From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 26 21:22:54 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA26831; Tue, 26 Aug 1997 21:22:54 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From walsmith@erols.com Tue Aug 26 21:03:14 1997 Message-Id: <34037BAB.7923@erols.com> Date: Tue, 26 Aug 1997 20:58:19 -0400 From: Walter Smith Reply-To: walsmith@erols.com To: cube-lovers@ai.mit.edu Subject: Got a new shape...? On 8/15/97 David Goyra asked for ideas for simulated puzzles. Obviously there are infinite possibilities. If you want a source of inspiration for simulated or real puzzles, I recommend the following book: Shapes, Space and Symmetry by Alan Holden Dover Publications, Inc. I got mine at Boarders Bookstores. It is a book about three dimensional shapes. It discusses symmetry and other properties with a minimum of mathematical terms. It gives instructions (and pictures) on constructing many shapes from cardboard or wire. Any solid shape could be cut (or cuts) parallel to the sides, between opposite corners, between opposite edges, along edges or any combination of the foregoing. You will see the shapes of the common puzzles and ideas for hundreds more. Walt Smith WALSMITH@EROLS.COM From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 27 14:39:14 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA02051; Wed, 27 Aug 1997 14:39:14 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Wed Aug 27 14:33:56 1997 Message-Id: <199708271830.OAA29527@life.ai.mit.edu> Date: Wed, 27 Aug 1997 14:36:47 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: minimal maneuvers for "continuous" isoglyphs i finished computing minimal maneuvers for the "continuous" isoglyphs. some may be in a different orientation than herbert gave them. i also give a maneuver that is simultaneously minimal in both the quarter turn metric and the face turn metric when there is such a maneuver. *.* *** (type 01) *** 1. (girdle 3-cycle) F R' L U' R' U R L' B' R F' B (12q, 12f) 2. (distorted girdle 3-cycle) U R U D' F2 U' D R U' (10q, 9f) *.* .** (type 02) *** 3. edge hexagon of order 2 U B2 U' F' U' D L' D2 L U D' F D' L2 B2 D' (20q, 16f) 4. edge hexagon of order 3 U' D L' B D B' U2 D' B' R' B R U' L D' (16q) F L B U F2 B2 R F2 B2 L' U' B' L' F' (14f) 5. (off-girdle 3-cycles) B' U F2 L' F2 U' F' B L B2 U B2 L' F (18q, 14f) 6. (distorted off-girdle 3-cycles) F L B R D' F B2 L' F' B L' F' D R F R' F' (18q) U R2 D F' L U2 D2 R' U2 D2 F D' R2 U' (14f) *.* **. (type 03) *.* 7. (plummer's C's) F U' F B' D2 B' U' D R B2 R L' B R' F U' D R' (20q) L2 U2 R' B' U' D B2 D' R' D L D2 F U2 D L2 (16f) *.* .*. (type 04) *.* 8. pons asinorum U2 D2 F2 B2 R2 L2 (12q, 6f) 9. checkerboards of order 3 F B2 R' D2 B R U D' R L' D' F' R2 D F2 B' (20q, 16f) 10. checkerboards of order 6 R' D' F' D L F U2 B' L U D' R' D' L F L2 U F' (20q) R2 L2 U B L2 D' F B2 R L' F' B R D F2 L' U' (17f) *** *** (type 10) **. 11. meson U F' D F U' F' L' U' L D' L' U L F (14q) D F2 D' R B2 R' D F2 D' R B2 R' (12f) *.* *** (type 11) **. 12. (meson & girdle 3-cycle) F' L' B' D2 B' D' B D' R F' R F R2 B L F (18q, 16f) *** **. (type 12) *.. 13. two twisted peaks F B' U F U F U L B L2 B' U F' L U L' B (18q) F D2 B R B' L' F D' L2 F2 R F' R' F2 L' F' (16f) 14. exchanged peaks F U2 L F L' B L U B' R' L' U R' D' F' B R2 (19q) F2 R2 D R2 U D F2 D' R' D' F L2 F' D R U' (16f) *.* .** (type 12) **. 15. (meson & girdle 3-cycles) F B' R F' U L U' F B' D' B D L' B D' R' D F' (18q, 18f) *.* **. (type 13) *.. 16. (plummer's Y's) R U' R B' R F R' U D' R L' B' L F L' F' R F' (18q) L F B' U' R' B' R' L' U2 L D' R F R2 B2 L' F (17f) *.* .*. (type 14) *.. 17. (plummer's cluster & girdle 3-cycles) R U' F U F' D' R F D' R L' F B' D' R F' L F' (18q) F B2 U R L2 B' L F D' L' B L B' U L' U' D2 (17f) 18. (christman's cluster & girdle) DL DB DR DF UL UB UR UF LB LF RB RF DLB URB UBL ULF DRF DFL UFR DBR F U R' U' R U2 R' B' R' F R' D R' L U F D' F B' R' (21q) F2 U R2 L' U2 D' F2 U' B R L' B' U2 B U' R B' L (18f) .*. *** (type 30) .** 19. (plummer's rabbits) F L' F R' U R U' F' L U R' U' R F' (14q, 14f) .*. .** (type 31) .** 20. twisted cube edges, orthogonal bars F L' U L U' R' U F' L F L' U' R F' (14q, 14f) ... .** (type 32) .** 21. cube in a cube F L F U' R U F2 L2 U' L' B D' B' L2 U (18q, 15f) .*. **. (type 32) ..* 22. twisted duck feet U2 F' B D B' U D2 L U2 F L F U' R' B' R F' (20q, 17f) 23. exchanged duck feet U F R2 F' D' R U B2 U2 F' R2 F D B2 R B' (21q, 16f) ... **. (type 33) ..* 24. (plummer's bend) F R B' R U R F D' L' F2 R U' F R' B' R F' (18q) F' R U2 L' F B U B2 R' U R2 D' R2 U' L' U (16f) ... .*. (type 34) ..* 25. twisted chicken feet D2 R U L' F2 R F' U F' U' B' U F D' L F' (18q, 16f) 26. exchanged chicken feet, cherries F L' D' B' L F U F' D' F L2 B' R' U L2 D' F (19q, 17f) .*. *** (type 40) .*. 27. christman's cross U R L' F2 U2 F2 R' L U2 F2 U (16q, 11f) 28. plummer's cross U' D2 R B2 D' R' U D' R L' D R F2 D' R2 L (20q, 16f) ... *** (type 41) .*. 29. four way street L U2 F' U F L2 U L F' D' F2 L' D' L D2 F' (20q, 16f) ... .** (type 42) .*. 30. exchanged rings B' U' B' L' D B U D2 B U L D' L' U' L2 D (18q) F U D' L' B2 L U' D F U R2 L2 U' L2 F2 (15f) 31. twisted rings F D F' D2 L' B' U L D R U L' F' U L U2 (18q, 16f) 32. anaconda, worm L U B' U' R L' B R' F B' D R D' F' (14q, 14f) .*. .*. (type 43) ... 33. six U's type 6 U D' F' U R L' B' U F U D' R' (12q, 12f) ... .*. (type 44) ... 34. six spot, six O's U D' R L' F B' U D' (8q, 8f) mike From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 1 22:33:57 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA05860; Mon, 1 Sep 1997 22:33:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Mon Sep 1 16:35:03 1997 From: SCHMIDTG@iccgcc.cle.ab.com Date: Mon, 1 Sep 1997 16:32:10 -0400 (EDT) To: cube-lovers@ai.mit.edu Message-Id: <970901163210.20217b13@iccgcc.cle.ab.com> Subject: Re: Open and Closed Subgroups of G I'd like to thank Jerry for taking the time to put together his message discussing basic group theory as it applies to the cube as well as the basics of Thistlewaite's algorithm. Although I consider myself somewhat beyond the "layman" level in this area, I'm not always able to follow the various posts to this group. Besides, it's also helpful to read a little "refresher" every now and then to help reinforce and clarify previously digested concepts. It might also be helpful for someone to cover the basics of cube parity. Although I think I understand the basic group theoretic concepts of permutation parity, the asymmetry of the marked faces of the cube have never quite left me feeling comfortable about how this concept is applied to the cube. Hofstadter, covers this, but does not discuss it in enough detail for one to fully grasp the concept. Regards, -- Greg Schmidt From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 1 23:26:56 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA05957; Mon, 1 Sep 1997 23:26:56 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Mon Sep 1 16:50:08 1997 From: SCHMIDTG@iccgcc.cle.ab.com Date: Mon, 1 Sep 1997 16:46:33 -0400 (EDT) To: cube-lovers@ai.mit.edu Message-Id: <970901164633.20217b13@iccgcc.cle.ab.com> Subject: Re[2]: Open and Closed Subgroups of G Oh, and I forgot to mention... My ultimate goal of understanding parity would be such that someone could hand me an arbitrary permutation puzzle and I'd be able to examine it and determine from the set of legal moves both the parity constraints and also be able to construct a parity test valid from any given puzzle state. I find it interesting that the method seems to differ across puzzles. For example, 15 puzzle parity can be determined by the number of pairwise exchanges required to solve the puzzle, whereas with the cube, it seems a more direct approach is possible by examining cubie orientations with respect to marked cubicles. Still, I'm somewhat mystified. Regards, -- Greg Schmidt From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 2 11:08:15 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA07826; Tue, 2 Sep 1997 11:08:14 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Tue Sep 2 09:46:52 1997 Date: Tue, 2 Sep 1997 08:44:30 -0400 (EDT) From: Nicholas Bodley To: SCHMIDTG@iccgcc.cle.ab.com Cc: cube-lovers@ai.mit.edu Subject: Parity (Was Re: Re[2]: Open and Closed Subgroups of G) In-Reply-To: <970901164633.20217b13@iccgcc.cle.ab.com> Message-Id: If I understand parity, Greg's examination would reveal whether someone had reassembled a Cube (or other mathematically-related puzzle) into a state that can't be solved. |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* Waltham is now in the new 781 area code. |* Amateur musician *|* 617 will be recognized until the end of 1997. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 3 18:01:12 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA17413; Wed, 3 Sep 1997 18:01:11 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From lvt-cfc@servtech.com Wed Sep 3 13:05:50 1997 From: "christopher f. chiesa" Message-Id: <199709031702.NAA20567@cyber1.servtech.com> Subject: Re: Open and Closed Subgroups of G To: cube-lovers@ai.mit.edu Date: Wed, 3 Sep 1997 13:02:11 -0400 (EDT) Greg Schmidt (SCHMIDTG@iccgcc.cle.ab.com) mentions discomfort about how concepts of "parity" are applied to the Cube. I second the notion! :-) I assume that by "parity" we mean that which is conserved as the "twist" of corner cubies or the "flip" of edge cubies. I myself have a HELL of a time determining a particular corner cubie's precise amount (N/3, N an integer) of "twist," or a particular edge cubie's precise amount (N/2, N an integer) of "flip," other than in the case of an observable change in ONLY that particular cubie -- and moreover, ONLY in its ORIENTATION. Any change in a cubie's POSITION, relative OR absolute, renders my notions of "twist" and "flip" rather fuzzy. F'rinstance, start with a Cube in the "solved" state and perform the sequence (generator?): R' D2 R F D2 F' U2 F D2 F' R' D2 R U2 You will find that "FRU has been twisted -1/3 ("one 'notch' CCW"), and BLU has been twisted +1/3 ("one 'notch' CW")," relative to their previous orientations (i.e., relative to "solved") -- and that this is easy to assess largely because the "solved" state of the rest of the Cube makes it very clear how the corner cubies' orientations have changed (and their positions have NOT). The sequence/generator would produce the same net effect (twisting FRU -1/3, and BLU +1/3) when performed on the Cube in ANY state; it would merely be more difficult for the casual observer to identify against the background of a "scrambled" Cube state. But, back to the start-from-"solved" example. If I now make the single turn B' I no longer find it so easy to characterize the corner-twist parity state of the Cube, because (all of) the corner-cubies affected by this particular Cube-state-change have left their previous positions, leaving me to wonder, "RELATIVE TO WHAT" their twist is to be assessed. How is it done? What can now be said about the "twist state" of, say, the former BLU (now BRU) cubie? What about the former BLD (now BLU) cubie? My efforts to "reason it out," within the limitations of my group-theory background (which is now infinitely broader thanks to Jerry Bryan!), lead to what almost seems a paradox. For what it's worth, I present it for your discussion, and will be very interested to hear what you Cubemeisters are able to contribute! Observe that the orientations of all corners in the F layer remain unchanged by the B' operation last performed. In particular, the FRU cubie retains its -1/3 twist relative to (what's left of) the "solved" state. Assuming that the "twist" of a cubie which "hasn't moved" REMAINS THE SAME, as opposed to being, say, "implicitly redefined" by the movement of OTHER cubies, I can still say a few things -- though not as many things as I would like! -- about the twist-states of the corner-cubies in the "B layer" after that B' face turn. Invoking twist-parity-conservation (let's just say "twist-conservation," okay?), I assert that "the TOTAL twist of all corner cubies in the B layer must still be 'some integer plus 1/3,'" so as to "cancel out" the -1/3 twist remaining on FRU. The B' turn thus imparted "some integer" TOTAL twist, which is to say, a total of 0 "net" twist, to the corner cubies in the B layer -- but was it e.g. "0, 0, 0, 0" or "+1/3, +1/3, -1/3, -1/3?" (I believe all other combinations reduce to these.) Note that this boils down to asking, "does a face turn, if it twists corner-cubies AT ALL, twist ALL FOUR the SAME WAY (i.e. apply the same "net twist" to all four), or NOT?" Is there a definitive answer? A standard assumption? Proof or disproof of either? It seems there would _have_ to be, in order to have "meaningful" discussions of "twist" at all. For a while I thought I could prove that it was the "0, 0, 0, 0" case, but it turned out that one of my working assumptions was equivalent to STATING that it was the "0, 0, 0, 0" case. I was only "proving" my own ASSUMPTION. Glad I didn't post THAT. :-) Naturally, analogous issues and questions will arise when discussing edge-cubie "flip" and the conservation thereof. :-) All in all, I'd be VERY interested in seeing the professional theoretical dissection of this issue! ... That's all I have today on the subjects of "twist," "flip," and "parity/ conservation thereof." But before I go, I'll leave you with two more demented, blue-sky thoughts. Beware; this is what I get for reading Star Trek novels before bed, and again at breakfast... 1) At the edge of my intuition, beyond my ability to formalize, I fancy I sense that there might be a way of looking at the Cube, perhaps through the use of additional spatial dimensions or their mathemati- cal equivalents, in which the Cube is in some sense "always" in the "solved" state, or at least in which it is trivially obvious where lies the "direct path" back TO the "solved" state. I'm visualizing some sort of extra-spatial "rubber bands," or "strings" (in those higher spatial dimensions specifically so as to avoid "tangling" issues) that "trace" the route (or "net" route) taken by each cubie, or arbi- trary collection of cubies, from its/their position(s)-and-orienta- tion(s) in the "solved" state, to its/their p(s)-and-o(s) in a "scram- bled" Cube. In such a perception, one could simply "tug on the strings" and "pull" the Cube back to "solved." Does this make ANY kind of sense to ANYBODY else here? I feel as though I can "almost see it." 2) Is there a notion, has anybody done any work, on Cube states which are each other's "duals?" I define the "dual" of a Cube state X as that Cube state reached by performing, on a "solved" Cube, the same sequence of turns/moves which "solve" Cube state X. In other words, define a sequence of turns which transforms the Cube from state X to "solved," then apply that sequence again to the "solved" cube to arrive at state Y. State Y is then the "dual" of state X. Ques- tions abound: - does each state have EXACTLY ONE dual? Or many, depending on the specific sequence (as we know, there are many) of moves performed in solving state X ? (My gut feeling is that each state has exactly one dual. This would seem to be pretty easy to prove using the group-theory math at the disposal of many readers here.) - are there states which are their OWN duals? (Yes, clearly; the trivial "checkerboard" pattern arising from a single 180- degree turn of each face, is its own dual) - a state which is its own dual, is a "two-cycle" with the "solved" state: perform the generating sequence on either and get to the other. Are there "three-cycles?" "Four-cycles?" etc.? Looking forward to the followups, Chris Chiesa lvt-cfc@servtech.com From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 3 18:42:04 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA18297; Wed, 3 Sep 1997 18:42:04 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Wed Sep 3 13:55:04 1997 Date: Wed, 03 Sep 1997 13:51:02 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Open and Closed Subgroups of G In-Reply-To: <970901163210.20217b13@iccgcc.cle.ab.com> To: SCHMIDTG@iccgcc.cle.ab.com Cc: cube-lovers@ai.mit.edu Message-Id: On Mon, 1 Sep 1997 SCHMIDTG@iccgcc.cle.ab.com wrote: > It might also be helpful for someone to cover the basics of cube > parity. Although I think I understand the basic group theoretic > concepts of permutation parity, the asymmetry of the marked faces > of the cube have never quite left me feeling comfortable about > how this concept is applied to the cube. Hofstadter, covers this, > but does not discuss it in enough detail for one to fully grasp > the concept. > I'll take your question as literal, assuming you mean just parity and not twist and flip, and assuming you know the basic group theoretic concepts of permutation parity. Parity of the cube is best described (I think) as applying to whole cubies rather than to facelets. As such, a quarter turn of any face is a 4-cycle on the corner cubies and a 4-cycle on the edge cubies. A 4-cycle is odd, which is to say that it can be decomposed into an odd number of 2-cycles. The "obvious" way to decompose a 4-cycle is into three 2-cycles. Although decomposition of a 4-cycle into 2-cycles is not unique, any such decomposition will contain an odd number of 2-cycles. Start is even for both the edges and the corners (the identity consists of zero 2-cycles). If you any quarter turn from Start, both edges and corners become odd. Make another quarter turn, both edges and corners become even. Make another quarter turn, both edges and corners become odd. Etc. Edges and corners are either both even or both odd. In the constructable group, you can have odd corners with even edges or vice versa. For example, remove two edge cubies from a cube and exchange them without moving any of the other cubies around. You will be changing the parity of the edges without changing the parity of the corners. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Thu Sep 4 17:02:06 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA24202; Thu, 4 Sep 1997 17:01:53 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Thu Sep 4 12:54:09 1997 Date: Thu, 04 Sep 1997 12:50:11 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Open and Closed Subgroups of G In-Reply-To: <199709031702.NAA20567@cyber1.servtech.com> To: "christopher f. chiesa" Cc: cube-lovers@ai.mit.edu Message-Id: On Wed, 3 Sep 1997, christopher f. chiesa wrote: > 2) Is there a notion, has anybody done any work, on Cube states which > are each other's "duals?" I define the "dual" of a Cube state X as > that Cube state reached by performing, on a "solved" Cube, the same > sequence of turns/moves which "solve" Cube state X. In other words, > define a sequence of turns which transforms the Cube from state X > to "solved," then apply that sequence again to the "solved" cube to > arrive at state Y. State Y is then the "dual" of state X. Ques- > tions abound: The concept of "dual" which you are describing is standard in group theory (and be extension, in cube theory). A "dual" is properly called an inverse. If you have a sequence of turns which creates a position, the inverse sequence consists of writing the turns in reverse order, and converting clockwise turns to counterclockwise turns and vice versa. So the inverse of FRU' is UR'F'. If there are multiple sequences for a position (and most typically there are), you can do the same thing for any such sequence. Also, a position can be described in terms of which cubies have gone where. For example, you might have something like flu --> fur fur --> frd frd --> fdl fdl --> flu (flu is the front-left-up cubie etc. Standard Singmaster notation uses lower case letters for cubies and upper case letters for the moves themselves.) You could get the inverse by reversing the arrows like so. flu <-- fur fur <-- frd frd <-- fdl fdl <-- flu More commonly, you would write the inverse by swapping the cubie designations between the left and right side of the arrows like so. fur --> flu frd --> fur fdl --> frd flu --> fdl I don't know what you mean by "any work", but here are some standard information about inverses. The length of a position X is the same as the length of its inverse X', where length is the minimum number of moves to create the position. If X' is the inverse of X, then X is the inverse of X'. The symmetry of an inverse X' is the same as the symmetry of a position X (see Symmetry and Local Maxima in the archives for a discussion of symmetry). A local maximum is a position such that no matter which move you make, you will be one move closer to Start. It is not necessarily the case that the inverse of a local maximum is also a local maximum. > > - does each state have EXACTLY ONE dual? Or many, depending on > the specific sequence (as we know, there are many) of moves > performed in solving state X ? Yes, inverses are unique, both for groups in general, and for cubes in particular. > > - are there states which are their OWN duals? (Yes, clearly; > the trivial "checkerboard" pattern arising from a single 180- > degree turn of each face, is its own dual) You have answered your own question. Many positions are their own inverse. Some of them are much more complicated than the one which you describe. > > - a state which is its own dual, is a "two-cycle" with the > "solved" state: perform the generating sequence on either and > get to the other. Are there "three-cycles?" "Four-cycles?" > etc.? > The proper term for the concept you are describing is order. If you repeat a maneuver n times from Start and return to Start, then the position is of order n. (Strictly speaking, the order of a position is the smallest n which will work. Obviously, if n will work then so too will 2n, 3n, etc.) There are many different orders for which there are cube positions of that order. One of David Singmaster's early Cubic Circulars (I don't have the reference handy) had a table of possible cube orders and how many positions there were of each order. The term cycle is also very important in group theory (and by extension in cube theory). Suppose you look at a scrambled cube and determine that cubie a has gone to cubie b's place, cubie b has gone to cubie c's place, and cubie c has gone to cubie a's place, then a, b, and c form a 3-cycle. The way I have defined this particular 3-cycle, you could write it as (a,b,c), as (b,c,a), or as (c,a,b). This so-called cycle notation is circular, so it does't really matter which you write first. However, (a,c,b) is a different cycle than (a,b,c). In fact, (a,c,b) is the inverse of (a,b,c). Just for emphasis, (a,b,c) is not like an ordered pair (or really an ordered triple in this case). (a,b,c) means a goes to b, b goes to c, c goes to a. As an example of a cycle in purely cube terms, the cycle for the example I gave earlier would be (flu,fur,frd,fdl), so it is a 4-cycle. