwise, as seen from the top) one notch Left-arrow 2 rotate the ring left two notches Right-arrow rotate the ring right one notch 2 Right-arrow rotate the ring right two notches All moves assume the lonesome plunger is toward the puzzle solver. All move sequences begin assuming the plungers are in the up position. There is a great deal of room for improving this notation. Particularly, the boxes should be dispensed with in favor of some single-printed-line way of differentiating between top and bottom rings. The current plunger position is not needed. Because many terminals or typewriters do not have four directional arrows in their character sets, these should be replaced. I like thinking in terms of left and right rotations rather than clockwise, since this puzzle is usually solved without turning the puzzle as a whole. Position notations are needed. For these reasons I propose the following: Move notations: U move the plungers Up D move the plungers Down T rotate the Top ring right (counter-clockwise from the top) one notch B rotate Bottom ring right one notch -T rotate Top ring left one notch -B rotate Bottom ring left one notch T2, B2, -T2, -B2 two notch versions of the above four respectively Positional notation: Same naming for "rings", "layers", "notches", "plungers", except call the top layer "5". The ball positions in layer one are, clockwise, as seen from the top, beginning with the one under the lonesome plunger: 1F (front), 1L (left), 1LL, 1RR, 1R. Correspondlingly for the other layers. There is no 5L or 5R, because of the orientaion of the puzzle. During a move, at times when the plungers are down, there will temporarily exist positions 0F, 0RR, 0LL. Any suggestions or arguments about the notation are welcome. Nourse's solution has two typos in it. Page 33: upper rightmost move on page should be in the bottom ring, not top. Page 36: second move on the page should have a 2, not single notch. If people are interested, I can summarize Nourse's approach to solving. /Don Lynn  Date: 8 Dec 1981 1952-EST From: ELF at MIT-DMS (Eric L. Flanzbaum) Reply-to: ELF at MIT-DMS To: Cube-Lovers at MIT-MC Subject: That's incredible! Message-id: <[MIT-DMS].217487> Last night on the show "That's Incredible!" they held the finals of the Rubik's Cube competition. The contest was for the U.S.A. and the winner wa then to go onto world competition. If I remember correctly, the fastest time (by the winner, of course) was ~26 seconds (it was actually between 27 and 26, but I can't remember that part). The show also mentioned that there were 43 quintillion combinations possible (is this true?). The second place time was between 28 and 29. -- Eric *******  Date: 9 December 1981 19:23-EST From: Alan Bawden Subject: Luks To: CUBE-LOVERS at MIT-MC I don't know any more about this, but it sounds like it might be interesting to some of us cube group hackers: MSG: SEMINA 3 DISTRIB: *DM, *MC, *ML, *AI EXPIRES: 12/17/81 16:21:19 BJ@MIT-ML 12/09/81 16:21:19 Re: Luks Gene Luks from the Mathematics Department at Bucknell University will give a talk on Tuesday, December 15 at 4:00 (refreshments at 3:45) in room NE43-512A. The talk is "ALGORITHMS FOR PERMUTATION GROUPS."  Date: 10 Dec 1981 1903-PST From: ISAACS at SRI-KL Subject: Ten Billion Puzzle (the Barrel) To: CUBE-LOVERS at MIT-MC It just happens that the day before I got the Nourse solution to the Ten Billion (Barrel, Magic Barrel, etc), I got another solution by Naoaki Takashima, that he presented to the ARMJ (??) meeting in Japan. His notation: Columns and stages are defined 5- x x x as shown on the right. s 4- x x x x x t 3- x x x x x a 2- x x x x x g 1- x x x x x e s | | | | | 1 2 3 4 5 columns He moves the drums, rather than the plunger. U--move drums up. D--move drums down. Ru (R sub u) Rotate upper drum right one column; superscripts for number Rl (R sub l) " lower " " " " of columns R for upper and lower together to the right L for left similarly. At any rate, a cumbersome notation for computer writing. What's more, unit move people (who don't like slice moves or half turns on the cube) won't like combining the two drums (rings). BUT since I hold my barrel by the bottom plunger, I do find it easier to think of moving the rings up and down, rather than pushing the plunger. I like rings better than drums, but columns better than notches. And I would rather take standard numbering to start from what is above defined as column 2, ie, the one under the neighborless plunger. So, all in all, I would like to propose accepting Don Lynns' notation with the following changes: U and D would refer to the rings moving up and down, instead of the plunger. "Notches" would become "columns". The picture above, moved to reflect the new columns, could be used when it is needed for clarity. * * * * Small spoiler * * The Takashima solution is really based on one move, which cycles 3 balls more-or-less vertically: let X = U T D -T / U B D -B / (TB) U -(TB) D then X2 moves 3F TO 4F TO 2L TO 3F; ie, the 3-cycle (3F,4F,2L). * * * * End of small spoiler * * * I already propose apostrophe instead of -, so the above becomes: UTDT'-UBDB'-(TB)U(TB)'D which I think is a little clearer. Notice that the above can be simplified to: UTDT'UBDTUB'DT'UBDTUTBD (I think. I find it hard to think about in such small pieces; I like the less efficient chunks at first.) * * * * * * * * SPOILER WARNING * * * * * * * * * * *SPOILER WARNING * * * * Anyway, with this move alone (plus some playing at the beginning), I can solve the puzzle as follows: First get the three black balls at the top, under the plungers, any old way. Then put 5 different colors in the first layer, moving only the bottom ring (to leave the black balls alone). Then use the move above to move in the matching second layer from the top (layer 4). When necessary, use the same move to get from 3 to 4 first, or even from 2 to 4. Using the inverse move can speed things up, but just doing it twice is easier on the memory. Move the space in layer 2 you want to fill, to column 5 (new numbering; col. 1 in above diagram, or "2L" in notation), by moving the lower ring; move the