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Fri Sep 5 21:03:58 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA04289; Fri, 5 Sep 1997 21:03:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Fri Sep 5 21:08:00 1997 Date: Fri, 5 Sep 1997 21:07:48 -0400 Message-Id: <199709060107.VAA04503@sun30.aic.nrl.navy.mil> From: Dan Hoey To: lvt-cfc@servtech.com Cc: cube-lovers@ai.mit.edu In-Reply-To: <199709031702.NAA20567@cyber1.servtech.com> (lvt-cfc@servtech.com) Subject: Re: Open and Closed Subgroups of G (fwd) Chris Chiesa , among other things, writes > If I now make the single turn > B' > I no longer find it so easy to characterize the corner-twist parity state of > the Cube, because (all of) the corner-cubies affected by this particular > Cube-state-change have left their previous positions, leaving me to wonder, > "RELATIVE TO WHAT" their twist is to be assessed. At the risk of being repetitious, the answer is, "relative to the home orientation of the position they find themselves in". You choose a special facelet for each corner cubie. When the cubie is in its home position, its twist is the position of its special facelet relative to the home of the special facelet. When cubie X is in cubie Y's home position, the twist of cubie X is the position of X's special facelet relative to the home of Y's special facelet. The edges are done the same way, except mod 2. Cube-lovers can find this in Vanderschel's article (6 Aug 1980) and the extension by Saxe (3 September 1980). I mentioned (23 September 1982) that the choice of special facelets is arbitrary, and that a conservation of twist occurs for a set of pieces of any puzzle that 1. have an Abelian orientation group, and 2. are moved in untwisted cycles by the generators. This is true even if not all the cycles have the same length. For instance, we could have a Rubik's cube in which generators move corners in permutations like (FTR,FRD,FDL,FLT)(BRT,BTL,BLD), and twist would be preserved. The key is that for each piece, the minimum power of the generator that returns that piece to its home position must also return it to its home orientation. I'm quite uncertain about what orientation constraints can arise in puzzles with non-Abelian orientation groups. For instance, the hypercorners of a Rubik's tesseract have the symmetry group A4, and any orientation is achievable up to a constraint imposed by an Abelian quotient of A4 of type 3 (See 22 Oct 1982). Does every group have a unique maximal Abelian quotient? Is that the only orientation constraint that can occur? Dan Hoey Hoey@AIC.NRL.Navy.Mil [ Moderator's Note: Cube-lovers will be down Saturday and Sunday due to major electrical work at MIT. ] From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 8 09:47:22 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id JAA07291; Mon, 8 Sep 1997 09:47:22 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Sun Sep 7 17:51:08 1997 Message-Id: <3411D734.6471@hrz1.hrz.th-darmstadt.de> Date: Sun, 07 Sep 1997 00:20:36 +0200 From: Herbert Kociemba To: cube-lovers@ai.mit.edu Subject: Number of maneuvers with n face turns The number of maneuvers with 1, 2, 3,.. face turns for Rubik's cube are of course well known and are 18, 243, 3240... But I did not see a closed formula for these numbers before, so maybe you find the following formula interesting: Let r:= sqrt(6), then you have with n face turns P(n) = [(3+r)*(6+3r)^n + (3-r)*(6-3r)^n]/4 maneuvers. Because the second part in brackets is much smaller than the first, asymptotically you have (3+r)*(6+3r)^n /4 maneuvers. Even for small n, this approximation is very good. So for n=3 you get 3240.33 instead of 3240. The asymptotic branching factor P(n+1)/P(n) is therefore (6+3r), which is about 13.348469 . Herbert From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 9 11:01:44 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA15886; Tue, 9 Sep 1997 11:01:43 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Tue Sep 9 00:17:33 1997 Message-Id: <199709090413.AAA00748@life.ai.mit.edu> Date: Tue, 9 Sep 1997 00:20:27 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: maximal abelian quotients dan asks > Does every group have a > unique maximal Abelian quotient? yes. let G be a group. it's not difficult to show that 1) the commutator subgroup G' is normal, 2) the quotient group G / G' is abelian, and 3) if G --> A is a homomorphism to any abelian group A , then G' is in the kernel, so there is a unique homomorphism G / G' --> A such that the original homomorphism is the composite G --> G / G' --> A . this last one is kind of technical, but in the special case where A = G / N for some normal subgroup N , it says that if G / N is abelian, then N contains the commutator subgroup. thus, G / G' is the maximal abelian quotient of G . the quotient G / G' is sometimes written G^ab (the "abelianization" of G). as you might guess, this is an important construction in group theory, and it's one of the reasons why commutator subgroups are important. mike From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 9 14:56:02 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA17811; Tue, 9 Sep 1997 14:56:01 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Tue Sep 9 11:06:36 1997 Date: Tue, 09 Sep 1997 11:02:32 -0400 (EDT) From: Jerry Bryan Subject: Re: Open and Closed Subgroups of G In-Reply-To: <199709060107.VAA04503@sun30.aic.nrl.navy.mil> To: Cube-Lovers Cc: lvt-cfc@servtech.com Message-Id: On Fri, 5 Sep 1997, Dan Hoey wrote: > Chris Chiesa , among other things, writes > > > If I now make the single turn > > > B' > > > I no longer find it so easy to characterize the corner-twist > > parity state of the Cube, because (all of) the corner-cubies > > affected by this particular Cube-state-change have left their > > previous positions, leaving me to wonder, "RELATIVE TO WHAT" their > > twist is to be assessed. > > At the risk of being repetitious, the answer is, "relative to the home > orientation of the position they find themselves in". You choose a > special facelet for each corner cubie. When the cubie is in its home > position, its twist is the position of its special facelet relative to > the home of the special facelet. When cubie X is in cubie Y's home > position, the twist of cubie X is the position of X's special facelet > relative to the home of Y's special facelet. The edges are done the > same way, except mod 2. Dan's response (plus his references in the Cube-Lovers archives) pretty well covers it. I would just like to add a couple of points. 1. There is a reference in the archives to a way of demonstrating conservation of twist without first establishing a frame of reference, but I can't find the reference. The best I can recall, the same technique did not work for edges. But I prefer the frame of reference technique anyway because it is closely tied to some of the more usual ways of representing the cube in a computer. 2. For example, number the corner facelets from 1 to 24. Each facelet has two companion facelets which are bound to it on the same cubie. By knowing where one of the three facelets of a cubie is in a computer program, you automatically know where the other two facelets are, so you only have to store one of the three facelets. The one that you store can be the "special" facelet that Dan described for the purposes of determining conservation of twist. The collection of eight "special" facelets for the corners have been described in the archives as constituting a supplement for the group, but I have yet to find a discussion group supplements in any group theory book. As Dan says, your choice of "special" facelet is totally arbitrary for each cubie, but most typically you choose the Front and Back facelets, or the Right and Left facelets, or something equally well organized. 3. For another example, number the corner cubies from 1 to 8, and for each of the cubies describe the twist with a number from 0 to 2. This is essentially a wreath product representation of the cube. The numbers from 0 to 2 which describe the twist can be used to describe whether a cubie is twisted when it is not home, and can therefore be used to prove conservation of twist. Without knowing any more than I do about supplements, it seems very likely that it should be easy to represent any group which can be representated as a supplement as a wreath product and vice versa. The isomorphism seems obvious. I wonder if anybody out there can shed any light on this issue? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Fri Sep 12 17:53:00 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA20354; Fri, 12 Sep 1997 17:52:59 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Fri Sep 12 17:08:04 1997 Date: Fri, 12 Sep 1997 17:07:47 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: isoglyphs In-Reply-To: <199708182216.SAA00604@sun30.aic.nrl.navy.mil> To: cube-lovers@ai.mit.edu Reply-To: Jerry Bryan Message-Id: On Mon, 18 Aug 1997, Dan Hoey wrote: > A "chiral isoglyph" is one in which the handedness of the glyph is > taken into account in testing for isoglyphy,* so that the glyph > appears only in one variety. > > Mike used "achiral" for an isoglyph that fails to be a chiral > isoglyph, though I would tend to use "non-chiral". I would rather use > "achiral" for a situation that lacked chirality, as in an isoglyph of > a mirror-symmetric glyph. Let me see what I can do to muddy these waters. It seems to me that we might ought to consider the chirality of an isoglyph as being a different issue than the chirality of a glyph. I think the two are clearly related, but I am not sure that the one necessarily derives from the other. As to a glyph, it seems to me that a glyph is chiral only if conjugating the position by each of the four reflections of the square yields a different set of positions than does conjugating the position by each of the four rotations of the square. Hence, you can have a glyph which occurs in right-handed or left-handed forms, or one that doesn't. This is the simple part. I think the situation with isoglyphs is a little more complicated. For example, form an isoglyph using both the right-handed and the left-handed forms of a chiral glyph. You might have 6 right-handed glyphs and 0 left-handed glyphs, 5 right-handed glyphs and 1 left-handed glyph, etc. If there are unequal numbers of right-handed and left-handed glyphs, then it seems natural to define the handedness of the isoglyph as being that of the dominate glyph. But what if there are three right-handed glyphs and three left-handed glyphs? Up to symmetry, there are only two ways to partition the six faces of a cube into two sets of three faces. For example, the F, U, and B faces can be of the same chirality, or the F, U, and R faces can be of the same chirality (or any conjugates of these choice of faces). In the first case, the cube is partitioned like a universal joint, or maybe like a cubic baseball. Such a position seems to me to lack chirality. In the second case, three faces with the same chirality cluster around a common corner. Again, such a position seems to me to lack chirality. So an isoglyph which lacks chirality can contain chiral glyphs. On the other hand, even on an isoglyph consisting of three right-handed and three left-handed glyphs, you still might be able to find a distinguishing characteristic of the right hand part that was different from the left-handed part. For example, the glyph boundaries which were internal to the right-handed part of the isoglyph might be continuous whereas the glyph boundaries which were internal to the left-handed part of the isoglyph might not be continuous. Or for another example, the rotations of the three right-handed faces relative to each other might be different than the rotations of the left-handed faces relative to each other. (By the way, I have not verified that any of these positions I have described are actually in G. I guess I am thinking in terms of the constructible group of the facelets -- conceptually, peeling all the facelets off and reattaching them.) On the other hand, two glyphs which lack chirality when placed side by side can be chiral. For example, XOOXXX (the base glyph is XXX XXXOXO OXO XOOOXO OXO ) I really haven't thought through the implications of using six glyphs instead of two, but it seems to me quite likely that an isoglyph could be constructed using six glyphs which lack chirality and which are the same pattern, and where the we could attribute chirality to the isoglyph as a whole. I have thought about this in terms of Herbert's Cube Explorer 1.5 program. The pattern editor has a check box for continuous. If you don't check the box, the program finds both continuous and non-continuous isoglyphs. If you do check the box, it finds only continuous ones. So I have considered what would happen if the program had a check box for chiral. What should it do? The obvious thing would be that in normal operation, it would consider conjugates of both rotations and reflections of the square when building an isoglyph from a glyph, but that if the chiral box were checked it would consider only conjugates of rotations of the square. But is that sufficient to satisfy our various definitions of chiral, achiral, and/or non-chiral? I'm not sure. Maybe Dan or Mike would be kind enough to clarify further their thoughts on this issue. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Sun Sep 14 22:54:52 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA00285; Sun, 14 Sep 1997 22:54:52 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Sat Sep 13 21:32:35 1997 Message-Id: <199709140132.VAA09760@life.ai.mit.edu> Date: Sat, 13 Sep 1997 21:33:59 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: optimal solver is available for those who are interested in my optimal cube solver, you can now get it from the web page http://www.math.brown.edu/~reid/rubik/optimal_solver.html i've reduced the size of my transformation tables, so now i think there's a reasonable chance that it will run within 80Mb of RAM. enjoy the program. if you make any new exciting discoveries, please share them with the entire mailing list. mike From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 29 13:08:14 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA14569; Mon, 29 Sep 1997 13:08:14 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Sun Sep 28 14:45:54 1997 Message-Id: <199709281845.OAA08068@life.ai.mit.edu> Date: Sun, 28 Sep 1997 14:46:39 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: idea for smaller optimal solver a number of people have told me that they don't have 80Mb of RAM on their computers, so that my optimal solver won't work on their machine. here's an idea for an optimal solver that uses much less memory; it should fit within 16Mb, or 20Mb at the most. of course, it's a space/time tradeoff, but perhaps will still be fairly good. in my current program, i use distances to the subgroup H = as my "heuristic" function. there is another subgroup, H' , which contains H as a subgroup of index 8. H' is the subgroup of all elements of H composed with all (valid) flips of U-D slice edges. another way to describe H' is the subgroup of all elements where the U face has only the colors U and D, and the same for the D face. from this latter description, we see that if H1' , H2' and H3' are the three orientations of this subgroup, then their intersection is the subgroup of elements that "look like" they're in the square group. this is the same target subgroup that my current program has. the subgroup H' also has 16 symmetries. using this to reduce the size of the pattern database, and storing each entry with 4 bits, it should take about 8.5Mb. my current program also has about 8.5Mb of transformation tables (but 3Mb of these are not used while searching). the transformation tables will probably be slightly smaller (certainly no larger), so it seems plausible that this could run with 16Mb of RAM. what about running time? in his paper, rich korf hypothesizes that the number of nodes generated should be roughly proportional to the inverse of the size of the pattern databases. this suggests that using the smaller tables above would result in about 8 times as many nodes as my current program. this isn't bad, especially given that the branching factor (6 + 3 * sqrt(6) = 13.348469 for face turns, 9.3736596 for quarter turns) is larger than this. so this approach would be within 1 turn of my current program. i don't foresee having enough spare time anytime soon to program this, so i'll just post it here and maybe someone who is interested will program this. mike From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 30 12:12:19 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA19848; Tue, 30 Sep 1997 12:12:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From C.McCaig@Queens-Belfast.AC.UK Tue Sep 30 05:39:44 1997 From: C.McCaig@queens-belfast.ac.uk Date: Tue, 30 Sep 1997 10:27:55 GMT To: cube-lovers@ai.mit.edu Message-Id: <009BB113.FAA1EEF6.44@a1.qub.ac.uk> Subject: 4x4x4 solution i recently borrowed a friends 4x4x4, and i know the basic method for solving it. ie get the 6 centres, pair up all the edges, and then solve for the normal cube. however, about half the time i end up with a single edge pair inverted and cant figure out a move for reorientating the single edge pair. usually i break a few pairs and try and reorientate them this way, but this seems rather longwinded... does anyone have a move for this?. for example, say the green edge is on the blue face, and the blue edge is on the green face... thanks. clive --- clive mccaig queens university belfast northern ireland From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 30 17:44:19 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA21436; Tue, 30 Sep 1997 17:44:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Cube-Lovers-Request@ai.mit.edu Tue Sep 30 15:39:10 1997 Date: Tue, 30 Sep 1997 17:43:57 -0400 Message-Id: <30Sep1997.174357.Cube-Lovers@AI.MIT.EDU> From: Cube Lovers Moderator Sender: Cube-Lovers-Request@AI.MIT.EDU To: Cube-Lovers@AI.MIT.EDU Subject: 4x4x4 solution -- [Digest v23 #159] Cube-Lovers Digest Tue, 30 Sep 1997 Volume 23 : Issue 159 Today's Topic: 4x4x4 solution [ I have gathered together several similar messages on a single topic, putting them in digest format. It would be nice to get an explicit process for this problem, though. --Moderator. ] ---------------------------------------------------------------------- Date: Tue, 30 Sep 1997 13:39:46 -0400 (EDT) From: der Mouse To: C.McCaig@queens-belfast.ac.uk Cc: cube-lovers@ai.mit.edu Subject: Re: 4x4x4 solution > i recently borrowed a friends 4x4x4, and i know the basic method for > solving it. [...] however, about half the time i end up with a > single edge pair inverted and cant figure out a move for > reorientating the single edge pair. Make a single 90-degree inner-slice turn, then solve as before. This introduces an odd permutation on the edge pairs, which gets you back into easily solvable space. (It's usually easiest if you make sure that the two swapped edge cubies are part of the slice turn, by placing on the same slice beforehand if necessary.) I'm not sure quite what the parity constraint here is. There is some kind of even-parity constraint on the edge cubies, it appears, with a linked constraint on the face centres, but it's not as simple as the parity of the edge and face permutations