column containing the ball you want in that space to col. 1 (F), apply the move once or twice, and move the rings back. Rest position has the barrel down (and the plunger up). Once the second layer is complete (or, for efficiency, complete except for 1), do similar things for layers 3 and 4, using the second layer for temporary storage. * * * * * * * END OF SPOILER * * * * * * * * * * * * * * * * * * * * * * * * * Don Lynn: I would be very interested if you would summarize Nourse's approach, and either send it to me or to cube-lovers. I have not had the time to break it down into chunks to see what basic moves he really uses. I find that to be the main trouble with most of the solution books to the cube as well as this - that they offer long sequences to memorize, with little thought to learning what is going on, and what the logic of the moves are. In Nourse's solution, what are the basic pieces of his moves. Can they be expressed (as above) as a "simple to comprehend" set as well as a "shortest number of unit moves" way. By use of parenthesis (or maybe slashes), U and D could probably be eliminated; at each slash, switch from up to down or vice versa. Would this be useful notationally. What are some basic algorithms to exchange 2 balls on a single layer or in a single column? To cycle a layer or column? Enough questions for tonight. When's the first Barrel contest? --- Stan -------  Date: 11 Dec 1981 1643-EST From: ROBG at MIT-DMS (Rob F. Griffiths) To: cube-lovers at MIT-MC Subject: Re: Ultimate Cube Message-id: <[MIT-DMS].217732> I have seen the 'Ultimate Cube', except here in Boulder, it is done a little differently. They are all Black cubes, and come in the standard (Ideal standard) container. And (no offense from me, this is what they are called) they are labeled 'Polish Cubes' with 'No wrong move'. Also, we have a Pyraminx puzzle (which is rather trivial compared to the cube) and it has a problem: if you twist the top cube on a certainm side too many times, (like 3 twists) it comes off and the whole mechanism falls apart. Has anyone else had this problem? How can I correct it? -Rob.  Date: 11 December 1981 18:20-EST From: Alan Bawden Subject: fowarded To: CUBE-LOVERS at MIT-MC Date: 11 Dec 1981 at 1630-CST From: korner at UTEXAS-11 Subject: Pyramid To: ALAN at MIT-MC Someone mentioned the pyramid recently. Are solutions really worth discussing? I had the solution within 15 min. and the group structure seemed trivial. Is this as simple as it seems or have I missed something? -Kim Korner -------  Date: 12 December 1981 02:12-EST From: Alan Bawden Subject: Puzzle-Lovers? To: CUBE-LOVERS at MIT-MC Richard Pavelle has suggested that since cubism seems to be stagnating, we might consider expanding the scope of this discussion to include other sorts of puzzles. He wondered (in a letter to this list a few days ago) if anyone was interested in the puzzle/book called "MASQUERADE". The response he got was enthusiastic. The discussion on this list has never been exclusively about cubes, but it has never really strayed very far either. Also, recently the discussion has slowed somewhat. So we were wondering how people would feel about changing the purpose of this list from cube hacking to more general puzzle hacking. This would still be the place to talk about cubing, but everyone would have to at least tolerate other puzzles as well. Note that there is another option for general puzzle hackers which is the formation of a separate Puzzle-Lovers (or something) mailing list. This second option would be the thing to do if there is enough interest, but also some objections from people currently on Cube-Lovers. (If a new list is formed, they will have to find someone else to be responsible for it. I won't deal with two!) To decide this tricky question I would like to hear from anyone who objects to the expansion idea. With enough objections, we simply won't do it. Don't bother to send me a note if you are simply in favor of, or don't mind the idea. I really only need to hear from the NO votes. I'd be interested to hear any other thoughts you have on the subject as well. Mail to Alan@MIT-MC.  Date: 13 December 1981 00:27 est From: Schauble.Multics at MIT-Multics Subject: Russian views To: cube-lovers at MIT-AI "Some assert that the Rubik Cube reflects the philosophy of the Reagan Administration -- to build and destroy aimlessly in a futile search for a solution to the world situation. Others see Americans' efforts to solve the puzzle as the desire to escape from a disordered life." Izvestia reporter Melor Sturua, reporting the views of unidentified Soviet sociologists that the Rubik's cube fad in the U.S. is a "new psychosis" July 12, 1981  Date: 13 December 1981 0150-EST (Sunday) From: Paul.Haley at CMU-10A To: cube-lovers at mit-mc Subject: Leaving the mailing list CC: Paul.Haley at CMU-10A Message-Id: <13Dec81 015008 PH71@CMU-10A> Please remove me from the cube-lovers mailing list. Paul  Date: 16 Dec 1981 1055-PST From: ISAACS at SRI-KL Subject: general puzzles To: cube-lovers at MIT-MC In order to perhaps start some discussion of puzzles in general, I will include some first steps in attempting to come up with some kind of classification scheme for puzzles. I have included both my own, and Jerry Slocums', though I haven't put in the details of either at this time (I'm lazy - if there is enough interest, I will fill out the outlines a little more.) Anyway, any suggestions for diferent ways, or changes, will be appreciated. Can you put all YOUR favorite types of puzzles somewhere in the two lists? -------  Date: 16 Dec 1981 1057-PST From: ISAACS at SRI-KL Subject: inclusion for previous message To: cube-lovers at MIT-MC PUZZLES I. LIFE A. Technical (science, social science, etc) B. Psychological (ie, people problems) II. CREATED A. Mental 1. Language (word) puzzles Crosswords, Acrostics, word squares, etc. 2. Logical puzzles Bridge crossing, truth tellers, etc,etc -- all those pencil and paper puzzles that appear in most non-word puzzle books. Mostly mathematical. B. Physical (Also sometimes called "Mechanical Puzzles") (THESE ARE THE KINDS I AM MOST INTERESTED IN, AND PROPOSE GOING INTO MORE DETAIL ABOUT.) 1. Geometrical (including 2 and 3 dimensions; jig-saws; things like that.) 2. Topological (rope and string puzzles, that involve deformation. Possibly including wire puzzles.) 3. Combinatorial (box filling puzzles; Rubics Cube) 4. Manipulation (dexterity?) (mazes, rolling balls, ballance puzzles) 5. Physical? Miscellaneous? (Other types, like puzzle jugs; optical puzzles, centrifical force.) I'll try to expand on this classification at a later time - I just want to get something started now. You can see that there are many problems in this - a lot of puzzles don't fit easily in any of these, and others fit clearly into more than one. The trouble is that you can look at puzzles by their form, or by how they are constucted, or how they are solved. And no one way seems satisfactory for all. Anyway, following is another classification, just of Mechanical (Physical) puzzles, by Jerry Slocum, who has one of the biggest puzzle collections in the world. 1. PUT-TOGETHER PUZZLES - Putting the object together is the puzzle Tangrams, jigsaw, soma, instant insanity, magic squares, puzzle rings 2. TAKE-APART PUZZLES - Taking the object apart or open is the puzzle Trick or secret opening 3. INTERLOCKING SOLID PUZZLES - Disassembly and Assembly is the puzzle Burrs, 3-D jigsaws, keychain, geometric object and figures (the Japenese wooden puzzles) 4. DISENTANGLEMENT PUZZLES - Disentanglement and entanglement is the puzzle Wire puzzles, nails, string and rope puzzles 5. SEQUENTIAL MOVEMENT PUZZLES - Moving Parts of the Object to a Goal While Following Rules is the Puzzle Solitaire, sliding block puzzles, rotating cube (Rubik) puzzles, maze and route. 6. DEXTERITY PUZZLES Rolling Ball, cup and ball, etc 7. MISCELLANEOUS Puzzle jugs, folding, paper and card, matchstick, trick, vanish puzzles, optical puzzles, find-the-object, rebus, reversibles, etc. -------  Date: 16 December 1981 22:47-EST From: Alan Bawden Subject: puzzle-lovers To: CUBE-LOVERS at MIT-MC Please note that despite the previous message we are NOT all agreed that cube-lovers should devolve (?) into a discussion of puzzling in general. Those of you who object to this idea are still welcome and encouraged to mail me an objection. Currently I have recieved only a couple of replies, so if you don't say something now, this is likely to become the Puzzle-Lovers mailing list in a week or so!  Date: 16 December 1981 22:58-EST From: "Martin B. Gentry, III" To: CUBE-LOVERS at MIT-AI Considering the recent suggestion that this list be used for discussing puzzles of all types, but not trying to force it in that direction, the following is submitted for your interest: In the Games section of the January OMNI there is an article about some currently unsolved puzzles such as \Masquerade/, \The Will: A Modern Day Treasure Hunt/ and the Beale ciphers. Included is a listing of Beale Cipher #1.  Date: 16 December 1981 23:32 cst From: VaughanW at HI-Multics (Bill Vaughan) Subject: puzzles of a different kind Sender: VaughanW.REFLECS at HI-Multics To: cube-lovers at MIT-MC the puzzle classification is interesting - but i propose another main heading: game puzzles (chess problems, bridge puzzles etc.) These probably fit in somehow but I don't know exactly where. (my 2 cents' worth: vivat puzzle-lovers, & keep up the digest format.)  Date: 18 December 1981 00:37-EST From: Alan Bawden Subject: Merry Christmas To: CUBE-LOVERS at MIT-MC While I am visiting my relatives in Philadelphia and while Dave Plummer is doing something-or-other similar in honor of the Christmas break, we have put the Cube-Lovers list back on direct distribution. -Alan  Date: 18 Dec 1981 1550-PST From: ISAACS at SRI-KL Subject: Authors' signing To: cube-lovers at MIT-MC Don Frederick, author of "Solve That Crazy Mixed-Up Cube", will be at the OLD MILL Shopping Center in Palo Alto/Mountain View Friday evening from 5 pm to 9 pm, and Saturday, Dec. 19 from 11 am till about 5 pm. His is one of the few books that at least tries to help you remember a solution to the cube. He'll be selling and signing his book. He claims plain language, cartoons, solving other shapes, and a diploma in the back of the book. -- Stan -------  Date: 21 Dec 1981 1522-PST From: ISAACS at SRI-KL Subject: new cube game To: cube-lovers at MIT-MC I just got a new cube puzzle, called "Color Cube Mental Game". Not a Rubik's cube, but still combinatorial, and a lot easier. It consists of 8 cubes in a 3x3 box, arranged around the perimeter (so there is a cubical hole in the center). The cubes are colored on all 6 sides (identically in mine). They roll by tilting them over an edge to an adjacent vacant space, and they change colors in doing so. The puzzle, of course, is to roll them and randomize the pattern, then restore the original color (or a new color). What patterns can be made? What is the minimum number of rolls to solve from some pattern to another? This type of puzzle was talked about in Journal of Recreational Mathematics, and similar types in Mathematical Games by Martin Gardner (I can give more accurate references if anyone wants). Here's the last paragraph on the "instruction" sheet: "It should be finished. please don't dishearten and try your best till you can complete the all test. Good Luck!!!" --- Stan ------- Happy Holidays to all -------  Date: 29 December 1981 23:48-EST From: Alan Bawden Subject: Happy New Year! To: CUBE-LOVERS at MIT-MC With this message we become a digest again.  Date: 30 Dec 1981 2017-EST From: ELF at MIT-DMS (Eric L. Flanzbaum) To: Cube-Lovers at MIT-MC Subject: Query Message-id: <[MIT-DMS].218800> Cube Fans; Why on the package of the original Rubik's cube, does it say "Over 3 billion combonations ... just one solution" when in reality there are over 43 quintillion? Is it because the manufacturers of it thought nobody would believe them? Happy Cubing, the ELF  Date: 31 December 1981 16:14-EST From: David C. Plummer Subject: Query To: ELF at MIT-DMS cc: CUBE-LOVERS at MIT-MC Why on the package of the original Rubik's cube, does it say "Over 3... Something like that. I believe there is a small discussion about this somewhere in the archives.  