being both even or both odd, because the single slice turn introduces two nonoverlapping 4-cycles on the face centre cubies - which is, overall, an even permutation on them. I do notice, though, that a slice turn produces a 4-cycle on the edges and two 4-cycles on the face centres; a face turn produces a 4-cycle on the face centres and two 4-cycles on the edges (and a 4-cycle on the corners, which may or may not be relevant). I wonder if there's a multiple-of-three constraint lurking. Doubtless some group theorist has long ago worked out exactly what the constraints are, but I haven't heard. (I tried to work through a group-theory text recently, got stalled along about the time it got to cosets, quotient groups, normal subgroups, etc.) der Mouse mouse@rodents.montreal.qc.ca 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B ------------------------------ Date: Tue, 30 Sep 1997 14:11:28 -0400 (EDT) From: Allan Wechsler To: C.McCaig@queens-belfast.ac.uk Cc: cube-lovers@ai.mit.edu Subject: 4x4x4 solution [C. McCaig:] i recently borrowed a friends 4x4x4, and i know the basic method for solving it. ie get the 6 centres, pair up all the edges, and then solve for the normal cube. however, about half the time i end up with a single edge pair inverted and cant figure out a move for reorientating the single edge pair. usually i break a few pairs and try and reorientate them this way, but this seems rather longwinded... does anyone have a move for this?. for example, say the green edge is on the blue face, and the blue edge is on the green face... What's happened here is that you've got those two edge-cubies _exchanged_. Here's how it works. Look at any face. You will see eight edge stickers, arranged around the face like eight square dancers (or Irish set dancers, if you prefer). Now I hope you are familiar with one or the other of these kinds of folk-dancing, because otherwise what I am going to say won't make sense. Those eight decals are either men or women, and no matter how they dance around the cube, they will never change sex. Every edge-cubie on the 444 has one permanently male and one permanently female sticker. If I haven't clarified things, at least I've spiced them up a bit. House around to home. - -A ------------------------------ Date: Tue, 30 Sep 1997 14:18:17 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: 4x4x4 solution To: C.McCaig@Queens-Belfast.AC.UK Cc: cube-lovers@ai.mit.edu On Tue, 30 Sep 1997 C.McCaig@Queens-Belfast.AC.UK wrote: > i recently borrowed a friends 4x4x4, and i know the basic method for > solving it. ie get the 6 centres, pair up all the edges, and then > solve for the normal cube. however, about half the time i end up > with a single edge pair inverted and cant figure out a move for > reorientating the single edge pair. usually i break a few pairs > and try and reorientate them this way, but this seems rather longwinded... > does anyone have a move for this?. for example, say the green edge > is on the blue face, and the blue edge is on the green face... > Your problem is one of parity. You have two edges cubies swapped (this swap is visible) and two face center (centre) cubies of the same color swapped (this swap is invisible). You have to have an even number of swaps in the total cube. If you want an even number in the edges (and you do), then you also have to have an even number in the face centers, even if swaps in the face centers are invisible. There is probably a more elegant solution, but the following will work. If you encounter the situation you describe, make any middle slice quarter turn. This will disturb the centers. The centers will now have an even numbers of swaps. Solve the centers again without simply undoing the middle slice you just made. The parity of the edges will then be ok. (I'm assuming that your solution for the face centers will maintain their parity after you correct it as described.) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 30 18:10:34 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA21599; Tue, 30 Sep 1997 18:10:33 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From roger.broadie@iclweb.com Tue Sep 30 18:05:38 1997 From: roger.broadie@iclweb.com (Roger Broadie) To: Subject: Re: 4x4x4 solution Date: Tue, 30 Sep 1997 23:02:48 +0100 Message-Id: <19970930230037.AAA21244@home> C.McCaig@queens-belfast.ac.uk wrote: > ...I recently borrowed a friends 4x4x4, and I ... can't figure out a > move for reorientating the single edge pair.... It is possible to solve the problem with a sequence based on a quarter turn of a central slice, since that, like a swap of two edge pieces, involves an odd-parity cycle of the edge pieces. Thus r2 U2 r U2 r2 (where r is the turn of the inner slice next to R in the direction parallel to R) puts a 4-cycle of edges onto the top face, but leaves you with the task of restoring the centres. It was the desire to find something less cumbersome that first lead me to investigate the archives of this list, and there the answer was: Date: Fri, 20 Oct 95 12:46:32 -0400 (EDT) From: Georges Helm Subject: Re: Old question about 2 adj edges how to flip 2 adj. edges (and nothing else) in 4x4x4 cube? r^2 U^2 r l' U^2 r' U^2 r U^2 r l U^2 l' U^2 r U^2 l r^2 U^2 Georges geohelm@pt.lu It does indeed contain an odd number of turns of the central slices to give the desired parity. Roger Broadie From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 11:53:15 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA25358; Wed, 1 Oct 1997 11:53:15 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From dokon@MIT.EDU Tue Sep 30 19:30:29 1997 Message-Id: <3.0.32.19970930192820.006ce8ac@po9.mit.edu> Date: Tue, 30 Sep 1997 19:28:21 -0400 To: cube-lovers@ai.mit.edu From: Dennis Okon Subject: God's Number I just found out that Keith Randall for the theory group of LCS (Lab for Computer Science) at MIT gave a talk Monday about God's number for the rubik's cube. He upped the lower bound 24 and gave "evidence" that it is 24. I don't know what moves he was counting (e.g. slice, quarter). Unfortunately, I missed it. Does anyone have any information on this? I'll see what I can find out. -Dennis Okon dokon@mit.edu From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 13:18:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA25712; Wed, 1 Oct 1997 13:18:15 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From ERCO@compuserve.com Wed Oct 1 01:56:54 1997 Date: Wed, 1 Oct 1997 01:52:24 -0400 From: Edwin Saesen Subject: Re: 4x4x4 solution -- [Digest v23 #159] Sender: Edwin Saesen To: CUBE Message-Id: <199710010152_MC2-225C-6120@compuserve.com> jbryan@pstcc.cc.tn.us wrote: >Your problem is one of parity. You have two edges cubies swapped >(this swap is visible) and two face center (centre) cubies of the >same color swapped (this swap is invisible). You have to have an >even number of swaps in the total cube. If you want an even number >in the edges (and you do), then you also have to have an even number >in the face centers, even if swaps in the face centers are invisible. I've had this problem as well. If I understand you correctly, this problem simply doesn't occur anymore as soon as you number (or mark in any other way) the center pieces which a) makes solving the cube a bit more difficult b) makes sure that you'll always get back to the original configuration of center pieces. I've had a similar problem on my 5x5x5 as well, and I assume that marking the nine center pieces might solve the problem as well. On my 4x4x4 I also had a problem of having two pairs of edges exchanged which simply can't happen on a 3x3x3. By experimenting with 3x3x3 moves I found a 24move solution to this, and I wonder if that's also sort of automatically solved by marking center pieces. Can anyone confirm this? Michael Ehrt --------------------------------------------- ERCO@compuserve.com From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 13:55:36 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA25872; Wed, 1 Oct 1997 13:55:35 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From darinh@ldr.com Wed Oct 1 13:22:36 1997 Message-Id: <34328707.3792@ldr.com> Date: Wed, 01 Oct 1997 10:23:24 -0700 From: Darin Haines Organization: Litho Development & Research To: Cube Subject: Piece for a Rubik's Revenge Hi Everyone, Does anyone know of someone wanting to sell a BROKEN Rubik's Revenge? or maybe a center piece from the same? Did anyone else have problems with the center pieces breaking on their RR? or am I the only one? My RR has been sitting useless on the shelf since '84. Hey, I was young and careless. ;-) -Darin [Moderator's note: I'll be away from cube-lovers from 2 Oct to 5 Oct. Messages received during that time will be distributed on the 6th. ] From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 15:49:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA26363; Wed, 1 Oct 1997 15:49:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From bagleyd@americas.sun.sed.monmouth.army.mil Wed Oct 1 14:41:30 1997 From: bagleyd@americas.sun.sed.monmouth.army.mil (David Bagley x21081) Message-Id: <199710011842.OAA21977@asia.sed.monmouth.army.mil> Subject: Piece for Alexander's Star To: Cube-Lovers@ai.mit.edu Date: Wed, 1 Oct 1997 14:42:15 -0400 (EDT) In-Reply-To: <34328707.3792@ldr.com> from "Darin Haines" at Oct 1, 97 10:23:24 am > > Hi Everyone, > > Does anyone know of someone wanting to sell a BROKEN Rubik's Revenge? > or maybe a center piece from the same? > That reminds me: If anyone needs a piece or 2 for the Alexander's Star let me know. They seem to break pretty easily IMHO. Please specify colors. I have a center piece too. Mine broke a while back and I have since gotten another one. -- Cheers, /X\ David A. Bagley (( X bagleyd@bigfoot.com http://wauug.erols.com/~bagleyd/ \X/ xlockmore ftp://wauug.erols.com/pub/X-Windows/xlockmore/index.html From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 16:57:31 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA26692; Wed, 1 Oct 1997 16:57:30 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Cube-Lovers-Request@ai.mit.edu Wed Oct 1 16:53:46 1997 Date: Wed, 1 Oct 1997 16:53:46 -0400 (EDT) Message-Id: <01Oct1997.165346.Cube-Lovers@AI.MIT.EDU> From: Cube Lovers Moderator To: Cube-Lovers@AI.MIT.EDU Subject: 4x4x4 solution -- [Digest v23 #165] Cube-Lovers Digest Wed, 1 Oct 1997 Volume 23 : Issue 165 Today's Topic: 4x4x4 solution ---------------------------------------------------------------------- Date: Wed, 1 Oct 1997 08:20:34 -0400 (EDT) From: Assoc Prof W David Joyner To: C.McCaig@queens-belfast.ac.uk Cc: cube-lovers@ai.mit.edu Subject: Re: 4x4x4 solution On Tue, 30 Sep 1997 C.McCaig@queens-belfast.ac.uk wrote: > i recently borrowed a friends 4x4x4, and i know the basic method for > solving it. ie get the 6 centres, pair up all the edges, and then > solve for the normal cube. however, about half the time i end up > with a single edge pair inverted and cant figure out a move for > reorientating the single edge pair. usually i break a few pairs > and try and reorientate them this way, but this seems rather longwinded... > does anyone have a move for this?. for example, say the green edge > is on the blue face, and the blue edge is on the green face... The idea is on the www page http://www.nadn.navy.mil/MathDept/wdj/solve4.txt Try L2^2*D1^2*U2*F1^3*U2^3*F1*D1^2*L2^2*L1*U1*L1^3*U2^3*L1*U1^3*L1^3 (due to Jeff Adams). - David Joyner ------------------------------ Date: Wed, 1 Oct 1997 14:13:50 -0400 (EDT) From: Nichael Cramer To: Edwin Saesen Cc: CUBE Subject: Re: 4x4x4 solution -- [Digest v23 #159] On Wed, 1 Oct 1997, Edwin Saesen wrote: > On my 4x4x4 I also had a problem of having two pairs of edges > exchanged which simply can't happen on a 3x3x3. I find it most convienent to think of this situation as the following: One of the "center-slices" containing one of the "swapped" edge pieces is rotated by 90 degrees. (This is roughly analogous to the the 3X case where the whole cube is solved except for two corners and two edge pieces being --respectively-- swapped. The problem is that the unfinished face is 90dg out of phase.) Rotate that center-slice by 90 degrees and re-solve from there. This is surely not the most efficient (i.e. shortest) solution; but it is conceptually straight forward. Nichael Cramer work: ncramer@bbn.com home: nichael@sover.net http://www.sover.net/~nichael/ ------------------------------ Return-Path: Date: Wed, 1 Oct 1997 16:23:14 -0400 From: Jim Mahoney To: ERCO@compuserve.com Cc: CUBE-LOVERS@ai.mit.edu Subject: Re: 4x4x4 solution -- [Digest v23 #159] >Your problem is one of parity. You have two edges cubies swapped >(this swap is visible) and two face center (centre) cubies of the >same color swapped (this swap is invisible). You have to have an Edwin> I've had this problem as well.... If I understand you Edwin> correctly, this problem simply doesn't occur anymore as Edwin> soon as you number (or mark in any other way) the center Edwin> pieces which a) makes solving the cube a bit more difficult Edwin> b) makes sure that you'll always get back to the original Edwin> configuration of center pieces. This isn't quite true, at least not on the 4x4x4. While it is true that parity is the question at hand, and also that on the 4x4x4 cube a quarter of a central slice performs an odd permutation on the edges which is otherwise "invisible", it is *not* true that marking the centers will help. The reason is that a quarter turn on a center slice of the 4x4x4 performs a cyclic rearrangement of 4 edges - an odd permutation - while at the same time rearranges *two* sets of 4 central pieces - an even permutation of the centers. Thus parity does not prohibit swapping two edges while leaving the centers untouched. Moreover, in fact there are move sequences which will exchange two edges without disturbing the position of any other piece, corner or center - though I don't have any on hand which are short. If there's interest, though, I can produce a move sequence to exchange two 4x4x4 edges while leaving all corners and centers in their original positions. A cross-section looks like this. A quarter turn cycles the four E's, the four C1's, and the four C2's. This is an odd permutation of the E's but an even permutation of the C's. (All the C's are corners, and can be put into each other's positions with a combination of face and center turns.) E C1 C2 E C2 C1 C1 C2 E C2 C1 E A full account of parity and possible 4x4x4 moves gives 4x4x4 type , how many , parity after: 1/4 face turn , 1/4 center turn - --------------------------------------------------------------------- corners | 8 | odd | even (untouched) edges | 24=2x(12 edges) | even (8 move) | odd centers | 24=4x(6 faces) | odd | even (8 move) Thus to solve a 4x4x4 cube you must have made both (1) an even total number of moves on the faces (to restore the corners and centers to even parity), as well as (2) an even total number of moves on the center slices, to restore the edges to even parity. The parity constraints on the 5x5x5 are a bit different. In that case there are two types of edges (the one in the middle of an edge vs the ones next to the corners) and three types of centers. Each has its own parity change under each different slice. A bit of playing around shows that any central slice move which cycles 4 edges must also cycle several kinds of centers. At least one of those center cycles is odd. Therefore on the 5x5x5 you cannot exchange a pair of edges without also exchanging two centers somewhere. So marking where the centers go will help on the 5x5x5. Regards, Jim Mahoney mahoney@marlboro.edu Physics & Astronomy Marlboro College, Marlboro, VT 05344 ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 17:46:05 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA26907; Wed, 1 Oct 1997 17:46:04 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Wed Oct 1 16:48:58 1997 From: SCHMIDTG@iccgcc.cle.ab.com Date: Wed, 1 Oct 1997 16:48:44 -0400 (EDT) To: cube-lovers@ai.mit.edu Cc: ljl@basmark.com Message-Id: <971001164844.2023493c@iccgcc.cle.ab.com> Subject: New cube program available I've recently finished my implementation of Kociemba's algorithm and it is now available from the cube-lovers ftp site at: ftp.ai.mit.edu /pub/cube-lovers/contrib/kcube1_0.zip The .zip files contains a README.TXT file, commented C++ source, and an executable program that runs on Win95/NT. Here's a brief description of the program that appears in the README file within the "contrib" directory: File: kcube1_0.zip Author: Greg Schmidt Description: A cube solver that implements Kociemba's algorithm. This program was written for the express purpose of understanding the algorithm in sufficient detail for me to implement it. The source code is included and commented with the hope of providing others with a similar understanding. I welcome feedback concerning any aspects of this program. Many thanks to Dik Winter and especially Herbert Kociemba for answering some of my detailed questions as well as allowing me to use their ideas and offer them to cube-lovers in the form of this program. Regards, -- Greg From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 19:01:21 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA28239; Wed, 1 Oct 1997 19:01:20 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From joemcg3@snowcrest.net Wed Oct 1 18:42:12 1997 Message-Id: <3432D0DB.4B19@snowcrest.net> Date: Wed, 01 Oct 1997 15:38:19 -0700 From: Joe McGarity Reply-To: joemcg3@snowcrest.net To: "Mailing List, Rubik's Cube" Subject: My Revenge is Complete How strange that I both have a broken Rubik's Revenge and need a piece to an Alexander's Star. What are the odds? I haven't looked at it for quite some time, but I think my Revenge is complete. The problem is the ball in the center. One of the corners is broken (if you have seen a dissasmbled RR it makes sense for a ball to have corners) and has resisted all attempts at being glued. So it may be that my broken cube will not be of any help to Darin Haines, but the rest of the cubies are intact if anyone needs any of them. Which brings up that I have a fairly large collection of broken or otherwise incomplete puzzles. I suspect that this is true for many of us. I would be more than willing to do some trading if anyone has any particular needs. Let me know. Joe McGarity From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 20:02:37 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA28462; Wed, 1 Oct 1997 20:02:36 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From randall@theory.lcs.mit.edu Wed Oct 1 19:49:18 1997 Date: Wed, 1 Oct 1997 19:46:08 -0400 Message-Id: <199710012346.TAA06162@hemp> From: Keith H Randall To: reid@math.brown.