Date: 3 January 1982 02:11-EST From: Alan Bawden Subject: nuts To: CUBE-LOVERS at MIT-MC There is a puzzle that has been around for several years called something like "Nuts To You" or "Drives You Nuts" or some variation thereof. It consists of 7 hexagonal pieces (each resembling a nut, hence the name) each with the numbers 1 through 6 inscribed around its perimeter in some order. The idea is to play a kind of hexagonal dominoes with these pieces; a solution is found when all 7 pieces are arranged in a hexagon (the puzzle comes complete with a hexagonal frame for arranging them, it also resembles a nut) such that every pair of adjacent pieces are labeled the same along their common edge. For example one piece might look like this: _____ / 6 \ /3 1\ / \ \ / \2 5/ \__4__/ 6 which I shall abbreviate by: 3 1 2 5 4 which might participate in the following solution: 5 6 4 6 1 2 3 3 1 3 2 4 2 5 3 1 6 4 5 1 5 4 2 6 5 5 2 4 6 1 3 1 4 3 2 6 1 3 4 5 2 6 Each piece can be rotated (of course), but it cannot be flipped over. (This restriction is enforced by simply not printing the numbers on the other side!) The seven pieces I have represented in my sample solution are definitely NOT the ones that make up the commercially sold puzzle. I don't have that anymore, I just made these up. It did have the property that no two pieces were alike, just like my hypothetical set. The commercial set of pieces seemed to have the property that there was JUST ONE possible solution, although I cannot be 100% certain that this was the case. Now the puzzle itself is a bit dry. But what I wonder about is the following meta-problem: Given that there are many (how many?) different sets of 7 pieces that can be chosen, how "interesting" a property is it for there to be only ONE solution? Do you have to try hard to achieve that property? Or do most of them have it? Indeed, do ANY of the have it (remember I only SUSPECT that the commercial set has it!). How about if we relax the restriction about duplicate pieces? -Alan  Date: 3 Jan 1982 0232-EST From: ELF at MIT-DMS (Eric L. Flanzbaum) To: Cube-Lovers at MIT-MC Subject: The Pyramid Message-id: <[MIT-DMS].219009> Is there any written (i.e., a book like "The Simple solutio....") solution for the pyramid? Do any of you people know a solution? People tell me the pyramid is easier than the cube (that is of course if you know how to do both). Is this true? Happy Cubing, the ELF  Date: 3 January 1982 17:00-EST From: Alan Bawden Subject: The Pyramid To: ELF at MIT-DMS cc: CUBE-LOVERS at MIT-MC I know of no books about the pyramid, but I can vouch for the fact that the pyramid is much easier than the cube. I have essentially ONE tool that is sufficient for all manipulations. It isn't even a very long one. It is in fact such an obvious tool to try that many people have discovered it independently. I'll tell you what it is after I insert the following spoiler warning: *** SPOILER WARNING !!! *** If you wish to solve the pyramid yourself, stop reading now! First off, notice that the points of the pyramid can be rotated independently of all the other pieces. Thus we can safely ignore those pieces and simply twist them into position as the last step. So from here on in, when I say to twist about a particular point, I mean to grasp the larger sub-pyramid that shares that point with the entire pyramid and rotate it. This motion always leaves the face opposite the designated point fixed. This is not analogous to the way we think about the cube. When dealing with the cube we rotate a face, leaving the rest of the cube fixed. When dealing with the pyramid I am proposing to always rotate everything BUT a face, a move that arranges to leave the four points in the same position. Having how described what I mean by a move, the actual tool is easy. It's only four twists long. Chose any two points, call them A and B. The tool is simply AB'A'B. It permutes three edge pieces that share a common face, it flips two of those pieces over, and everything else is untouched. Since everything except the edge pieces can be placed in position trivially (both the point pieces and the third kind of piece I haven't mentioned yet), this tool might be sufficient (after arranging everything else). I don't know if it is in fact sufficient since I also employ some conjugates (like CAB'A'BC') in my solution. I leave the details of actually applying this tool for you to discover by yourself. -Alan  From: Woods.pa @ PARC-MAXC Date: 3-Jan-82 18:34:41 PST Subject: Re: nuts In-reply-to: ALAN's message of 3 January 1982 02:15-EST To: Alan Bawden cc: Cube-Lovers at MIT-MC I've seen the "Nuts" puzzle; I think it is indeed "Drive you nuts", but I'm not sure. I seem to recall that not every (any?) piece had all six numbers on it; that is, I think one or more numbers were duplicated on one or more pieces. This of course opens up the search space even more in trying to find a set of pieces with exactly one solution. Also, rather than consider which sets of pieces (or what fraction of possible sets of pieces) have unique solutions, I think it makes more sense to choose a solution and see how likely it is that the pieces forming that solution have another solved orientation (other than trivial rotations of the whole puzzle). That way you at least eliminate all the sets that have NO solution. -- Don.  Date: 3 January 1982 23:05-EST From: Alan Bawden To: CUBE-LOVERS at MIT-MC, Woods.pa at PARC-MAXC Well, my memory is very clear on the point that each nut had all 6 numbers since my solution depended on that fact. But sure, if we can answer your extended problem all the better. You mention rotations as a symmetry of a solution, which reminds me of another kind of symmetry that is relevant to exactly what is meant by "a set of nuts". Clearly the values of the integers inscribed on the pieces have no effect of the character of the solution. If someone broke into your house one night an erased all the 1's and replaced them with 2's and replaced all the 2's with 1's, they wouldn't have damadged your puzzle in any way (ignoring that you may have memorized the solution by number, but that's your fault for chosing a bad way to remember the solution). So applying any of the 6! permutations of 6 things to the numbers leaves your "set of nuts" fixed (relative to eachother). The manufacturers's choice of numbering is thus at least partially arbitrary.  Date: 5 Jan 1982 09:38 PST From: Lynn.ES at PARC-MAXC Subject: Re: The Pyramid In-reply-to: ELF's message of 3 Jan 1982 0232-EST To: Cube-Lovers at MIT-MC Nourse's new book "The Simple Solutions to Cubic Puzzles" has a pyramid solution. Aside from the (almost) obvious stuff about the little and big portions of each corner, it has specific macros for moving edge pieces from any edge position to the top-front (he solves it with a flat surface up), and for flipping an edge in place (actually a pair of edges, but the second one is by design an edge not yet solved). Then there are two macros that jumble the edges of the bottom large corner until it is right. The solution is simple to follow as a cookbook operation, but not so simple to memorize, involving several macros rather than one or two that many people are using. /Don Lynn  Date: 6 Jan 1982 1024-PST From: ISAACS at SRI-KL Subject: drive ya nuts To: cube-lovers at MIT-MC The numbers on my "Drive Ya Nuts" are as follows (starting from 1 and reading clockwise: 123456, 143652, 162453, 164253, 165432, 165324, and 146235. As you can see, each "nut" is unique. There are several related puzzles, including: "Japanese Mind Bender", which is logically identical to "Drive Ya Nuts", e except it used colors instead of numbers, and each hexagon looks like a small circus tent. "Super Dominos", 4x6 squares, each divided diagonally into four sections, and colored in all possible ways with 3 colors (yellow, orange, and brown). Object is to arrange the squares in a 4x6 array, with adjacent edges of matching colors, and the border to be only one color. It claims there are about 12000 different solutions, a computer generated number. I find it difficult to find even one. "Colored Squares Puzzle", A magnetic version of Super Dominos. They also suggest a second puzzle as above, but with 2 colors around the edge. "Colored Triangles Puzzle", also magnetic (a match for "Colored Squares"), this one has the 3 colors on 24 triangles, arranged in a hexagon. Same problem, though. "Try Nine" is octagonal pieces with numbers (similar to "Drive Ya Nuts"), to be arranged in a 3x3 lattice such that numbers match vertically, horizontally , and diagonally. That is, the square spaces between the octagons have opposite edges with the same number. "Square Crazy" or "Le Carre fou! fou!", a new puzzle from Montreal (If anyone wants one, I can give you the address). It is 9 cardboard square "cards", each of which has half a fish on each edge (either the head or the tail), and each fish is one of 3 or 4 colors. So direction and color has to match up at each edge of the 3x3 square. I met the inventer, and he says he used a computer to ensure there is only one solution. "Its Knot Easy" has 16 square plastic pieces to be arranged in a 4x4 array, so that the picture of a rope running through the pieces forms a continuous loop. And many more. Including a dodecahedron whose faces turn, to try to get a continuous loop running through it, and a version of 3x3 squares which uses heights instead of colors. I think all the "Instant Insanity" type of puzzles are also related to this. What we need are some non trial-and-error methods of solution, as well as algorithms to tell if the solution is unique. By the way, I am planning to make a Rubik's cube with 9 colors, with a solution having 9 different colors on each face. The question is, how to design it so it is solvable (preferably not just by trial-and-error), and so there is a unique solution. --- Stan Isaacs -------  Date: 6 January 1982 19:09-EST From: Alan Bawden Subject: fowarding this... To: CUBE-LOVERS at MIT-MC Date: 6 Jan 1982 0955-PST From: ISAACS at SRI-KL Subject: Re: Masquerade To: RP at MIT-MC, TRB at MIT-MC, mike at UCLA-SECURITY, SWG at MIT-DMS, HAGERTY at RUTGERS, CC.Clive at UTEXAS-20 cc: ALAN at MIT-MC In-Reply-To: Your message of 6-Jan-82 0404-PST How should we do this? List all the things we see? By "chapter", or in some other order. I don't have my book here right now, so I will start from memory, by listing some of the obvious, just to make sure we all start from the same place. 1. Each picture has a hare in it. Sometimes hidden, sometimes in the open. I've always wondered if some of the obvious hares also have a second, hidden one somewhere. The man in the rabbit ears and with rabbit feet bothered me the most. 2. Each picture has around the outside edge, letters in red, which can be re-arranged to make some appropriate word. In addition, there are letters marked with a spike, which form a second word. 3. Several of the chapters have riddles; I think I know the answers to all but one. Besides these characteristics of each picture, there are many symbols. We should just start to make a list of what they are, and guesses as to what they mean. Again, I don't have my book handy, but two examples: 1. Many pictures have a number square. What is its' significance? If I remember correctly, all but one of them were missing the same number. 2. In one of the pictures, the moon(?) has her fingers arranged to say "love" in sign language. Somebody should put in the answers to the riddles and pictures in 1-3 above; just to have them on record, and to make sure we all agree. --- Stan Isaacs -------  Date: 6 January 1982 19:10-EST From: Alan Bawden Subject: Puzzle-Lovers To: CUBE-LOVERS at MIT-MC Well, it looks like the discussion is going to start wandering a bit. It is not to late to register your complaints with me if you don't like this turn of events. Where can one obtain this "Masquerade" object anyway? Is this something I find in any bookstore?  