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <199710012120.AA25636@theory.lcs.mit.edu> (message from michael reid on Wed, 1 Oct 1997 17:19:02 -0400) Subject: Re: God's Number Don Dailey, Aske Plaat, and myself have a program that will do a complete 22-ply search in about 24 hours on an 8 processor Sun machine. The program measures distance in the QT (quarter-turn) metric. I've run some experiments on random cubes, summarized as follows: 112 random odd cubes: 20 depth 19 92 depth 21 57 random even cubes: 41 depth 20 16 depth 22 >From this random sample, it seems as if less than 1% of cubes are depth 23, let alone more than depth 24. In fact, the only depth 23 cubes I know of so far are the twelve cubes 1 move away from the superflip. This fact gives some evidence that God's number is probably 24. By the way, below are solutions and depths for all of the symmetric cubes enumerated by Hoey and Saxe in their message of Sun, 14 Dec 80. These are obvious cubes to try because they are local maxima, and they are all depth 22 or less except for the superflip. Only one representative from each of the 26 conjugacy classes is given. All solutions were obtained from the program, except for the superflip solution which is absconded from a post from Reid on Tue, 10 Jan 95. All depths are exact minimal depths, i.e. no shorter solutions exist. M-symmetric cubes 0 solved -- 12 pons asinorum F F B B L L R R U U D D 24 superflip R' U U B L' F U' B D F U D' L D D F' R B' D F' U' B' U D' 20 pons asinorum * superflip F' U' B' R' F R L' D' R L' U D' L' U D' F R B U F T-symmetric cubes 22 girdleflip F F U F F B' U R' L B U F D' F F B D' R L' B' D' F 19 girdleswap F U F R U' L' U' B U' B' R' F' R' L' F' R L L F' 21 girdleflip * girdleswap F U' L U F' U' B B D B U B' D' R D' R' B' R' D R B' 22 girdleflip * pons asinorum F F U L F L' D' R L' U' L L U U R F' B D' F' U R' D' 17 girdleswap * pons asinorum F R F B R' F' B' L D D F F D D R' L' F' 21 girdleflip * girdleswap * pons asinorum F R' L B R U' R R U' D F' R F' B L B R' F B' U' L' 20 girdleflip * superflip F U U F' R' U' L F' D F B' L U' L U' F' L U D' F 21 girdleswap * superflip F R F B U D' F' B R R U F B D' R L D' F' B' U F 21 girdleflip * girdleswap * superflip F U D B' R' F' D' R' U R' L' B R F U F D B D L' B' 20 girdleflip * pons asinorum * superflip F F B R' F U' B' R' L D L U' R' U' D F L B' D F 21 girdleswap * pons asinorum * superflip F U U B D' L' U F F B R' U R B U D' L B U D' L 21 girdleflip * girdleswap * pons asinorum * superflip F B U F' U' F R B' R' F' U R' U F B U' F' B' U R U' H-symmetric cubes 22 plummer F F R B' U L U R F L U' L L B R' D F D B L F D' 16 six-H F F R R F B' R R L L F' B R R B B 20 plummer * six-H F F U F' R' B' D' F' R U D L B' U' F' L' B' U' F F 20 plummer^2 * six-H F F U F R B U F R' U' D' L' B D F L B U' F F 20 plummer^2 * pons asinorum F R U F D' B B L F L' F' L F R R D' L U B L 20 plummer^2 * superflip F B U F R L' U' D' L U' D L B L' B' U F D L U 18 six-H * superflip F R' U D D F' B R F R' L D' F' R' L U L F' 22 plummer * six-H * superflip F U D F' R L F U D R R L L B' U D F' R L B' R' L 22 plummer^2 * six-H * superflip F B U F' B' D R' L' U F F B B R' L' D' F' B' D R' L' D' 22 plummer * pons asinorum * superflip F B R' U' D' R' F' B' R U' D' F F B B L U' D' R' F' B' L' reference for cube names: pons asinorum W B W B W B W B W O R O G Y G R O R Y G Y R O R Y G Y O R O G Y G O R O G Y G R O R Y G Y B W B W B W B W B superflip W Y W O W R W G W O W O G W G R W R Y W Y Y O G O G R G R Y R Y O O B O G B G R B R Y B Y B G B O B R B Y B plummer Y W Y W W W G W G W O W O G R W R W R Y O O O O G G G R R R Y Y Y B O B O G R B R B R Y O G B G B B B Y B Y six-H W W W B W B W W W O O O G Y G R R R Y G Y R O R G G G O R O Y Y Y O O O G Y G R R R Y G Y B B B W B W B B B girdle flip (about ULF-DRB axis) W Y W W W R W W W O O O G G G R W R Y W Y Y O O G G R G R R Y Y O O B O G B G R R R Y Y Y B G B O B B B B B girdle swap (about ULF-DRB axis) R B B W W B W W Y B O O G G B O O G O G G R O O G G Y O R R Y Y G R R Y R Y Y W R R Y Y W W W O W B B G B B -Keith From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 20:49:17 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA28649; Wed, 1 Oct 1997 20:49:16 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From roger.broadie@iclweb.com Wed Oct 1 19:12:26 1997 From: roger.broadie@iclweb.com (Roger Broadie) To: Subject: Re: 4x4x4 solution Date: Thu, 2 Oct 1997 00:09:47 +0100 Message-Id: <19971002000735.AAA23683@home> I'm tempted to try a little more analysis of the parity constraints on the 4x4x4 cube, though no doubt it's all been done before. As der Mouse said, A slice turn produces a 4-cycle on the edges and two 4-cycles on the face centres; a face turn produces a 4-cycle on the face centres and two 4-cycles on the edges (and a 4-cycle on the corners, which may or may not be relevant). I think it is very relevant. We can set the effects out as follows: Turn Piece Cycle(s) Parity ------ ------- -------- ------- Slice edge 1x4 odd centre 2x4 even Face edge 2x4 even centre 1x4 odd corner 1x4 odd The consequence is that the parity of the centre pieces depends entirely on the number of face turns - any slice turns do not affect the parity of these pieces since the changes they introduce will be of even parity. For face turns, the changes to the parity of the corner pieces and the centre pieces are the same. Hence if the corner pieces are in place, the centres will be in an even permutation, and that will not be changed even if the edge pieces are in an odd permutation, which was the essence of Clive McCaig's original question. Nor will that be changed by any turn of a central slice to bring them back to an even permutation. I the corners are correct (which I guess is the normal situation when the problem with the swapped edge pieces shows up) then, though I say so with some hesitation, I do not think Jerry Bryan is right in saying that the pair of swapped edge pieces will be matched by a pair of swapped centre pieces. For example, the process I quoted switches edge pieces, and though it has no visible effect on the centre pieces, it does in fact change the positions of the centre pieces on the front face (if I have correctly identified the results of a bit of hasty work with little Post-it stickers). However, the whole block of four rotates through 180 degrees, which is two 2-cycles and thus of even parity. Edwin Saesen could mark the centre pieces, get them back to their original position and still find the edge pieces swapped, but that will not prevent his correcting the edge pieces, and then, if he wants to, correcting the centre pieces with even-parity processes. Luckily, for the 4x4x4, we do not have to worry about twists for the edge pieces or the centre pieces, since that is fixed geometrically for each position they can occupy. When an edge piece is in its home position it must be the right way round. When it moves to its next-door position it must flip. I imagine this is the point behind Allan Wechsler's charming square-dancing analogy. The centre pieces always present the same corner to the central intersection of the face. Roger Broadie From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 19:55:23 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA29256; Mon, 6 Oct 1997 19:55:23 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Goyra@iol.ie Wed Oct 1 20:07:28 1997 Message-Id: <199710020007.BAA10545@GPO.iol.ie> From: "Goyra (David Byrden)" To: Subject: For all cube programmers Date: Thu, 2 Oct 1997 01:02:23 +0100 When writing a program to manipulate the Cube, you're interested in your algorithm. The output usually looks like RLULRURL because you won't waste time programming any graphics. I will shortly release a freeware software component that displays a standard Rubik's Cube. You can incorporate it into your software and manipulate the cube directly. See your cube solutions executed in front of your eyes. For an idea of what this component will look like, take a Java browser to my pages at http://www.iol.ie/~goyra/Rubik.html The component will be a Java Bean, meaning you can use it in Java, and also in any Activex environment such as Visual C++ or Visual Basic. Anyone with suggestions about how the programmatic interface to the component should look, please mail me. David From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 21:04:33 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA29479; Mon, 6 Oct 1997 21:04:32 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Cube-Lovers-Request@ai.mit.edu Mon Oct 6 21:02:21 1997 Date: Mon, 6 Oct 1997 21:02:21 -0400 (EDT) Message-Id: <06Oct1997.210221.Cube-Lovers@AI.MIT.EDU> From: Cube Lovers Moderator To: Cube-Lovers@AI.MIT.EDU Subject: 4x4x4 solution -- [Digest v23 #170] Cube-Lovers Digest Mon, 6 Oct 1997 Volume 23 : Issue 170 Today's Topic: 4x4x4 solution ---------------------------------------------------------------------- Date: Wed, 1 Oct 1997 23:26:34 -0400 (EDT) From: Nichael Cramer To: Roger Broadie Cc: Cube-Lovers@ai.mit.edu Subject: Re: 4x4x4 solution Message-Id: Roger Broadie wrote: > A slice turn produces a 4-cycle on the edges and two 4-cycles on the > face centres; a face turn produces a 4-cycle on the face centres and > two 4-cycles on the edges (and a 4-cycle on the corners, which may or > may not be relevant). > > I think it is very relevant. We can set the effects out as follows: > > Turn Piece Cycle(s) Parity > ------ ------- -------- ------- > > Slice edge 1x4 odd > centre 2x4 even > > Face edge 2x4 even > centre 1x4 odd > corner 1x4 odd > > The consequence is that the parity of the centre pieces depends > entirely on the number of face turns - any slice turns do not affect > the parity of these pieces since the changes they introduce will be of > even parity. For face turns, the changes to the parity of the corner > pieces and the centre pieces are the same. Hence if the corner pieces > are in place, the centres will be in an even permutation, and that will > not be changed even if the edge pieces are in an odd permutation, which > was the essence of Clive McCaig's original question. Nor will that be > changed by any turn of a central slice to bring them back to an even > permutation. As one of the folks who advocated rotating a center slice, let me explain my (admittedly non-optimal) process for getting out of this fix and perhaps you can explain where my reasoning is wrong. 1] Imagine a 4X which is completely solved except for two flipped (i.e. swapped) edge-pieces. 2] For simplicity's sake --and without loss of generality--, assume the 2 flipped/swapped pieces are adjacent and in the top front location. So the top of the cube will look like this: X X X X X X X X X X X X X 1 2 X (Here the numbers are meant to indicate only where the cubies are located, having nothing to do with their colors.) 3] I now rotate one of the center slices (say, the one on the right, i.e. the one containing the cubie "2") 90dg away from me. 4] The top of the cube now looks like: X X 2 X X X O X X X O X X 1 3 X 5] I can now perform the 3-cycle 1->3->2 (i.e. without affecting any of the rest of the cube). The top of the cube now looks like: X X 3 X X X O X X X O X X 2 1 X 6] In particular, note that "2" and "1" are now in their correct positions (and, of course, necessarily in their proper "flip" orientation). 7] Moreover, note that I now have exactly three edge cubes in the wrong place (i.e. "3" from above and the other two edge cubes which were misplaced during my original 90dg rotation of the center slice). I can now perform a 3-cycle on these edges pieces (similar to the one used in step 5 above) again without affecting any of the other locations on the cube. 8] My cube now has all the edge pieces in their correct location. 9] I now have only to "fix" the 8 central-face cubes which were misplaced during my initial 90dg twist. I can now do this is short order. QED[?] Nichael Cramer work: ncramer@bbn.com home: nichael@sover.net http://www.sover.net/~nichael/ ------------------------------ From: roger.broadie@iclweb.com (Roger Broadie) To: "Nichael Cramer" Cc: Subject: Re: 4x4x4 solution Date: Thu, 2 Oct 1997 23:12:39 +0100 > From: Nichael Cramer > To: Roger Broadie > Cc: Cube-Lovers@ai.mit.edu > Subject: Re: 4x4x4 solution > Date: 2 October 1997 4:26 > > As one of the folks who advocated rotating a center slice, let me > explain my (admittedly non-optimal) process for getting out of this > fix and perhaps you can explain where my reasoning is wrong. >[followed by a procedure in which a quarter turn of a centre slice is followed, first, by a 3-cycle of edges on the top to restore the two swapped pieces, second, by a 3-cycle of edges to restore the other displaced edges, and, third, by restoring the displaced centres] I absolutely agree with your reasoning. A quarter turn of a central slice must be at the heart of any procedure to perform an edge swap, because it is the only way to change the parity of the edges. That was what I said in my first post on 1 October 1997. In my second post I was trying to look at the effect of that quarter turn of the central slice on the centre pieces, and show that, as they had been subjected to an even permutation by reason of the centre-slice turn, the centre pieces could not have undergone an invisible swap of a single pair of centre pieces. Having made a single quarter turn of the central slice, all the other edge and centre pieces can be restored with processes of even parity, like your two 3-cycles. Roger Broadie ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 22:28:22 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA29717; Mon, 6 Oct 1997 22:28:22 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Cube-Lovers-Request@ai.mit.edu Mon Oct 6 22:27:32 1997 Date: Mon, 6 Oct 1997 22:27:32 -0400 (EDT) Message-Id: <06Oct1997.222732.Cube-Lovers@AI.MIT.EDU> From: Cube Lovers Moderator To: Cube-Lovers@AI.MIT.EDU Subject: 4x4x4 solution -- [Digest v23 #171] Cube-Lovers Digest Mon, 6 Oct 1997 Volume 23 : Issue 171 Today's Topic: Pieces of broken cubes: Rubik's Revenge (Clarified) My Revenge is Complete Piece for a Rubik's Revenge Piece for Alexander's Star ---------------------------------------------------------------------- Date: Thu, 02 Oct 1997 08:01:03 -0700 From: Darin Haines To: Cube Subject: Rubik's Revenge (Clarified) I guess my terminology was incorrect. The parts I need are actually the center cubies of which there are 4 on each side (for a total of 24). It sounds like Joe McGarity's broken RR will help me out just fine (as will a couple of other responses I've received). I got to looking last night and found that I actually need 3 (not just 1) of these center cubies. - -Darin ------------------------------ To: cube-lovers@ai.mit.edu From: "Bryan Main" Subject: Re: My Revenge is Complete Date: Thu, 02 Oct 1997 14:32:15 Eastern Daylight Time At 03:38 PM 10/1/97 -0700, you wrote: >I haven't looked at it for quite some time, but I think my Revenge is >complete. How stable are the 4x4x4 and 5x5x5? I was thinking on getting one but they cost quite a lot of money and was wondering how easy it is to break them. Also what kind of paint should I use to paint my cubes as I have 4 normal ones and would like to make different patterns on them to make them more intersting. Also any patterns would be helpful. bryan __________________________________________________________________ Bryan Main Cartographic Specialist http://caddscan.com ------------------------------ Date: Thu, 2 Oct 1997 22:42:11 -0400 (EDT) From: Nicholas Bodley To: Darin Haines Cc: Cube Subject: Re: Piece for a Rubik's Revenge On Wed, 1 Oct 1997, Darin Haines wrote: {Snips} }Did anyone else have problems with the center pieces breaking on their }RR? or am I the only one? These pieces >are< rather fragile, as I remember. |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* Waltham is now in the new 781 area code. |* Amateur musician *|* 617 will be recognized until 1 Dec. 1997. ------------------------------ Date: Sun, 5 Oct 1997 18:40:17 -0400 (EDT) From: Nicholas Bodley To: David Bagley x21081 Cc: Cube-Lovers@ai.mit.edu Subject: Re: Piece for Alexander's Star They do break easily. I haven't had mine out of storage for some time, but I well remember that it needed conscious care when manipulating; nothing like a properly-lubricated deluxe Ideal 3^3 (the one with plastic color tiles, and changes to the shapes of the pieces that tend to make it self-align). |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* Waltham is now in the new 781 area code. |* Amateur musician *|* 617 will be recognized until 1 Dec. 1997. ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 23:26:59 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA29891; Mon, 6 Oct 1997 23:26:59 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Cube-Lovers-Request@ai.mit.edu Mon Oct 6 23:26:19 1997 Date: Mon, 6 Oct 1997 23:26:19 -0400 (EDT) Message-Id: <06Oct1997.232619.Cube-Lovers@AI.MIT.EDU> From: Cube Lovers Moderator To: Cube-Lovers@AI.MIT.EDU Subject: God's number -- [Digest v23 #172] Cube-Lovers Digest Mon, 6 Oct 1997 Volume 23 : Issue 172 Today's Topic: God's Number ---------------------------------------------------------------------- Date: Thu, 02 Oct 1997 17:04:33 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: God's Number To: Keith H Randall Cc: reid@math.brown.edu, cube-lovers@ai.mit.edu Message-Id: On Wed, 1 Oct 1997, Keith H Randall wrote: > Don Dailey, Aske Plaat, and myself have a program that will do a > complete 22-ply search in about 24 hours on an 8 processor Sun > machine. The program measures distance in the QT (quarter-turn) > metric. > > I've run some experiments on random cubes, summarized as follows: > > 112 random odd cubes: > 20 depth 19 > 92 depth 21 > > 57 random even cubes: > 41 depth 20 > 16 depth 22 Wow. I am impressed with how much data you have. For the case of random cubes and guaranteed optimal solutions, I believe this is the most data which has been posted to Cube-Lovers. It would be nice to examine enough cases to raise the probability that a few positions of length 17q would show up for odd cubes and of length 18q for even cubes. At this distance from Start, the branching factor for one level is about 9.3, so the branching factor for two levels (e.g., between level 17 and level 19) would be about 85 or so. So you are just at the edge of the sample size where you would expect the shorter lengths to show up. Notwithstanding that, I decided to play with the numbers to see if I could make any reasonable projection about the overall distribution of lengths in the quarter-turn metric. Here is what I have come up with. Consider the 19q case. Your results suggest that about 17.8% of odd positions, and hence about 8.6% or 8.7% of all positions are exactly 19q from Start. (The sample size does not support an estimate of that precision, of course, but let's continue anyway). It's easy to calculate that no more than about 8.4% of positions can be 19q from Start. From this, I would conclude two things. First, your results seem right on, well within the bounds of sampling error. Second, your results suggest that it is very unlikely that the branching factor drops below about 9.3 until you pass 19q from Start. Using the best available known results, plus using your results as an estimate, plus some other guessing, I would propose that the actual search tree for the q-turn case looks something like the following. Distance Number Branching Cumulative from of Factor Number of Start Positions Positions 0 1 1 1 12 12.000 13 2 114 9.500 127 3 1068 9.368 1195 4 10011 9.374 11206 5 93840 9.374 105046 6 878880 9.366 983926 7 8221632 9.355 9205558 8 76843595 9.347 86049153 9 717789576 9.341 803838729 10 6701836858 9.337 7505675587 11 62549615248 9.333 70055290835 12 5.838E+11 9.333 6.538E+11 13 5.449E+12 9.333 6.102E+12 14 5.085E+13 9.333 5.696E+13 15 4.746E+14 9.333 5.316E+14 16 4.430E+15 9.333 4.961E+15 17 4.134E+16 9.333 4.631E+16 18 3.859E+17 9.333 4.322E+17 19 3.601E+18 9.333 4.034E+18 20 1.546E+19 4.294 1.950E+19 21 1.657E+19 1.071 3.606E+19 22 6.035E+18 0.364 4.210E+19 23 12 0.000 4.210E+19 24 1 0.083 4.210E+19 Notice that my table does not quite reach |G|, so there are probably a few more positions than this at 20q, 21q, and 22q from Start (there can't be more any closer to Start than that). Also, the branching factor probably does not remain constant at 9.333 all the way out to 19q from Start; it probably declines slightly, maybe to 9.300 or so. Finally, the distribution is probably bimodal, with modes at 20q and 21q (it almost has to be bimodal because of odd/even parity considerations). (By the way, I am making no claim whatsover that the diameter of the cube group is 24q. This is only an educated guess based on the evidence at hand. In fact I tend to doubt it. I think the branching factor in the chart just drops off too sharply at levels 21q, 22q, and 23q for the chart to be real.) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 ------------------------------ Date: Sun, 5 Oct 1997 18:54:32 -0400 From: michael reid To: cube-lovers@ai.mit.edu, randall@theory.lcs.mit.edu Subject: Re: God's Number keith randall writes > Don Dailey, Aske Plaat, and myself have a program that will do a > complete 22-ply search in about 24 hours on an 8 processor Sun > machine. The program measures distance in the QT (quarter-turn) > metric. wow, that's quite a bit faster than my optimal solver! how about searches through other depths (20q, 21q, 23q, ... )? does the run time depend upon the input position? could you describe your searching algorithm? i'm sure that this would be of interest to many people on the cube-lovers mailing list. > By the way, below are solutions and depths for all of the symmetric > cubes enumerated by Hoey and Saxe in their message of Sun, 14 Dec 80. i already posted data for these positions, but it's always nice to have confirmation. however, ... > 22 girdleflip * pons asinorum > F F U L F L' D' R L' U' L L U U R F' B D' F' U R' D' this is solvable in 18q: ) 3. U R U' F D R L' B' L' F R F B' U' L' D B' D' (18q, 18f) although i gave it in a different orientation. > 22 plummer * six-H * superflip > F U D F' R L F U D R R L L B' U D F' R L B' R' L there's a slight typo here; the last twist should be L' . mike ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 7 13:06:59 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA03233; Tue, 7 Oct 1997 13:06:58 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nichael@sover.net Tue Oct 7 00:45:56 1997 Message-Id: In-Reply-To: <06Oct1997.222732.Cube-Lovers@AI.MIT.EDU> Date: Mon, 6 Oct 1997 23:03:54 -0400 To: Cube-Lovers@ai.mit.edu From: Nichael Lynn Cramer Subject: Re: 4x4x4 solution -- [Digest v23 #171] Cc: bmain@caddscan.com [ Moderator's note--The subject is misleading, because I erroneously titled Digest v23 #171 as "4x4x4 solutions". It was actually about "Pieces of broken cubes"--the discussion of breakability and the idea of trading pieces of cubes. I regret the error. ] >To: cube-lovers@ai.mit.edu >From: "Bryan Main" >Subject: Re: My Revenge is Complete >How stable are the 4x4x4 and 5x5x5? I was thinking on getting one but they >cost quite a lot of money and was wondering how easy it is to break them. 5Xs are pretty stable; each side has a fixed center piece (i.e. like a 3X). I've had three and never had any problem with any of them. 4Xs are another ballgame altogether. Since they don't have a fixed center, they depend on an internal configuration, consisting of a cluster of four plates, to hold the faces on. Each of these plates is held on with a screw and this adjustment is _critical_. Too tight and it can be all but impossible to twist the faces; too loose and the cube tends to dissolve in your hands. I've owned four; one was fine, one was OK/usable, one was too stiff to use and one couldn't be kept together. So, the "usability" rate was approx 1/3. (OTOH I picked them all up for $2/ea at a ToysRU clearance...) Nichael nichael@sover.net 6.501 http://www.sover.net/~nichael/ -- the ln of the Beast From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 7 17:04:52 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA06610; Tue, 7 Oct 1997 17:04:51 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Tue Oct 7 16:59:58 1997 Date: Tue, 07 Oct 1997 16:59:25 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Maximality Analysis Through 11q To: cube-lovers@ai.mit.edu Message-Id: Not too long ago, I reported that my Shamir program had completed searching through 11q from Start, that the results did confirm my previous results using tape spinning programs, that no local maxima were found 11q from Start, and that otherwise nothing new was found. I have come to realize that there is a small bit of new information. I really should post the maximality analysis in its entirety, because the whole row 11q from Start is new. The row 11q from Start does include the failure to find any new local maxima. As always, the local maxima are in the right-most column, where all 12 moves go closer to Start. Maximalility Analysis In Terms of Patterns (M-conjugacy classes) Number of Moves which go Closer to Start 1 1 1 0 1 2 3 4 5 6 7 8 9 0 1 2 |x| 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 2 3 0 0 0 0 0 0 0 0 0 0 3 0 20 4 1 0 0 0 0 0 0 0 0 0 4 0 182 34 2 1 0 0 0 0 0 0 0 0 5 0 1677 280 20 1 0 0 0 0 0 0 0 0 6 0 15642 2561 184 8 0 0 0 0 0 0 0 0 7 0 145974 23773 1721 61 0 0 0 0 0 0 0 0 8 0 1362579 222235 16241 663 1 3 0 3 0 0 0 0 9 0 12719643 2077549 153026 5954 74 15 2 3 0 0 0 0 10 0 118711701 19418503 1438825 58862 925 318 11 37 0 8 0 4 11 0 1107594690 181433604 13517370 576891 11843 3442 251 321 10 21 2 0 Maximalility Analysis In Terms of Positions Number of Moves which go Closer to Start 0 1 2 3 4 5 6 7 |x| 0 1 0 0 0 0 0 0 0 1 0 12 0 0 0 0 0 0 2 0 96 18 0 0 0 0 0 3 0 912 144 12 0 0 0 0 4 0 8544 1368 96 3 0 0 0 5 0 80088 12816 912 24 0 0 0 6 0 749376 120612 8640 252 0 0 0 7 0 7001712 1135104 82152 2664 0 0 0 8 0 65391504 10645824 777936 28200 48 56 0 9 0 610499652 99666528 7338720 280800 3048 624 96 10 0 5698027296 931905180 69049264 2796978 43800 12336 528 11 0 53164171632 8708296416 648777868 27618360 563880 159024 11904 1 1 1 8 9 0 1 2 |x| 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 6 0 0 0 0 0 7 0 0 0 0 0 8 27 0 0 0 0 9 108 0 0 0 0 10 1296 0 138 0 42 11 14856 408 828 72 0 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 8 12:48:49 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA12652; Wed, 8 Oct 1997 12:48:47 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From dzander@solaria.sol.net Tue Oct 7 19:21:35 1997 From: Douglas Zander Message-Id: <199710072320.SAA09034@solaria.sol.net> Subject: broken rubik's cube: help! To: cube-lovers@ai.mit.edu (cube) Date: Tue, 7 Oct 97 18:20:57 CDT Hello, I wonder if someone can suggest a way to fix my 3x3x3 cube. The screw that holds a center cubie to the spindle has stripped out of the spindle. I thought of just super-glueing it back in; would this work? Also, I wonder if there is to be any tension (compression) on the spring inside the center cubie when the screw is set in? I'm afraid that I will have to open up the center cubie and use a driver to screw the cube together again. I don't want to pry open my center cubie. Thanks for any suggestions. -- Douglas Zander | dzander@solaria.sol.net | Milwaukee, Wisconsin, USA | From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 14 12:51:26 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA15380; Tue, 14 Oct 1997 12:51:25 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Mon Oct 13 16:19:54 1997 Date: Mon, 13 Oct 1997 16:18:30 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: God's Number In-Reply-To: <3.0.32.19970930192820.006ce8ac@po9.mit.edu> To: Dennis Okon Cc: cube-lovers@ai.mit.edu Message-Id: On Tue, 30 Sep 1997, Dennis Okon wrote: > I just found out that Keith Randall for the theory group of LCS (Lab for > Computer Science) at MIT gave a talk Monday about God's number for the > rubik's cube. He upped the lower bound 24 and gave "evidence" that it is > 24. I don't know what moves he was counting (e.g. slice, quarter). > Unfortunately, I missed it. Does anyone have any information on this? > I'll see what I can find out. Was there ever any more information on this? The lower bound for the diameter of the cube group was raised to 24q on 19 February 1995. I would be very surprised if Keith Randall presented a position requiring 24f. I don't know of any published results in metrics which include both slice and face turns. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 14 18:44:16 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA18461; Tue, 14 Oct 1997 18:44:15 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From fb91@dial.pipex.com Tue Oct 14 08:00:33 1997 Message-Id: <199710141200.IAA10374@life.ai.mit.edu> From: "Richard Armitage" To: Subject: VRML puzzles/newsletter Date: Tue, 14 Oct 1997 12:59:59 +0100 We are shortly to create a full VRML site of cubes, and other similar puzzles a la Rubiks and spacecubes. It will contain both free and for sale items and will evolve as demand requests from people like you!! I am going to be publishing a monthly newsletter from November 1997, covering 3D puzzles (real and virtual) and SpaceCubes news. You can sign up from the first page of our website or by sending an e-mail to info@spacecubes.com with the subject newsletter. You will receive a SpaceCubes info standard letter for now until we set up all the right autoresponses but I wiil happily deal with all feedback. Thankyou and looking forward to giving you good challenges Richard Richard Armitage (SpaceCubes Marketing) tel:44 191 281 6011 US fax 2125048016 autorespond: or From cube-lovers-errors@mc.lcs.mit.edu Thu Oct 23 12:00:22 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA14162; Thu, 23 Oct 1997 12:00:22 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From whuang@ugcs.caltech.edu Thu Oct 23 07:05:58 1997 To: Cube-Lovers@AI.MIT.Edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Magic Make-a-cube Date: 23 Oct 1997 11:05:18 GMT Organization: California Institute of Technology, Pasadena Message-Id: <62nb1e$q4a@gap.cco.caltech.edu> I have just acquired what appear to be the components to Rubik's Magic Make-a-cube. Unfortunately, two color paper pieces are missing. Can anyone tell me what the color arrangements are, and in what order? Thanks for anything you can dig up. -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- "[Lucy's eyes] look like little round dots of India ink..." -- Charlie Brown From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 11:43:29 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA11742; Fri, 31 Oct 1997 11:43:29 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From MO374@cnsvax.albany.edu Wed Oct 29 11:06:48 1997 Date: Wed, 29 Oct 1997 11:02:53 -0500 (EST) From: Mary Osielski Subject: Where to buy one??? To: cube-lovers@ai.mit.edu Message-Id: <01IPDKFAISLE90NIU9@cnsvax.albany.edu> I'm trying to buy a regular, standard, run-of-the-mill Rubik's cube which I now realize is not so easy. Can you please direct me to a source? Are they no longer produced? I got your address from the mountains of material on the Internet about Rubik. Is there a store, a phone number, a person from whom I can buy one. I'm in Albany, NY but mail-order is fine. Thanks in advance for the help! Mary Osielski mo374@cnsvax.albany.edu From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 12:09:35 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA11837; Fri, 31 Oct 1997 12:09:33 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From mouse@Rodents.Montreal.QC.CA Thu Oct 30 12:30:33 1997 Date: Thu, 30 Oct 1997 12:29:32 -0500 (EST) From: der Mouse Message-Id: <199710301729.MAA08700@Twig.Rodents.Montreal.QC.CA> To: cube-lovers@ai.mit.edu Subject: Re: A* versus IDA* It's a little off-topic and rather old (June 1st) anyway, so I'll make this quick: > [...discussion of FreeCell and "Baker's game"...] Could someone interested in these contact me? I'd like to learn more about Baker's game, whatever that is, and discuss some empirical results with with Seahaven. der Mouse mouse@rodents.montreal.qc.ca 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 12:45:42 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA12020; Fri, 31 Oct 1997 12:45:41 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From mouse@Rodents.Montreal.QC.CA Thu Oct 30 19:12:43 1997 Date: Thu, 30 Oct 1997 19:11:51 -0500 (EST) From: der Mouse Message-Id: <199710310011.TAA10960@Twig.Rodents.Montreal.QC.CA> Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 8bit To: cube-lovers@ai.mit.edu Subject: Re: Categorization of cube solving programs This is a response to a pretty old message: > Date: Thu, 5 Jun 1997 22:56:56 -0400 (EDT) However, I kept the message around, which usually means I never did anything with it. If I already did, my apologies to the list for duplication. > Since I'm interested in such things, I came up with the following > categories of cube solving programs in general order of increasing > sophistication: > Class 1: Simply provide a simulation of the cube and allow the > user to manipulate the cube model [...]. Often these > programs have very nice 3D graphics. > Class 2: A program which solves the cube by implementing a > canned algorithm (or 'book procedure'). [...] > Class 3: A program that when given a specific instance of the > cube, attempts to 'discover' or learn a sequence which > will solve that particular instance. [eg, Kociemba] > Class 4: A program which attempts to discover an ALGORITHM to > solve ALL randomized cubes. [...] Korf wrote a > program to do this in the mid 1980s. [Such programs > generally produce Class-2-ish solutions.] I believe > Korf's program is the only program ever achieved that > can be placed in this category. I wish to speak to the last sentence of the Class 4 description. Back in my larval stage (mid-'80s), someone at a lab I worked for build a Class 4 program in Franz Lisp. It wasn't fast, but that was probably because it had nothing more than a VAX-11/780 to run on. (I remember it particularly as it was one of the most impressive pieces of hot-spot optimization I ever did; replacing about 20 lines of Lisp with about 20 lines of assembly got a speedup of between two and three orders of magnitude overall.) I have no idea whether the program still exists in any form. I do believe I can still reach its author, if anyone would like me to inquire. der Mouse mouse@rodents.montreal.qc.ca 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 21:19:56 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA13919; Fri, 31 Oct 1997 21:19:55 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From mouse@Rodents.Montreal.QC.CA Thu Oct 30 19:28:57 1997 Date: Thu, 30 Oct 1997 19:28:16 -0500 (EST) From: der Mouse Message-Id: <199710310028.TAA11074@Twig.Rodents.Montreal.QC.CA> To: cube-lovers@ai.mit.edu Subject: Re: 5x5x5 Stuctural Integrtity > Where are you able to find 5x5x5 cubes that don't instantly fall > apart? I've owned only one 5-Cube and have had no mechanical problem with it at all. I bought it in mid-December 1993, but unfortunately I don't know where it came from. I probably got it at a retail toy/game store here in the city called Valet de Coeur ("Jack of Hearts" in French), but (a) am not sure of even that by now (though I have trouble imagining where else might have had it) and (b) I have no idea where it was made or what distributor they got it from. > The orange stickers seem to have a habit of fleeing the cube in > terror. (It's always the orange ones on any cube that fall off > first. Has anyone else noticed this?) I sure have, with my 5-Cube. Three of the 25 have come off, and one has been completely lost (the other two are attached to the cube with a piece of masking tape, pending my doing something more permanent). I may do to it what I did to one of my 3-Cubes recently: take all the stickers off and use plastic-model paint to color the cubies. (I actually may do this to just the orange face, since that's the only problematic one.) der Mouse mouse@rodents.montreal.qc.ca 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 22:06:26 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA14062; Fri, 31 Oct 1997 22:06:26 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jferro@knave.ece.cmu.edu Fri Oct 31 13:50:54 1997 Date: Fri, 31 Oct 1997 13:50:10 -0500 From: "Jonathan R. Ferro" Message-Id: <199710311850.NAA26736@knave.ece.cmu.edu> Organization: Electrical and Computer Engineering, CMU To: cube-lovers@ai.mit.edu In-Reply-To: <01IPDKFAISLE90NIU9@cnsvax.albany.edu> (message from Mary Osielski on Wed, 29 Oct 1997 11:02:53 -0500 (EST)) Subject: Re: Where to buy one??? "Mary" == Mary Osielski writes: Mary> I'm trying to buy a regular, standard, run-of-the-mill Rubik's Mary> cube which I now realize is not so easy. Can you please direct me Mary> to a source? Are they no longer produced? There has been a new run (I'm not sure if it's by Ideal or not), and I saw two on the shelf under the Lego Brand Construction Blocks (tm) (Note to self: kill the lawyers) at K-Mart just last week. -- Jon From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 1 22:17:34 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA19000; Sat, 1 Nov 1997 22:17:33 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Sat Nov 1 20:01:52 1997 From: SCHMIDTG@iccgcc.cle.ab.com Date: Sat, 1 Nov 1997 20:01:05 -0500 (EST) To: cube-lovers@ai.mit.edu Message-Id: <971101200105.20201302@iccgcc.cle.ab.com> Subject: Re: Categorization of cube solving programs "der Mouse" wrote: >This is a response to a pretty old message: >> Since I'm interested in such things, I came up with the following >> categories of cube solving programs in general order of increasing >> sophistication: >[...Class 1 through Class 2...] > Class 3: A program that when given a specific instance of the > cube, attempts to 'discover' or learn a sequence which > will solve that particular instance. [eg, Kociemba] > Class 4: A program which attempts to discover an ALGORITHM to > solve ALL randomized cubes. [...] Korf wrote a > program to do this in the mid 1980s. [Such programs > generally produce Class-2-ish solutions.] I believe > Korf's program is the only program ever achieved that > can be placed in this category. In retrospect, Class 4 programs are not necessarily more sophisticated than Class 3 programs especially when one considers that the latter should be be able to produce a macro-table solution by solving for a sufficient set of specific sequences. Perhaps, I'm overly fascinated by a learning program which, in essence, outputs a solving program but I don't want to discount the fact that there are some very interesting and sophisticated Class 3 programs out there. Richard Korf points out a suggestion by Jon Bently that the learning program can be be interleaved with the solving program, as co-routines, and only running the learning program when a new macro is needed to solve a particular problem instance. Thus, the specific entries required in the macro-table do not have to be planned out in advance. >I wish to speak to the last sentence of the Class 4 description. Back >in my larval stage (mid-'80s), someone at a lab I worked for build a >Class 4 program in Franz Lisp. It wasn't fast, but that was probably >because it had nothing more than a VAX-11/780 to run on. (I remember >it particularly as it was one of the most impressive pieces of hot-spot >optimization I ever did; replacing about 20 lines of Lisp with about 20 >lines of assembly got a speedup of between two and three orders of >magnitude overall.) >I have no idea whether the program still exists in any form. I do >believe I can still reach its author, if anyone would like me to >inquire. It would be interesting to compare the approach of this program to Korf's learning program. If the program is still available I suggest it would make a quite excellent addition to the cube lovers archive. Regards, -- Greg Schmidt From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 12:42:36 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA29514; Mon, 3 Nov 1997 12:42:36 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From mouse@Rodents.Montreal.QC.CA Sun Nov 2 06:52:25 1997 Date: Sun, 2 Nov 1997 06:51:36 -0500 (EST) From: der Mouse Message-Id: <199711021151.GAA27954@Twig.Rodents.Montreal.QC.CA> To: cube-lovers@ai.mit.edu Subject: Re: Categorization of cube solving programs >> Class 3: A program that when given a specific instance of the >> cube, attempts to [solve it] [eg, Kociemba] >> Class 4: A program which attempts to [find an algorithm to solve >> arbitrary cubes]. > In retrospect, Class 4 programs are not necessarily more > sophisticated than Class 3 programs especially when one considers > that the latter should be be able to produce a macro-table solution > by solving for a sufficient set of specific sequences. Sure...but who picks the specific instances for them? > Richard Korf points out a suggestion by Jon Bently that the learning > program can be be interleaved with the solving program, as > co-routines, and only running the learning program when a new macro > is needed to solve a particular problem instance. This means that the solving program has to imagine macros, try to choose a useful one, determine whether it's actually possible (you gotta keep the program from trying to produce, for example, a single edge flipper). You also have to decide when it's worth trying for a macro and when it's better to just hit the (sub)problem with brute force. I would expect all these problems to be quite hard. >> I wish to speak to the last sentence of the Class 4 description. >> Back in my larval stage (mid-'80s), someone at a lab I worked for >> build a Class 4 program in Franz Lisp. [...] >> I have no idea whether the program still exists in any form. I do >> believe I can still reach its author, if anyone would like me to >> inquire. > It would be interesting to compare the approach of this program to > Korf's learning program. If the program is still available I suggest > it would make a quite excellent addition to the cube lovers archive. I'll send off a missive to the author. der Mouse mouse@rodents.montreal.qc.ca 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 13:18:15 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA29659; Mon, 3 Nov 1997 13:18:15 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Sun Nov 2 20:32:15 1997 Date: Sun, 2 Nov 1997 20:31:16 -0500 (EST) From: Nicholas Bodley To: der Mouse Cc: cube-lovers@ai.mit.edu Subject: 5^3 orange stickers In-Reply-To: <199710310028.TAA11074@Twig.Rodents.Montreal.QC.CA> Message-Id: I think these might be made of plastic instead of paper, and they seem to have a different adhesive. I thought mine were loose because I had tried several lubricants on my cube, and the lube. had interacted with the adhesive; apparently not. Someone who's smart with solvents might be able to remove all the adhesive, and reattach them with a better adhesive. CA ("Krazy Glue"; cyanoacrylate) might be good, as might plastic-model cement. However, one should be careful; a 5^3 is not something to mistreat! My regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* Waltham is now in the new 781 area code. |* Amateur musician *|* 617 will be recognized until 1 Dec. 1997. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 14:03:20 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA29888; Mon, 3 Nov 1997 14:03:20 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From bmain@caddscan.com Mon Nov 3 10:09:08 1997 To: cube-lovers@ai.mit.edu From: "Bryan Main" Subject: Re: Where to buy one??? Date: Mon, 03 Nov 1997 10:07:11 EST Message-Id: <19971103100711.0054df7a.in@caddscan.com> At 01:50 PM 10/31/97 -0500, you wrote: >"Mary" == Mary Osielski writes: >Mary> I'm trying to buy a regular, standard, run-of-the-mill Rubik's >Mary> cube which I now realize is not so easy. Can you please direct me >Mary> to a source? Are they no longer produced? > >There has been a new run (I'm not sure if it's by Ideal or not), and I >saw two on the shelf under the Lego Brand Construction Blocks (tm) (Note >to self: kill the lawyers) at K-Mart just last week. The new ones, at least the ones that I've gotten in the last year or less, are made by Oddz-on (sp?). I think that they still make them but I haven't looked in a few months. I called them a few months ago to see if they had plans to make a 4x4x4 but they said no. Also they did make 2x2x2's for awhile but I don't think they do anymore, plus the 2's were hard to rotate and fell apart eaisly. bryan __________________________________________________________________ Bryan Main Cartographic Specialist http://caddscan.com CADDScan Engineering Inc. NOAA Site Number: 301-713-0388 X 110 From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 19:04:36 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA02443; Mon, 3 Nov 1997 19:04:34 -0500 (EST) Message-Id: <199711040004.TAA02443@mc.lcs.mit.edu> Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Mon, 3 Nov 1997 13:48:19 -0700 (MST) From: cube-lovers-request@ai.mit.edu To: cube-lovers@ai.mit.edu Reply-To: Paul Hart Subject: Auction on Rubik's Revenge (4x4x4) cubes Paul Hart has announced he has 6 unopened Rubik's Revenge cubes for sale to the highest bidder. The Cube-lovers list will not include details of the offer; I am passing this information on only because a number of persons on this list have asked about finding Rubik's Revenge cubes, apparently without success. Contact hart@iserver.com for any further information. As always, beware of fraud. Dan Hoey, Interim moderator Cube-Lovers-Request@AI.MIT.Edu From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 19:40:01 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA02611; Mon, 3 Nov 1997 19:40:00 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From dzander@solaria.sol.net Mon Nov 3 18:52:31 1997 From: Douglas Zander Message-Id: <199711032351.RAA14876@solaria.sol.net> Subject: Re: Where to buy one??? To: bmain@caddscan.com (Bryan Main) Date: Mon, 3 Nov 97 17:51:42 CST Cc: cube-lovers@ai.mit.edu (cube) In-Reply-To: <19971103100711.0054df7a.in@caddscan.com> from "Bryan Main" at Nov 3, 97 10:07:11 am Can you comment how good the new cubes from Oddz-on rotate? Are they smooth and slick like the original Rubik's Cubes were or hard to turn like the knock-offs were? -- Douglas Zander | dzander@solaria.sol.net | Milwaukee, Wisconsin, USA | From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 4 14:24:17 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA08790; Tue, 4 Nov 1997 14:24:17 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Tue Nov 4 01:38:07 1997 From: SCHMIDTG@iccgcc.cle.ab.com Date: Tue, 4 Nov 1997 1:37:40 -0500 (EST) To: cube-lovers@ai.mit.edu Message-Id: <971104013740.202034d6@iccgcc.cle.ab.com> Subject: Re: Categorization of cube solving programs "der Mouse" wrote: >>> Class 3: A program that when given a specific instance of the >>> cube, attempts to [solve it] [eg, Kociemba] >>> Class 4: A program which attempts to [find an algorithm to solve >>> arbitrary cubes]. > >> In retrospect, Class 4 programs are not necessarily more >> sophisticated than Class 3 programs especially when one considers >> that the latter should be be able to produce a macro-table solution >> by solving for a sufficient set of specific sequences. > >Sure...but who picks the specific instances for them? See below... >> Richard Korf points out a suggestion by Jon Bently that the learning >> program can be be interleaved with the solving program, as >> co-routines, and only running the learning program when a new macro >> is needed to solve a particular problem instance. > >This means that the solving program has to imagine macros, try to >choose a useful one, determine whether it's actually possible (you >gotta keep the program from trying to produce, for example, a single >edge flipper). You also have to decide when it's worth trying for a >macro and when it's better to just hit the (sub)problem with brute >force. I would expect all these problems to be quite hard. Although I haven't verified this with Richard Korf, I think there is a very simple approach to this. Consider each cubie to have one of two states, either "fixed" or "don't care". Initially, all cubies are in the "don't care" state. If a cubie state is "don't care" then that means we disregard it's position (i.e. location and orientation) in the target state for a particular macro. Number all 20 of the corner and edge cubies. Now perform the following "Pidgin C" algorithm: Mark all cubies[1 through 20] as "don't care" in current_cube_state for (i = 1 to 20) { target_state = cubies 1 through i in proper home cubicle position and marked as "fixed", all other cubies are in a "don't care" state Construct a unique macro index = f(IN = current_cubie_position[i], IN = desired_cubie_position[i]) if (the macro at "index" doesn't exist) { Class_3_Solve(IN = current_cube_state, IN = target_cube_state, OUT = macro) add the new "macro" to the macro table at "index" } Apply the macro to our current_cube_state Mark cubie[i] as "fixed" in current_cube_state } Note: Class_3_solve must be able to accept an initial and goal state augmented with the "fixed" and "don't care" markings and should honor the constraints implied by them. To put it another way, if a cubicle is marked as "don't care" then a valid target state allows this cubie to be placed in any other cubicle not currently occupied by a "fixed" cubie. Not really a big deal for any search procedure as we are simply relaxing the goal state condition to a partial match rather than requiring an exact match. So we start out by solving for one cubie only and ignore the effect this has on the remaining 19 cubies. We continue doing this, each time successively fixing another cubie and ignoring the rest, until all cubies are finally in place. For any valid cube configuration, we are always guaranteed to find a macro that can solve this subproblem. Actually, we will never fully iterate to all 20 cubies since it is impossible to move just a single cubie. For example, the very last subproblem for cubie #19 might be an edge flip. Once we've discovered and applied the appropriate macro for this particular edge flip we will have also flipped the #20 cubie and placed the cube in its solved configuration. Initially, the macros are very easy to find since most cubies can be relocated. At the very end, we can only move very few cubies, and the macros are more difficult. But a class 3 program can solve any cube and thus can find even the most difficult macros (e.g. an edge flip). Eventually, once we've solved enough cube instances, our macro table will be complete and all future cubes can be solved via macro table lookup without the aid of the solving portion of the program. Regards, -- Greg From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 6 18:53:55 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA22713; Thu, 6 Nov 1997 18:53:55 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From bmain@caddscan.com Thu Nov 6 10:15:24 1997 To: Richard E Korf From: "Bryan Main" Subject: Re: Where to buy one??? Cc: cube-lovers@ai.mit.edu Date: Thu, 06 Nov 1997 10:13:37 Eastern Standard Time Message-Id: <19971106101337.000d3578.in@caddscan.com> At 11:26 AM 11/5/97 -0800, you wrote: >Douglas, > I bought an Oddz-on Cube the other day, and although I don't have a > large basis for comparison, it seems to work pretty well. > -rich This got sent to me and I think that it was for the list so I'm forwarding it. On the same note I have three of these cubes and they work well but the stickers become old fast. They begin to come off around the edges and the protective cover sometimes comes off. __________________________________________________________________ Bryan Main Cartographic Specialist http://caddscan.com From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 6 19:32:59 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA22848; Thu, 6 Nov 1997 19:32:59 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From tenie1@juno.com Thu Nov 6 18:13:12 1997 To: Cube-Lovers@ai.mit.edu Subject: Better way to flip a middle edge? Message-Id: <19971106.151149.11046.0.tenie1@juno.com> From: tenie1@juno.com (Tenie Remmel) Date: Thu, 06 Nov 1997 18:10:26 EST Is there a short way to flip a middle edge cubie without disturbing the top layer or the other middle edges? I mean, better than replacing it with one from the bottom and then putting it back in the correct orientation, which takes 15 moves. --Tenie Remmel (tenie1@juno.com) From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 13:52:09 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA26675; Fri, 7 Nov 1997 13:52:07 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cubeman@idirect.com Thu Nov 6 23:15:29 1997 Message-Id: <34629664.2B5E@idirect.com> Date: Thu, 06 Nov 1997 23:17:40 -0500 From: Mark Longridge To: cube lovers Subject: Availablility of Rubik's Cube I've followed the thread about the availibility of Rubik's Cubes. Ideal Toy is once again manufacturing Rubik's Cube for the mass market. New packaging (with the correct number of permutations) has been purchased by myself in a Canadian Toys R Us store just recently. I know nothing of the Oddz-On cubes, but the new Ideal Toy cubes are wonderful. The new cubes sport a new logo and brighter colours, but they use the same colour arrangement as the Ideal Cubes of old. There is also an official site for Rubik's Cube at http://www.rubiks.com Unfortunately they are not answering their mail and are attracting a mostly younger crowd. They are also using Karl Hornell's rubik's cube java applet (sporting the incorrect colour arrangement I might add) without giving any mention of Karl's name. It is an exact byte for byte copy. Although Karl does give out the Rubik's Cube java applet as freeware, I think he deserves credit from the Ideal web site. As for my own web site (which does sport Karl's java applet with the correct standard colouration, and also BEFUDDLER support!) I intend to record the entire chronology of all the cube contests from every country, including all the records from the World Championships. My Rubik's Cube web page is currently http://web.idirect.com/~cubeman If anyone has any information about the cube contests I have missed, please email me. Thanks! -> Mark Longridge <- The Cubeman of the Internet From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 14:22:06 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA26766; Fri, 7 Nov 1997 14:22:05 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From C.McCaig@Queens-Belfast.AC.UK Fri Nov 7 05:33:32 1997 From: C.McCaig@queens-belfast.ac.uk Date: Fri, 07 Nov 1997 10:28:08 GMT To: cube-lovers@ai.mit.edu Message-Id: <009BCEF0.4DC8B4D9.41@a1.qub.ac.uk> Subject: Re: Where to buy one??? i've noticed that here in northern ireland, there are a couple of places selling cubes. one is a standard copy of the rubik's cube, and the other is called "magic cube" which has holographic stickers on it, and the cubies are much squarer making it very difficult to take apart.. it comes with a locking key which allows you to remove one of the faces.. the turning mechanism is _really_ loose, too loose in fact, but mine hasnt fallen apart. as an aside, i have an original cube, that my grandmother bought me 16 or 17 years ago, and it's still got all it's original stickers! clive --- Clive McCaig Dept. Applied Mathematics Queens University Belfast Northern Ireland From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 14:46:43 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA26878; Fri, 7 Nov 1997 14:46:43 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From davidbarr@iname.com Fri Nov 7 02:09:32 1997 Sender: davidb@davidb.concentric.net Message-Id: <3462BE64.3F20493A@iname.com> Date: Thu, 06 Nov 1997 23:08:20 -0800 From: David Barr Organization: Medweb To: Tenie Remmel , Cube-Lovers@ai.mit.edu Subject: Re: Better way to flip a middle edge? References: <19971106.151149.11046.0.tenie1@juno.com> Tenie Remmel wrote: > Is there a short way to flip a middle edge cubie without disturbing the > top layer or the other middle edges? I mean, better than replacing it > with one from the bottom and then putting it back in the correct > orientation, which takes 15 moves. > > --Tenie Remmel (tenie1@juno.com) Take a look at http://ssie.binghamton.edu/~jirif/Mike/middle.html. I think this is the sequence you want: 2) R2 D2 F' R2 F D2 R D' R This sequence flips the cubie on the front-right edge without disturbing the upper face or the other middle edges. -- mailto:davidbarr@iname.com http://www.concentric.net/~Davebarr/ From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 18:43:53 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA28008; Fri, 7 Nov 1997 18:43:52 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Michael.Swart@switchview.com Fri Nov 7 09:33:48 1997 Message-Id: <199711071431.JAA05030@support.switchview.com> From: "Michael Swart" To: , "Tenie Remmel" Subject: Re: Better way to flip a middle edge? Date: Fri, 7 Nov 1997 09:26:36 -0500 > Is there a short way to flip a middle edge cubie without disturbing the > top layer or the other middle edges? I mean, better than replacing it > with one from the bottom and then putting it back in the correct > orientation, which takes 15 moves. F' L' R F L' R T' B2 T L R' F L R' D2 F' 18 q turns 16 h turns. Guess it wasn't any better but you may notice that this sequence also leaves the bottom intact except for one flipped cubie at the back down edge. Michael Swart From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 9 14:40:21 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA28882; Sun, 9 Nov 1997 14:40:20 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From charlied@erols.com Sun Nov 9 01:06:22 1997 Message-Id: Date: Sun, 9 Nov 1997 01:05:27 -0500 To: Cube-Lovers@ai.mit.edu From: Charlie Dickman Subject: A 4 Dimensional Rubik's Cube About 18 months ago I sent this group an email about a Mac based program I have written that simulates a 4 dimensional (3x3x3x3) Rubik's Cube based on an unpublished paper by Harry Kamack and Tom Keene. Some of you who were interested in the paper that describes the model and the program had difficulty with the copies I sent you and, I suspect, were unable to read it after you received it. Someone suggested that I translate the document into HTML and this email is to let you know that I have done that and will send either a ZIP or STUFFIT archive of the document to anyone interested. I know that maybe I should get a web site and put the paper there but I'm not up for designing a web page or maintaining it. If you would like a copy of the document and would also like to put it on your web site, let me know that too. The HTML version of the document consists of 36 fairly small GIFs that illustrate the words. The STUFFIT archive is 328K and the ZIP file is 320K. The documentation for the Mac based ZIP program claims that the file can be successfully unZIPped on non-Mac platforms. The STUFFIT archive is self-extracting if you're Mac enabled. Send me an email if you are interested in either the program or the HTML document or both. If you just want the document, tell me which format you want. If you ask for the program I will assume you have a Mac and will send everything in a STUFFIT sea. Regards to all... Charlie Dickman charlied@erols.com From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 9 15:59:22 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA29870; Sun, 9 Nov 1997 15:59:22 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From whuang@ugcs.caltech.edu Sun Nov 9 12:26:11 1997 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Where to buy one??? Date: 9 Nov 1997 17:25:11 GMT Organization: California Institute of Technology, Pasadena Message-Id: <644rln$45r@gap.cco.caltech.edu> References: C.McCaig@queens-belfast.ac.uk writes: >and the other is called "magic cube" which has holographic stickers >on it, and the cubies are much squarer making it very difficult to >take apart.. it comes with a locking key which allows you to remove >one of the faces.. the turning mechanism is _really_ loose, too >loose in fact, but mine hasnt fallen apart. Ah... this is a recent Taiwanese invention. -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- "[Lucy's eyes] look like little round dots of India ink..." -- Charlie Brown From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 10 10:58:12 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA13166; Mon, 10 Nov 1997 10:58:11 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From whuang@ugcs.caltech.edu Sun Nov 9 12:36:08 1997 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Better way to flip a middle edge? Date: 9 Nov 1997 17:35:21 GMT Organization: California Institute of Technology, Pasadena Message-Id: <644s8p$4er@gap.cco.caltech.edu> References: "Michael Swart" writes: >> Is there a short way to flip a middle edge cubie without disturbing the >> top layer or the other middle edges? I mean, better than replacing it >> with one from the bottom and then putting it back in the correct >> orientation, which takes 15 moves. >F' L' R F L' R T' B2 T L R' F L R' D2 F' >18 q turns 16 h turns. Guess it wasn't any better but you may notice >that this sequence also leaves the bottom intact except for one flipped >cubie at the back down edge. This can be slightly improved: D R' F D' R' L B D' R B' D R L' F' 14 quarter turns; does exactly the same thing. -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- "[Lucy's eyes] look like little round dots of India ink..." -- Charlie Brown From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 10 11:32:49 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA14072; Mon, 10 Nov 1997 11:32:48 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cubeman@idirect.com Sun Nov 9 23:05:52 1997 Message-Id: <34668899.248A@idirect.com> Date: Sun, 09 Nov 1997 23:07:53 -0500 From: Mark Longridge To: cube lovers Cc: joyner.david@mathnt1.sma.usna.navy.mil Subject: Megaminx, the 10-spot and GAP First of all, the STANDARD colour arrangement used by Ideal Toy is as follows: UP = White DOWN = Blue FRONT = Yellow BACK = Green LEFT = Red RIGHT = Orange All the official Ideal Toy 2x2x2, 3x3x3 & 4x4x4 cubes used this arrangement. Even my 5x5x5 cube is the same. Secondly, I have at last resolved the 10-spot pattern for the megaminx in GAP. I created the process m1a which is the sequence of operators to generate the 10-spot. I had no C_U operator, so it was more difficult than I thought it would be. To see all the gory details surf to the following URLs (These are all GAP text files) http://web.idirect.com/~cubeman/dodeca.txt describes the megaminx http://web.idirect.com/~cubeman/megaop.txt describes operators http://web.idirect.com/~cubeman/spot.txt generates the 10-spot Note that after executing spot.txt (which loads the other necessary files) in gives the order of process m1a correctly as 5. This generator uses all of the megaminx operators except the top and bottom faces, so it is a pretty good test of the correctness of the all of dodeca.txt, megaop.txt, and spot.txt I believe this is the first simulation of the megaminx generating the 10-spot although Dr. David Joyner is very close! His work is more graphically interesting (using Maple to generate 3d pics of the megaminx) but his operators to rotate the whole megaminx are cooked. However, we have both verified that processes m2, m3 and m3a are correct and have been graphed correctly using Maple. -> Mark <- From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 11 20:07:56 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA24028; Tue, 11 Nov 1997 20:07:56 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From tim@mail.htp.net Tue Nov 11 00:07:26 1997 From: tim@mail.htp.net (Tim Mirabile) To: cube lovers Subject: Re: Megaminx, the 10-spot and GAP Date: Tue, 11 Nov 1997 05:06:29 GMT Organization: http://www.webcom.com/timm/ Message-Id: <3467e58b.881450@mail.htp.net> References: <34668899.248A@idirect.com> In-Reply-To: <34668899.248A@idirect.com> On Sun, 09 Nov 1997 23:07:53 -0500, Mark Longridge wrote: >First of all, the STANDARD colour arrangement used by Ideal Toy is as >follows: > >UP = White >DOWN = Blue >FRONT = Yellow >BACK = Green >LEFT = Red >RIGHT = Orange >... I remember having one if the early Ideal cubes (at least I think it was), and green was opposite blue. Recently I bought one of those "odds-on" cubes, and within a week I wore the plastic coating off the faces, so I decided to peel all the stickers off and paint it using model paint. I decided to keep the most similar colors opposite each other (blue-green, red-orange, yellow-white). I find this arrangement makes things easier when cubing under dim lighting. :) -- Long Island chess -> http://www.webcom.com/timm/ TimM on ICC and A-FICS The opinions of my employers are not necessarily mine and vice versa. From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 11 20:43:51 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA24200; Tue, 11 Nov 1997 20:43:51 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Anders.Larsson@hvi.uu.se Tue Nov 11 02:25:24 1997 Message-Id: <346807C1.ACFE4D68@hvi.uu.se> Date: Tue, 11 Nov 1997 08:22:41 +0100 From: Anders Larsson To: cube lovers Subject: Colour arrangements (Was: Re: Megaminx, the 10-spot and GAP) References: <34668899.248A@idirect.com> Mark Longridge wrote: > First of all, the STANDARD colour arrangement used by Ideal Toy is as > follows: > > UP = White > DOWN = Blue > FRONT = Yellow > BACK = Green > LEFT = Red > RIGHT = Orange In front