Date: 6 Jan 1982 2328-PST From: Alan R. Katz Subject: Folding puzzle To: cube-lovers at MIT-MC cc: katz at USC-ISIF There is a puzzle out which looks to be a book of paper. The purpose seems to be to fold over the paper to make various patterns or shapes (various pages are folded over but the patterns or two dimensional). Does anyone know anything about this and is it any good?? Alan -------  Date: 8 Jan 1982 1009-PST From: ISAACS at SRI-KL Subject: FOLDING PUZZLE To: CUBE-LOVERS at MIT-MC The folding puzzle book (I can look up the name and reference at home if anyone is interested) is original, and a "cute" idea, but not very difficult to solve. It is a pleasant pastime, and good to do with older children, I found. It seems to be solved almost purely by trial and error. I think with some work, a truly interesting and difficult puzzle could be made using the idea of paper folding to make patterns. --- Stan -------  Date: 25 Jan 1982 0952-PST From: ISAACS at SRI-KL Subject: SILENCE To: CUBE-LOVERS at MIT-MC HELLO! Anybody still there? Its been awfully quiet lately. --- Stan -------  Date: 8 February 1982 20:36-EST From: Alan Bawden Subject: St. Valentine's Day Massacre? To: CUBE-LOVERS at MIT-MC I wondered out loud to some of the Masquerade-Lovers why no one was discussing Cubes or Masquerade or anything. And got the following response which I foward to you all: Date: 8 Feb 1982 1639-CST From: Clive Dawson Subject: Re: Masquerade Masquerade is still on my stack, but actually I've come across another amusement which has been taking up my time during the last couple of weeks. Does anybody know about the St. Valentine's Day Massacre? It is a cross-country road rally which takes place on paper--the Rand McNally road atlas to be precise. You are given hundreds of very detailed and tricky instructions and puzzles as you make your way from the Golden Gate Bridge to the Statue of Liberty. You must answer questions along the way about what you see, time and distance between various points, etc. I've found the whole thing to be a lot of fun and quite challenging, even though the entry fee was $24.00 (includes a copy of the Road Atlas). Trophies will be given out in 3 different classes of competition (Class C= first timers, Class B= previous participants, Class A= Previous trophy winners). If anybody is interested, I'll be happy to provide more info--deadline for entry is Feb. 14, answers must be sent in by March 1. Also, I'd like to hear from anybody already involved if they'd like to compare notes, etc. Clive P.S. Alan--go ahead and forward to Cube-Lovers if you wish. -------  Date: 9 Feb 1982 1624-CST From: Clive Dawson Subject: More on St. Valentine's Day Massacre To: cube-lovers at MIT-MC Several people have inquired about this further, so I thought I'd go ahead and send more info to the group... Write to: St. Valentine's Day Massacre P.O. Box 53 La Canada, CA 91011 Since time is short, you may want to give them a call between noon and 3PM PST at (213)790-4937. Deadline for entering is Feb. 14. For $24, they will send you the Rand McNally Road Atlas, and a 50-page booklet with all of the rules, course instructions, questions, and answer sheet. Here's a sample of what it's like: 1. Begin the 1982 St. Valentine's Day Massacre at the interchange north of the Golden Gate Bridge (NC-19 on page 11), where you find two cars from which to choose your first Massacre conveyance: a 1960 Falcon and a 1930 L-29 cabriolet. Hmmm: Ford or Cord. Ponder the selection for something under a second, then hop in and head south on state highway. Q1. How many "San Gregorio"s do you see? a) 0 b) 1 c) 2 2. Left on 152 Q2. How many of these do you see? Bell Bells a) 0 d)3 Bell's Station b) 1 e)4 "Bells Station" c) 2 3. South onto 33. Q3. Do you see Devils Den? a) yes b) no 4. Left on 58 Q4. How many of these do you pass? Calloway Oak a) 0 b) 1 c) 2 5. South on 99 6. Southeast on Intersate 5 Q5. Which of these do you see first? a) Los Angeles b) Burbank c) Gorman - - - - - - - - By the time you start answering these questions, you will have read through the 6 pages of rules which spell out VERY PRECISELY just what it means to "see" something (be within 1/4 inch of it on the map), the difference between going ONTO a highway and ON a highway, and the difference between "Bell" and Bell. The course following rules can be very tricky, telling you when to try to stay on a given road, when you can turn, when you can switch maps, when to start and stop consideration of a given question, etc. Later in the course you will be given various puzzles (visit all 6 National Monuments in 4 Arizona counties by travelling only on U.S. and state highways with no U-turns, for example.) At one point you actually travel by balloon at 120,000 feet--this turns out to be about 1 inch above the paper for purposes of "see"! The instructions are filled with various tricks and traps which will take you miles off course if you're not careful. But they are also cleverly constructed to eventually get you back on the right track again so that you don't realize your mistakes. One of their favorite tricks is exemplified by Question 5 above: you're tempted to say Gorman since it is north of L.A., until you realize that you see Los Angeles COUNTY first! They give you plenty of help on the first leg so that you get an idea of what it's all about. You can stop after 4 legs and compete for the Class C trophies, or else go onto legs 5-7 to try for a Class B trophy as well as one for Class C. Legs 8 and 9 are for the Class A trophies, and are the hardest of all. I've entered a couple of similar contests put on by the same people (The Great Maltese Circumglobal Trophy Dash) and even though they're a little slow with scoring and sending results, I've found them to be very fair and reasonable. As soon as the contest deadline passes, they will send you an answer booklet with a complete explanation of the entire course and reasons for each answer. You then have the right to challenge their reasoning if you disagree with an answer. If they agree with your challenge, they will eliminate that question from the scoring. After the protest period ends, they do the final scoring, sending out a complete list of scores for all participants as well as trophies. The top 10% in each class all get something, which gets progressively fancier for the higher percentiles and classes. Depending on how careful you are, you can expect to spend anywhere between 30 and 100 hours if you complete the whole course. Personally I find the entry fee a little high, but considering the free road atlas and the number of hours of tortu...er, entertainment, I guess it's worth it! If you have any other questions, let me know. --Clive ------- -------  Date: 16 Feb 1982 0931-PST From: ISAACS at SRI-KL Subject: MAGIC OCTAHEDRON To: CUBE-LOVERS at MIT-MC The Magic Octahedron (Octahedron Cube?) is out. A friend got one for me over the weekend. Each face is 9 triangles; it works sort of like two tetrahedrons base to base, except each tetra is really a 4-sided pyramid. The solution is not too hard, and not too different from the tetrahedron. Like the tetra, the corner and next-to-corner pieces don't travel-they only twist. The remaining edges are isomorphic to the edges of a cube. In fact, if you corner-center a Cubes coloring (ie each face has 4 colors, each corner has one; see the Scientific American article), and then peel off the labels from the corners, you have an exact isomorphism of the Octahedron. By the way, this version twists around the corners, using the same type mechanism as the original Cube (I think). It should be possible to use the mechanism of the Tetrahedron (with its 4 axes of rotation) to build an Octahetron whose faces twist. Any mechanical engineers out there to do so? --- Stan -------  Date: 14 Mar 1982 0703-PST Sender: OLE at DARCOM-KA Subject: Poison From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Message-ID: <[DARCOM-KA]14-Mar-82 07:03:13.OLE> Folks, You may love your cubes, but don't kiss them or lick your fingers after having twisted them! According to a recent article in one of the big papers here, the colotrtabs contain exceptionally high levels of lead. The worst is the yellow face, closely followed by the green, orange, red and blue. You have been warned.  Date: 14 March 1982 10:32-EST From: Richard Pavelle Subject: MASQUERADE- too little, too late To: CUBE-LOVERS at MIT-MC We were beaten by a dog. According to an article on Page 2 of the Sunday Globe (3/14/82) the Golden Hare was found "by a man whose dog accidentally stumbled across it"... "The hare was buried just beneath the surface in the shadow of an ancient cross at Ampthill which commemorates Catherine of Aragon..". One of the major clues in the book pointed to her.  Date: 23 Mar 1982 1032-PST From: ISAACS at SRI-KL Subject: new book To: CUBE-LOVERS at MIT-MC "Handbook of Cubik Math", by Alexander Frey and David singmaster, is now out from Enslow Publications. Most of the contents, but not all, are from Singmasters other opus, but this is better organized and clearer. I have not finished it yet, so I can't give it a full review. By the way, over the weekend I found that Meffert, inventer of the Pyraminx Tetrahedron has also invented about TWENTY (20) other variations! Diferent versions of Icosahedra and dodecahedra; a triangular prism, a cube cut along the three planes that make hexagonal cross-sections, etc. I have high hopes that they will be available in this country soon. We need a name for this type of puzzle. Clearly, "Cube" is not very satisfactory any more - not with spheres, tetrahedra, octahedra, etc. "group theory puzzles" is too broad - it also includes sliding block puzzles and others. Perhaps something like "Axial" or Axially Rotation or some such. David Singmaster is publishing a "Cubic Circular" (nice name) now. If anyone wants his address, I'll be glad to supply it. He will also be selling Mefferts puzzles, I understand. Has anyone seen a 4x4x4 or 5x5x5 cube yet? Anynew information on them? Does anyone know if the Masquerade clues have been published now that the hare has been found? The Subject of this message is out of date. Oh well. ---Stan -------  Date: 23 March 1982 15:51-EST From: David C. Plummer Subject: rumor confirmation To: CUBE-LOVERS at MIT-MC The 4x4x4 has been confirmed by two stores in the Boston area (Games People Play [Harvard Square] and The Name of the Game [Quincy Market]). Unfortunately Ideal is only taking orders now; delivery is not expected until the fall (****sigh****). I have heard nothing of the 5x5x5. ALAN thinks we should solve the beast before we actually get one. This could done in our heads, on paper, by machine assistance, or by machine. I have thought about it a little, and I think I have all the necessary tools to do it. It is not as easy as saying, "Well if we just consider these two planes as one, it looks like the 3x3x3, and..." That doesn't work for the majority (the hardest parts) of the puzzle, for reasons I'll be happy to explain. Before people start asking a lot of questions about the properties of the 4x4x4, they should look in MC:ALAN;CUBE 4X4X4 which contains the parts of the archives that discuss the 4x4x4. If there is any interest I can try and create a discussion...  Date: 25 Mar 1982 1714-PST Sender: OLE at DARCOM-KA Subject: Singmaster and IBM From: Ole at DARCOM-KA (Ole J. Jacobsen) To: Cube-Lovers at MIT-MC Message-ID: <[DARCOM-KA]25-Mar-82 17:14:12.OLE> Singmaster has as you probably know set up his own company selling cubes and other puzzles of all shapes and sizes. The latest price- list includes Rubik's Calendar Cube, a true Executive Toy. The object is, if I understand it correctly, to get one or more faces displaying the correct date/month. He also sells Braille cubes,- Ideal cubes with raised dots. His "Cubic Circular" as mentioned by Stan, is well worth getting, as it contains alot of interesting information. A Marketing Consultant from IBM gave me a cube which has IBM4331 printed on all the six center faces and the words: Value,Function, Support,Reliability,Performance,Productivity on each facie of each face respectively. The orientation of the IBM 4331 relative to the other 8 repeated words on each face makes it a supergroup variety. In my opinion, this cube is the best product to come from IBM in years!  Date: 31 Mar 1982 1821-PST From: ISAACS at SRI-KL Subject: 4**3,5**3 To: cube-lovers at MIT-MC I should think you could solve a 4x4x4 cube by applying 3x3x3 moves, using different combinations for the "center slice" - ie, rotating the center two slices together allows all the regular corner moves; it also allows switching adjacent edge pieces (in pairs) by using whatever you use to flip edges in 3x3x3. Using only one of the center slices as "center", and turning the other with a face should allow flipping a pair of L-R pieces, so that L becomes R and vice versa. The center 4 pieces (which I haven't thought about carefully) probably can be changed around at (even parity) will, sometimes treating them as edges (since they can be carried along on certain of the "edge" moves), and sometimes as centers (for instance, to rotate a group of 4 of them halfway around in place). I don't know what new parity limitations exist; nor do I know if the same type of sequences are efficient for solving (ie, top-middle-bottom, etc), but I shouldn't think the new cubes will be so very much more difficult. I would assume the 5x5x5's could be handled similarly, except, of course, they have a real center. --- Stan -------  Date: 31 March 1982 22:26-EST From: David C. Plummer Sender: DCP0 at MIT-MC Subject: 4**3,5**3 To: CUBE-LOVERS at MIT-MC NO, NO, NO !!!! You CANNOT treat 1 of the center slices of a 4x4x4 as a center of a 3x3x3. Suppose you did this for one axis, and for the other two axes you treated both "centers" as a unit (and therefore the center slice of a 3x3x3). Now take one of the axes with a double width center, and rotate an outer slice 180 degrees. Suppose the front face looked like this: +####+####+####+####+ # # # | # # # # | # +####+####+####+####+ # # # | # # # # | # +----+----+----+----+ # # # | # # # # | # +####+####+####+####+ # # # | # # # # | # +####+####+####+####+ You rotate the top slice and the front face now looks like: +####+####+####+####+ # | # # # # | # # # +####+####+####+####+ # # # | # # # # | # +----+----+----+----+ # # # | # # # # | # +####+####+####+####+ # # # | # # # # | # +####+####+####+####+ Notice that the top layer does not go very well with the bottom 3 layers. The 5x5x5 has similar problems. I think the right way to solve both the 4x4x4 and 5x5x5 at first is to use mono-flips. Once conceptually understood, they are very powerful and easy to visualize.  Date: 9 Apr 82 15:00-PDT From: mclure at SRI-UNIX To: cube-lovers at mc Subject: Peking newspaper criticizes cube a058 0436 09 Apr 82 PM-China-Cube,100 Newspaper Decries Rubik's Cube Craze PEKING (AP) - Rubik's Cube, the colorful brain-teaser that has started to captivate the Chinese, is being criticized by a Peking newspaper as a dangerous pastime that can lead to divorce, abnormal behavior, high blood pressure and aching fingers. ''The Magic Cube possibly is beneficial to sharpening intelligence, but don't forget that its side effects might bring danger,'' the Peking Evening News said, using the puzzle's Chinese name. Last November, the Chinese press reported people were standing in line to buy the cube in Shanghai, and that city already had held its first Magic Cube contest although the toy hit the market only a few months earlier. ap-ny-04-09 0721EST **********  Date: 22 April 1982 1703-EST (Thursday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Cubebot Message-Id: <22Apr82 170345 DH51@CMU-10A> Caption from a photograph said to be from the Washington Post, April 13, 1982: University of Illinois engineering student Daniel Talken adjusts the mechanical hands of a robot built by students at the school in Urbana to solve the Rubik's Cube puzzle. The robot's computer brain can work out the solution in two-tenths of a second, but it takes the hands about 12 minutes to make average 110 moves to solve the puzzle. The photo shows a guy diddling a complicated machine in which an unsolved cube is visible. Eyes are painted on the front of the machine's support above a bulge that may be functional as well as vaguely resembling a nose. No camera equipment is apparent; presumably they tell the machine how the cube is scrambled, or cheat and have the machine itself scramble the cube.  Date: 3 May 1982 1724-PDT From: ISAACS at SRI-KL Subject: RUBIK'S CUBE CLUB To: CUBE-LOVERS at MIT-MC I just got a flyer from Ideal announcing their Rubik's Cube Club, address Box 72, Hollis, NY 11423, $5.00 for a year. They also have "merchandise" (no prices included), including cube shirt, tie, patch, button, bumper sticker, etc. They also have a poster with many of Ideal's cube-related products on it, including the 4**3 (called Rubik's Revenge), a 2**3, Rubik's Race, Rubik's Challenge, etc. I can't tell quite what all of them are. Also, there are several new group-theory puzzles on the market (here or in England), sort of related to the barrel. Orb-it is a sphere with beads that rotate around 4 parallels. In addition, the whole sphere rotates around a meridian, to bring different halves of the parallels into contact, changing from 4 separate circles of beads to one continuous "spiral", or two disjoint closed paths. Equator Puzzle is a sphere with 3 intersecting (of course) great circles, each consisting of 12 squares which move around the circles. The coloring is into 4 segments, sort of like orange slices. The Trillion Puzzle (will be available from Ideal) is a cross of colored pieces, 17 of them, four each of 4 colors plus an extra red piece. They can be arranged with each arm of the cross monochromatic, or in 4 circles around the center. "The cross lies in a circle divided into 3 concentric rings which are independently rotatable, though the outer ring cannot move when the plunger is pushed in. The middle ring is 2 pieces wide." (From the description in Singmasters catalogue). The plunger moves one arm (9 pieces) over 1 piece. About 1 billion distinguishable patterns. And many more - Hungarian Rings, and Gears, for two. I also saw some which used polarized light, so it was hard to tell where the pieces should go. The latest cube I've seen has Pac-man and other electronic game figures on it. --- Stan Isaacs -------