of me I hold a cube from one of the first batches from Hungary with the following colour arrangement: Up = white Down = yellow Front = blue Back = green Left = red Right = orange Does anybody know the history why this colour arrangement was changed? BTW: Even if Ideal Toys has their own local standard, it doesn't change the original ("correct") colour arrangement. /Anders -- Anders Larsson, PhD Institute of High Voltage Research Tel.: +46 (0)18 532702 Uppsala University Fax.: +46 (0)18 502619 Husbyborg E-mail: Anders.Larsson@hvi.uu.se S-752 28 Uppsala, Sweden http://www.hvi.uu.se/IFH/staff/Anders/Anders.html From cube-lovers-errors@mc.lcs.mit.edu Wed Nov 12 21:40:03 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA01451; Wed, 12 Nov 1997 21:40:03 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From ck1@home.com Tue Nov 11 22:26:29 1997 From: "Chris and Kori Pelley" To: Subject: Colors and other variations between brands Date: Tue, 11 Nov 1997 22:26:17 -0500 Message-Id: <01bcef1a$bcbe2f60$da460318@CC623255-A.srst1.fl.home.com> Most of the early "clone" cubes had the Blue/Green arrangement instead of the Blue/White. Most Ideal cubes seemed to have the Blue/White. There were exceptions, though... I remember there were several factories where Ideal had their cubes made. Some factories were better than others in terms of their quality. My favorites were the ones that said "Made in Korea" on a little peel-off gold sticker. Back in those days I would refer to "my Korean cube." Believe it or not, these all had the Blue/Green arrangement but they were genuine Ideal cubes! Their cubes were also the smoothest. I still have one of them that is in near perfect condition. It was the cube I used in the competitions. The other factories included Japan and Hong Kong. The Japanese cubes seemed more prevalent and I still have at least three of those-- all featuring the Blue/White arrangement. The earliest Rubik's Cube I ever saw had strange colors-- grey instead of white and the shades of green and blue were very different from later cubes. I don't think it was an Ideal cube. The Blue/White arrangement definitely won out as Ideal's "standard" arrangement since their 4x4x4 Revenge and 2x2x2 Pocket Cubes featured the identical coloring. Some Ideal 3x3x3 cubes were Blue/White but "non-standard" because the Yellow/Green would be reversed (mirror image). Who knows why these variations existed-- probably something as simple as some factory tech switching the sticker feeds accidentally? The new "Rubik's Cubes" made by Oddz-On are not all that great, in my opinion. They look shiny and great in the box, but after mild use the stickers get ruined. The Square-1 puzzles suffer the same fate. Also their turning mechanism is nowhere near the quality of the "Korean cubes." Their 2x2x2 "Mini-Cube" as it is now called also lacks in quality compared to the old Ideal Pocket Cubes. Still, it warms my heart to see them back in toy stores again! Much better are the "Magic Cube" clones that appeared last year. I have purchased several of these (only $3.99 at Walgreen's!) and they turn very smoothly. The holographic stickers are different, but they don't wear out like the Oddz-On cubes. Also, mine feature the Blue/White arrangement! I recently saw a post that Ideal is now making cubes again. This seems strange since I thought they went out of business, but I could be wrong. Anybody know the real scoop? Finally, Square-1 seems to have made a reappearance. I thought they only made one batch of these, but maybe they've made another lately? Chris Pelley ck1@home.com http://members.home.net/ck1 From cube-lovers-errors@mc.lcs.mit.edu Wed Nov 12 22:10:31 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA01766; Wed, 12 Nov 1997 22:10:31 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From chrono@ibm.net Wed Nov 12 00:37:48 1997 Message-Id: <34694088.5E3AC959@ibm.net> Date: Tue, 11 Nov 1997 21:37:12 -0800 From: "Jin 'Time Traveler' Kim" Organization: The Fourth Dimension To: cube lovers Subject: Rubik's Cube Color arrangements References: <34668899.248A@idirect.com> <346807C1.ACFE4D68@hvi.uu.se> I have serveral cubes spanning over a decade and a half in front of me. Here's the quick color arrangements: Rubik's Cube (circa 1982) Up = White Down = Yellow Front = Blue Back = Green Left = Orange Right = Red I've owned this cube for what feels like forever. I'm not even sure how I even found it again, because I dug it out of some old junk after not having a puzzle for about 5 years. This is slightly different from the one you described as being from Hungary. I suspect mine is from there too, so maybe production values weren't as high as they could be. Rubik's Cube "4th Dimension" (Golden Toys, circa 1988) - Poor quality in my opinion, as the stickers are paper with a clear plastic laminate, but despite only being taken out of the box only 4 Times ever, the plastic laminate is already peeling in spots. Rubik's Mini Cube (OddzOn, circa 1996) Rubik's Cube (OddzOn, circa 1997) All of the above have the same color arrangement as what you described below as being the "Ideal" solution, which I believe isn't the best. I still think that the opposite pairing of red/orange, white/yellow, and blue/green makes for the best balanced color combination. Not to mention it's also the best quality with plastic stickers instead of paper. Anders Larsson wrote: > Mark Longridge wrote: > > > First of all, the STANDARD colour arrangement used by Ideal Toy is as > > follows: > > > > UP = White > > DOWN = Blue > > FRONT = Yellow > > BACK = Green > > LEFT = Red > > RIGHT = Orange > > In front of me I hold a cube from one of the first batches from Hungary > with the following colour arrangement: > > Up = white > Down = yellow > Front = blue > Back = green > Left = red > Right = orange > > Does anybody know the history why this colour arrangement was changed? > > BTW: Even if Ideal Toys has their own local standard, it doesn't change > the original ("correct") colour arrangement. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@mc.lcs.mit.edu Wed Nov 12 22:54:25 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA01970; Wed, 12 Nov 1997 22:54:25 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From richard_morton@icom-solutions.com Wed Nov 12 04:56:12 1997 Date: Wed, 12 Nov 1997 17:31:37 GMT From: David Singmaster To: Anders.Larsson@hvi.uu.se Cc: cube-lovers@ai.mit.edu Message-Id: <009BD319.4B14FD66.61@ice.sbu.ac.uk> Subject: RE: Colour arrangements (Was: Re: Megaminx, the 10-spot and GAP) The colour arrangement on the early Hungarian cubes was quite random!! I even have two examples where two faces have the same colour!! It was not until about 1980 that the idea of having a standardised colour pattern was adopted and the most common was to have the opposite faces differ by yellow. That is the opposite faces were White - Yellow, Blue - Green, Red - Orange. Rubik went to some effort to select six colours that would be maximally distinct, but I think the yellow, red and orange tended to be too close in the sense that either the orange was too close to the red or too close to the yellow! However, this does not completely determine the colour pattern. Just as with a die, there are two possible arrangements. Conway and Guy etc. observed that Blue, Orange and Yellow meet at a corner and they can occur clockwise or counterclockwise, spelling BOY or YOB. Some people have expressly asked me for one form rather than the other! An early anecdote, from about 1979. A friend's son was trying to help another friend solve his cube over the telephone. This is a pretty formidable task at the best of times, but their two cubes had different colour patterns, so the son was making statements like: turn the red face, that's blue on your cube, .... DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 13 13:15:59 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA05230; Thu, 13 Nov 1997 13:15:58 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From richard_morton@icom-solutions.com Wed Nov 12 04:56:12 1997 Message-Id: <199711120955.EAA01537@life.ai.mit.edu> Date: Wed, 12 Nov 1997 04:55:28 EST From: "Richard M Morton" To: cube-lovers@ai.mit.edu Subject: Cube Colours Mark Longridge wrote: > First of all, the STANDARD colour arrangement used by Ideal Toy is as > follows: > > UP = White > DOWN = Blue > FRONT = Yellow > BACK = Green > LEFT = Red > RIGHT = Orange Is the orientation of the above fixed in some way or is it arbitrary ? My second cube (can't remember what happened to the first one) is a later edition (not sure if it is Ideal) with the same arrangement to above except, the orientation is different (UP is either RED or ORANGE) The reason I say this is that the LEFT,RIGHT,DOWN and FRONT faces have symbols printed in the centre as follows : YELLOW - signature of Erno Rubik WHITE - Rubik's CUBE tm GREEN - C*4**4 (actually uses superscript 4 for the power) BLUE - silhouette (of Erno Rubik) The symbols are designed to make the cube harder to solve - the challenge is to solve the cube with the centre cubes all in the correct orientation. I recall that there are sequences of moves that rotate pairs of centre cubes. This cube is definitely a lot stiffer than my original cube but the novelty of speed cubing has worn off anyway. Richard Morton (If my employers views are not necessarily those of my own, why am I still working here ?) Icom Solutions http://www.icom-solutions.com/offprods/default.htm From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 13 13:57:08 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA05391; Thu, 13 Nov 1997 13:57:07 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Geoffroy.VanLerberghe@ping.be Wed Nov 12 15:37:45 1997 Message-Id: <346A13C7.7C38@ping.be> Date: Wed, 12 Nov 1997 21:38:32 +0100 From: Geoffroy Van Lerberghe To: Cube-Lovers Subject: Cubes in London In one month I am going to London for a few days and I would like to know where I can buy brainteasers there (mainly Rubik's cubes and related puzzles). Could you help me and send me all the information you have? In Brussels, Belgium, you can (sometimes) find Magic Dodecahedron, Pyraminx, Skewb and "555" cube at Dedale Galerie du Cinquantenaire Avenue de Tervuren 32 1040 Brussels Thank you for your help. Geoffroy From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 13 14:26:35 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA05543; Thu, 13 Nov 1997 14:26:34 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From tenie1@juno.com Thu Nov 13 13:02:44 1997 To: Cube-Lovers@ai.mit.edu Date: Thu, 13 Nov 1997 10:01:41 -0800 Subject: 6x6x6 cube design Message-Id: <19971113.100159.5094.0.tenie1@juno.com> From: tenie1@juno.com (Tenie Remmel) I am attempting to design a 6x6x6 cube. My idea to make it structurally sound is to attach both the center cubies and the middle edge cubies to a ball in the center. Then all other pieces are wedged behind those. I think that extending from the 5x5x5 design the same way the 4x4x4 was extended from the 3x3x3 design would be way too flimsy, mainly because the centers would have to be attached via long, thin struts which are apt to break easily unless made out of metal, which would make the thing way too heavy. The width of the cubies probably could not be more than 14 or 15 mm; if they were larger, the cube would be quite big and so it would be difficult to manipulate. Unfortunately the ball would be quite complicated, with six or even nine tracks in it instead of just three as in the 4x4x4 cube. It might have to be made of metal instead of plastic (it shouldn't be too heavy if it is hollow). Also the 152 pieces will be a real pain to put together... Of course, even if it can be built, does anyone know how to solve it? Here is a rather crude diagram of a cross section through the center of the cube. Actually it is just a quarter of a cube. ------------------------------------------------------------------------ aaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc aaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc aaaa......aaaabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccc aaaa..............bbbbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccc aaaaaaaa................bbbbbbbbbbbbbbbbbbbbcccccccccccccccc aaaaaaaa....................bbbbbbbbbbbbbbbbcccccccccccccccc ................................bbbbbbbbbbcccccccccccccccccc ..................................bbbbbbccccccbbbbbbbbcccccc ....................................bbccccccbbbbbbbbbbcccccc ..............................cccc..ccccccbbbbbbbbbbbbbbbbbb ..............................ccccccccccbbbbbbbbbbbbbbbbbbbb ................................cccccc....bbbbbbbbbbbbbbbbbb ..................................cccccc....bbbbbbbbbbbbbbbb ....................................cccc......bbbbbbbbbbbbbb ..............................................bbbbbbbbbbbbbb ................................................bbbbbbbbbbbb ................................................bbbbbbbbbbbb ..................................................bbbbbbbbbb ..................................................bbbbbbbbbb ..................................................bbbbbbaaaa ....................................................bbbbaaaa ....................................................bbbbaaaa ....................................................aaaaaaaa ....................................................aaaaaaaa ......................................................aaaaaa ..............................................aaaa....aaaaaa ..............................................aaaa....aaaaaa ..............................................aaaaaaaaaaaaaa ..............................................aaaaaaaaaaaaaa ------------------------------------------------------------------------ BTW, Does anyone have experience with TurboCAD? Can it be used to design this type of thing? It sure would be easier to use a computer program than to use graph paper. I believe that the 6x6x6 is the largest mechanically possible, because with the 7x7x7 and higher cubes, the corner cubies aren't attached to anything at all! Is this correct? Also what is the mechanism for a 2x2x2 cube? Could it be extended to make a more stable 4x4x4 and/or 6x6x6 cubes... And how about a GigaMinx, a 5x5 version of the MegaMinx magic pentagonal dodecahedron, with five pieces on each edge, 31 pieces on each face (5 corners, 11 edges and 11 central pieces), 242 pieces total. I would draw a diagram if it wasn't so hard to make a pentagon out of chars... --Tenie Remmel (tenie1@juno.com) [ Moderator's note: The purported impossibility of a Rubik's 7^3 has been discussed and refuted repeatedly on this list, and several mechanisms have been proposed for it; see the archives. It is not true that the corner cubies "aren't attached to anything". Each corner will be attached to at least two edge cubies, though not always the same two edge cubies. You should also look in the archives to find descriptions of the 2^3, some as recently as 28 July. Unfortunately, I haven't been able to understand it. I'd like to see a clear description, as I haven't got a 2^3 handy to try myself. -Dan ] From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 14 10:37:37 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA10458; Fri, 14 Nov 1997 10:37:36 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cubeman@idirect.com Thu Nov 13 21:03:37 1997 Date: Thu, 13 Nov 1997 19:06:53 -0500 (EST) From: Mark Longridge To: Richard M Morton Cc: cube-lovers@ai.mit.edu Subject: Re: Cube Colours In-Reply-To: <199711120955.EAA01537@life.ai.mit.edu> Message-Id: On Wed, 12 Nov 1997, Richard M Morton wrote: > Mark Longridge wrote: > > > First of all, the STANDARD colour arrangement used by Ideal Toy is as > > follows: > > > > UP = White > > DOWN = Blue > > FRONT = Yellow > > BACK = Green > > LEFT = Red > > RIGHT = Orange > > ... Ok folks, one last bit of info about the cube colour controversy The colouring "standard" I was referring to was used by Canadian and the USA cube contests. Having said that there were probably contests where people brought there own cubes, and that would make it potpourri. Moreover, this was stipulated in the rules of the contest. I still have the form. The only difference between differ by yellow and the standard Ideal cube was the transposition of yellow and blue. There isn't really a standard orientation, save for the orientation I use in my own cube programs. All the Ideal cubes I have conform to White/Blue, Yellow/Green, Red/Orange for Top/Down, Front/BACK, Left/Right. So I suppose it is open to interpretation. I thought David Singmaster might mention what colour arrangement was used in the World Championship. So a case may be made for both "Differ by Yellow" and Ideal Contest Colours. Would someone like to pick one?? :-) -> Mark <- The Colourist From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 14 11:12:41 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA10598; Fri, 14 Nov 1997 11:12:41 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From chrono@ibm.net Thu Nov 13 22:03:44 1997 Message-Id: <346BBF7F.ADFDE0D4@ibm.net> Date: Thu, 13 Nov 1997 19:03:27 -0800 From: "Jin 'Time Traveler' Kim" Organization: The Fourth Dimension To: Cube-Lovers@ai.mit.edu Subject: Re: 6x6x6 cube design References: <19971113.100159.5094.0.tenie1@juno.com> Tenie Remmel wrote: > I am attempting to design a 6x6x6 cube. My idea to make it structurally > sound is to attach both the center cubies and the middle edge cubies to > a ball in the center. Then all other pieces are wedged behind those. I > think that extending from the 5x5x5 design the same way the 4x4x4 was > extended from the 3x3x3 design would be way too flimsy, mainly because > the centers would have to be attached via long, thin struts which are > apt to break easily unless made out of metal, which would make the thing > way too heavy. The width of the cubies probably could not be more than > 14 or 15 mm; if they were larger, the cube would be quite big and so it > would be difficult to manipulate. > Of course, even if it can be built, does anyone know how to solve it? If it can be built and scrambled, it can be solved. In fact, it could make for a very interesting puzzle since it could behave identically to a 3x3x3 if one wanted it to, just like a 4x4x4 can be manipulated like a 2x2x2. Heck, the 6x6x6 could also behave like a 2x2x2... One puzzle could take the place of two others. Sort of a "mix and match" difficulty setting. Regardless, I suspect that many would applaud the ingenuity of a 6x6x6 if it was executed elegantly and worked well, like the 5x5x5. > I believe that the 6x6x6 is the largest mechanically possible, because > with the 7x7x7 and higher cubes, the corner cubies aren't attached to > anything at all! Is this correct? The moderator of the mailing list stated that a 7x7x7 cube could be built, but I counter that it would require "cubes" of dissimilar size or some kind of groove type scheme, which actually isn't quite in the spirit of a cube. Even a 6x6x6 would require some careful engineering since the corner cubes just barely overlap. > Also what is the mechanism for a 2x2x2 cube? Could it be extended to > make a more stable 4x4x4 and/or 6x6x6 cubes... The mechanism of the 2x2x2 is similar to the 4x4x4, which makes both of them rather stiff. > And how about a GigaMinx, a 5x5 version of the MegaMinx magic pentagonal > dodecahedron, with five pieces on each edge, 31 pieces on each face > (5 corners, 11 edges and 11 central pieces), 242 pieces total. I would > draw a diagram if it wasn't so hard to make a pentagon out of chars... I'm sure supersets of many existing puzzles have been considered. I myself spent some hours contemplating and drafting the possibility of a pyraminx to the next level. I called it Tut's Curse as a sort of 'project' name, despite the fact that Tut was never buried in a pyramid. Maybe that's why I never completed the project. Oh well. The best laid plans of mice and men... -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 14 11:46:24 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA10792; Fri, 14 Nov 1997 11:46:23 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Nov 14 10:27:59 1997 Date: Fri, 14 Nov 1997 15:25:06 GMT From: David Singmaster To: tenie1@juno.com Cc: cube-lovers@ai.mit.edu Message-Id: <009BD499.F2FD74E5.202@ice.sbu.ac.uk> Subject: RE: 6x6x6 cube design First, regarding the 6^3 and 7^3. As noted, when you get to these sizes, the connection of the corners while turning becomes problematic. For the 6^3, the overlap is about 15% of the edge length of the cubie, probably too small to be practicable. One can imagine some clever mechanism to hold onto the corners, but it would be tricky and I've never seen one clearly described. However, if you think about it, there's no reason for all the levels of the cube to be the same size. That it, the parallel cutting planes of the entire cube do not have to be equally spaced. One can thus have the corner cubies be very large with much smaller centre cubies. The edge cubies will be cuboids, rather than cubes. Using this idea, one can make arbitrarily large cubes, but the interior pieces become impossible to manipulate. Now let me try my hand at describing three versions of the 2^3. I'll start with the simplest which was sent to me from Japan about 1980. This had a steel sphere in the middle and each cubie had a magnet in it. Although the sphere and the cubies were carefully machined, when one moved it quickly, a piece would catch against another piece and lift off and then fall off. Not very successful. The second version was patented by Ishige in Japan about 1977? and several versions were made. I received a batch of seven with different colouring patterns made by a German sports firm - three or four had broken just in the post! This version has a central sphere and six of what I call 'umbrellas' sticking out toward each face centre. Each of the pieces has a notch around the part that rest against the inner sphere. The umbrellas catch into these notches. One can also think of the cubies as having their own umbrellas, but of triangular form and concave. This is the same mechanism used in the Impossiball. The third version is the most common and is shown in Rubik's Hungarian patent, but is hard to interpret as I've never had the text translated. Basically, his 2^3 is a 3^3 with the edge and centre pieces concealed. I gather from earlier messages that there were several versions of this, but I only recall one, but I only ever took a few apart. At the very centre was a cube. On each face was a square rod extending almost to the face center. The ends of these had a + groove. Between the rods were pieces in the form of a quadrant with a groove on the outer, curved, edge. When all these pieces are in place, each of the midplanes of the cube is seen to contain a circle with a groove on its outer edge. The corner pieces are basically hollow, but each interior face is a layer ending in a quarter-circular curve, which fits as a tongue into the groove just mentioned. Where two of these meet, at the interior edge of the piece, a section is cut away to allow the piece to slide past the projections of the end of the square rods. In theory, one might be able to avoid the quadrant pieces, but I think they give the structure stability. A more serious problem is that the inner, concealed, pieces can get out of synch with the visible pieces, The early patent of Gustafson left gaps so one could see the inner pieces and move them. The method used by Rubik and in some similar puzzles is to fix one corner piece to the inner structure by some method. Rubik's 2^3 did this by making some of the rod ends solid rather than grooved (or perhaps they were fixed to the central cube so they couldn't rotate). One could also not notch one of the corner pieces. Whatever one does, it must have the effect of preventing one corner from moving in relation to the inner structure. I seem to recall that the 4^3 uses this idea also. Don't know how much this helps, but that's the best I can do off-hand. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 15 22:37:51 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA20977; Sat, 15 Nov 1997 22:37:51 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Nov 15 14:55:00 1997 Date: Sat, 15 Nov 1997 14:54:07 -0500 (EST) From: Nicholas Bodley Reply-To: Nicholas Bodley To: David Singmaster Cc: tenie1@juno.com, cube-lovers@ai.mit.edu Subject: RE: 6x6x6 cube design; also notes about the 2^3 innards. (Fairly long) In-Reply-To: <009BD499.F2FD74E5.202@ice.sbu.ac.uk> Message-Id: There's a short mention in passing in Douglas Hofstadter's (second major?) book (Metamagical Themas?) to the effect that a physical prototype exists for the 6^3, and a paper design for the 7^3. This was ca. 1982, iirc. On Fri, 14 Nov 1997, David Singmaster wrote: {Snips} } Now let me try my hand at describing three versions of the 2^3. } The third version is the most common and is shown in Rubik's }Hungarian patent, but is hard to interpret as I've never had the text }translated. Basically, his 2^3 is a 3^3 with the edge and centre }pieces concealed. The ones I had were quite difficult to take apart and reassemble; if they weren't made of a strong, resilient engineering plastic, they would not have been possible to make, I would say. } At the very centre was a cube. In mine, this cube was almost tiny; perhaps 15% (along an edge) of the size of a cubie as seen from the outside. } On each face was a square rod }extending almost to the face center. In mine, just about sure that three adjacent faces of this inner cube each had thin cylindrical rods extending toward the face centers. These were surrounded by square rods of the same width as the other three which were part of the center cube. The thin cylindrical rods served as pivots for the square rods of the same width. When you rotated one half of the Cube, these pivots allowed one half to rotate with respect to the other without prying anything apart. The three fixed square rods, which are extensions and "part of" the center cube, stayed fixed within their half of the Cube when the other half was rotated, much as the ball inside a 4^3 stays fixed. } The ends of these [rods -nb] had a + }groove. Between the rods were pieces in the form of a quadrant with a }groove on the outer, curved, edge. When all these pieces are in }place, each of the midplanes of the cube is seen to contain a circle }with a groove on its outer edge. The aforementioned rods are required to keep the quadrants from moving inward and therefore out of engagement with the inner, "cut-away" edges of the cubies. If that were to happen, the Cube would fall apart. (Please see the next paragraph.) (When I tried to describe the innards of a 2^3 a while back, I called these quadrants "clips". My hat's off to Mr. Singmaster for his fluency!) } The corner pieces are basically }hollow, but each interior face is a layer ending in a quarter-circular }curve, which fits as a tongue into the groove just mentioned. Where }two of these meet, at the interior edge of the piece, a section is cut }away to allow the piece to slide past the projections of the end of }the square rods. } In theory, one might be able to avoid the quadrant pieces, but }I think they give the structure stability. With all due respect, without them, the Cube would instantly fall apart! They are essential. } A more serious problem is that the inner, concealed, pieces }can get out of synch with the visible pieces. The natural tendency is to squeeze the cubies of each half together when maneuvering. Because the thin square rods molded along with the center cube are "attached" to adjacent faces of that cube, the other three faces of that cube carry the swiveling rods. No matter how you pick up the Cube, one half will contain a fixed rod. Squeezing the cubies together around that rod will make the center cube stay aligned with those four cubies that are squeezing one of its rods. (Actually, the cubies squeeze the quadrants, and the quadrants squeeze the rods.) Keeping that center cube aligned also means it will keep aligned the four rods that have their axes in the current shear plane. These rods will then keep the quadrants aligned with the half of the cubie that is squeezing the fixed rod. The four quadrants in the swiveling half will squeeze the hollow, swiveling rod, which will rotate around the thin cylindrical [rod] that extends from that face of the center cube. I'm indebted to Mr. Singmaster for his clarifying description. This mechanism seems to be a real challenge to describe solely in words! Here, a few images equal many kB of ASCII... }DAVID SINGMASTER, Professor of Mathematics and Metagrobologist } email: }zingmast or David.Singmaster @sbu.ac.uk * * * Here's another go, for those who have the patience: Imagine that each cubie is hollow. (They really are.) Imagine that they are separated from each other in 3-D space by moderate and equal distances, but still not tilted with respect to each other. In other words, there's a large gap between any two. Now, imagine a spherical rotary cutter, spinning in the center of the 3-D array of 8 cubies. Move the cubies toward the cutter, along radii of the cutter passing through their outermost corners. Don't tilt or rotate, just translate radially inward toward the cutter along a [45-degree] axis. Let the cutter machine a curved outline in each of the three inner faces. (The diameter of the cutter is maybe 80% of the edge of a complete Cube.) Make the cutter disappear, and you have a spherical cutaway inside the whole cluster of eight. (This is real, in essence.) The cubies are hollow, and they really have this curved "cut" in each of their concealed inside faces. Of course, this was molded in, not machined by a cutter. Now, you need something to hold the cubies together. If you've seen a radar corner reflector used by small boat owners, think of one made of three intersecting, mutually-orthogonal circles. They intersect at a common, center point. Make this corner reflector tiny, maybe 3,5 (3.5) cm (?) in diameter. Cut this apart into eight quadrants. Make them thick, if they aren't. Make a rectangular groove in each curved edge. Remove some material from the straight edges; line up the curved edges with a circle (same size as the original structure before you cut it apart) on your workbench). Space them equally apart. The gaps form a cross (or an "X", if you like 45-degree angles). The rods will go into those gaps. OK: These are now positioned the way they will be in one of the three shear planes in a Cube. [The radius of the corner reflector is somewhat bigger than that of the ball cutter.) Thinking back to the corner reflector, if you replace all 12 quadrants where they used to be in 3-D space, with gaps between them as they were on the bench, that is how they are positioned in an aligned Cube. To start assembling the Cube, you take four cubies, lay them down next to each other (touching) with colors properly aligned, but with their inside surfaces facing upward. Pick up four quadrants, placing the grooves you made (in the curved edges) onto the curved "cutaways" in the adjacent inside edges of the hollow cubies, where the cubies touch. As long as these quadrants don't move toward each other, they will keep the cubies together. This, in two dimensions, is what holds the Cube together. The next four quadrants fit into the remaining cutouts. They lie flat, and form a circle, the way they did when you laid them down on the workbench. To keep the quadrants away from the center, you now insert the center cube and its rods. However, assembling the remaining four cubies (and their four quadrants) to what you have so far, is, in the real world, a major struggle. It involves some worrisome distortions of the pieces! This "geometric interference" is also what makes it so hard to disassemble. Wonder how these are assembled at the factory? My best regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* 'T was the night before Xmas, and all through |* Amateur musician *|* the coffeehouse, not a creature was stirring. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 15 23:21:58 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA21060; Sat, 15 Nov 1997 23:21:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From roger.broadie@iclweb.com Thu Nov 13 19:09:04 1997 From: roger.broadie@iclweb.com (Roger Broadie) To: "Geoffroy Van Lerberghe" , "Cube-Lovers" Subject: Re: Cubes in London, plus OddzOn, clones and colours Date: Fri, 14 Nov 1997 00:07:33 -0000 Message-Id: <19971114000519.AAA4398@home> London's most famous toy-shop is Hamley's in Regent St. On the fourth floor they have a wall of OddzOn cubes at 39.00 pounds each, together with snakes and magics. I say OddzOn because that is the name given in the copyright notice at the back of the instruction booklet, although in this country that name appears nowhere else. The packaging (a rectangular cardboard casing with a clear central panel holding the cube at an angle) gives the distributors as Toybrokers Ltd. This cube sometimes appears in the British Toys 'R' Us, and my family bought one in Jenners, the big Edinburgh department store, this summer. I wouldn't rate it as highly as the Ideal cubes. I had a quick look for clones in the sort of shop in London that I have seen them in in the past, but found none. I bought a couple of Taiwanese clones in Dublin a few weeks ago - they came in a cube-sized cardboard box with a picture of a cube on the front with two yellow centre pieces, five green edge pieces and five red corner pieces. I did not complain that the cube inside did not match this picture. I tried both the sample on display and one of the cubes which I later bought. They turned quite well. I did not try the other one until I got it home, when to my annoyance I found it much stiffer. The colours are rather dull, but yellow and white are opposite, which I prefer, because then the colours of the opposite faces seem to have a sensible connection helping recognition of a piece that is in the opposite face to its home face. They cost 35.00 Irish pounds each . Among various other puzzle, Hamleys also has those from Meffert, including the skewb and an annoying dodecahedron - the colours are duplicated at opposite poles. So does Toys 'R' Us. What I have not been able to find is the 5x5x5 that is shown on the Meffert packaging. The OddzOn cube is the one that is associated with the www.rubiks.com site, which reproduces the instruction booklet and uses the same logo in chubby capitals. Rubik himself is clearly involved - he is quoted on the site. I had concluded that Ideal Toys (the US company) had gone out of business, having failed to find any reference to it currently. There is a British company Ideal Toys (UK) Limited, but that is a subsidiary of Triumph Adler AG. There was a company The Ideal Toy Company Limited, but that was dissolved, as was CBS Toys Limited, which may have been connected. OddzOn Products Inc appears to be a subsidiary of Hasbro Inc. I have a suspicion that Ideal may have deliberately adopted the yellow-opposite-green configuration to create a new colour arrangement that would help them expunge clones by relying on their trade dress. Roger Broadie From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 16 14:26:45 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA23966; Sun, 16 Nov 1997 14:26:44 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Nov 16 06:12:15 1997 From: roger.broadie@iclweb.com (Roger Broadie) To: "Cube-Lovers" Subject: Re: Cubes in London Date: Sun, 16 Nov 1997 11:07:06 -0000 Message-Id: <19971116110836.AAA18652@home> I apparently wrote > .. at 39.00 pounds .. Before people get the wrong idea about the price of cubes on this side of the Atlantic, I'd better say that the OddZon cube was 9 British pounds in Hamleys and the Dublin clone was 5 Irish pounds. I used the pound symbol and the conversion - I suspect both machine and human - went awry. I can add to my slightly meandering note on the Dublin clone that the central spider has now bust. Life is full of new hazards. Roger Broadie [ Sorry, you're a victim of moderator error. While replacing the pound symbol with the word "pounds", and I left in an extra 3. Thanks for the information. --Dan ] From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 16 14:57:58 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA24068; Sun, 16 Nov 1997 14:57:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Nov 16 07:24:33 1997 Message-Id: <199711161045.KAA30105@GPO.iol.ie> From: "Goyra (David Byrden)" To: Cube-Lovers@ai.mit.edu Subject: Re: Cubes in London, plus OddzOn, clones and colours Date: Sun, 16 Nov 1997 10:43:38 -0000 > From: Roger Broadie > What I have not been able to find is > the 5x5x5 that is shown on the Meffert packaging. Try writing to Dr. Christophe Banelow An Der Wabeck 37 D-58456 Witten Germany tel: 49 2302 71147 fax: 49 2302 77001 I have his catalogue here and he lists the 5^3, Skewb, Dedecahedron, Pyraminx, Octahedron, Magic Jewel, among others. David From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 00:07:22 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id AAA26194; Mon, 17 Nov 1997 00:07:22 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Nov 16 17:31:22 1997 Date: Sun, 16 Nov 1997 17:30:08 -0500 (EST) From: Nicholas Bodley To: "Goyra (David Byrden)" Cc: Cube-Lovers@ai.mit.edu Subject: Re: Cubes in London, plus OddzOn, clones and colours In-Reply-To: <199711161045.KAA30105@GPO.iol.ie> Message-Id: On Sun, 16 Nov 1997, Goyra (David Byrden) wrote: {Snips} } } Try writing to Dr. Christophe Banelow } An Der Wabeck 37 } D-58456 Witten } Germany } tel: 49 2302 71147 } fax: 49 2302 77001 I hope I'm not being rude to point out minor typos; his name should be "Christoph Bandelow". It's an easy slip to make. Regards, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* 'T was the night before Xmas, and all through |* Amateur musician *|* the coffeehouse, not a creature was stirring. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 21:35:42 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA01646; Mon, 17 Nov 1997 21:35:41 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Nov 17 04:26:22 1997 Message-Id: <34700D89.24426818@ibm.net> Date: Mon, 17 Nov 1997 01:25:29 -0800 From: "Jin 'Time Traveler' Kim" Organization: The Fourth Dimension To: Cube-Lovers@ai.mit.edu Subject: Color schemes revisited References: <5dlt1c$baq@gap.cco.caltech.edu> An interesting thing to note regarding the color patterns on cubes... on the Rubik's Cube home page (http://www.rubiks.com) a picture is displayed showing Dr. Rubik himself holding a mixed cube in his hand. On a whim I decided to figure out what the color scheme of the cube was. If you wish to figure out yourself without being told, or if you just want to try to refute my guess (I'm no stranger to being wrong) then don't read the "answer" to the puzzle that's below my .sig. Otherwise, if you can't be bothered with minor trivialities like this one (it's really not that difficult to figure out the colors anyway) then read on. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com I determined that the color scheme of the cube held in Erno Rubik's hand is: Front: Red Back: Orange Left: Green Right: Blue Top: White Bottom: Yellow From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 22:05:10 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA01784; Mon, 17 Nov 1997 22:05:10 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Nov 17 10:56:56 1997 Date: Mon, 17 Nov 1997 15:53:23 GMT From: David Singmaster To: chrono@ibm.net Cc: cube-lovers@ai.mit.edu Message-Id: <009BD6F9.65F6A7C1.425@ice.sbu.ac.uk> Subject: Re: 6x6x6 cube design I'm sure that this has been mentioned before, but the 6^3 etc. actually introduce no further complications than present on the 4^3 (and 5^3). There are just more types of center pieces, but they all behave in much the same way. In my message on notation and solution of the 4^3, I gave a method of producing a 3-cycle of center pieces and it can be used for each class of centre pieces - the puzzle doesn't get any more interesting, just longer! The 6^3 introduces a slightly interesting feature theoretically in that the center pieces break up into more classes than one might initially expect because the piece at the (1,2) location is not in the same class as the piece at the (2,1) location. (Taking a corner as (0,0).) DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 22:44:50 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA01939; Mon, 17 Nov 1997 22:44:49 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Nov 17 10:58:38 1997 Date: Mon, 17 Nov 1997 15:57:19 GMT From: David Singmaster To: cubeman@idirect.com Cc: cube-lovers@ai.mit.edu Message-Id: <009BD6F9.F2D7ACBC.209@ice.sbu.ac.uk> Subject: Re: Cube Colours According to what I wrote in my Cubic Circular 3/4 of Summer 1982, the cubes used in the World Championship were of the +/- yellow BOY pattern. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 18 23:50:41 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA08835; Tue, 18 Nov 1997 23:50:41 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Nov 18 23:43:40 1997 To: Cube-Lovers@ai.mit.edu Date: Tue, 18 Nov 1997 20:42:55 -0800 Subject: Rubiks Revenge moves Message-Id: <19971118.204255.7126.1.tenie1@juno.com> From: tenie1@juno.com (Tenie Remmel) Is there an easy way to cycle three adjacent top edges on the Rubiks Revenge? I can't find one shorter than 62 moves, but if there was a short one I could simplify my solution greatly. . b c . . a b . a . . . => c . . . . . . . . . . . . . . . . . . . Hopefully it won't mess up the corners, but it's ok if it does. I'd also like to see some short moves for the following 3-cycles: . * * . . . * . . . . . . . * . . . . . * . . * * . . * . . . * * . . . . . . . . . . * * . . . . . . . . . . . . . . . . . . . Is there a good source anywhere for moves, pretty patterns, etc. for the Rubiks Revenge? It's quite difficult to find information about it. Also is there an automatic move generating program for the higher order cubes like 'Cube Explorer' is for the 3x3x3? --Tenie Remmel (tjr19@juno.com) From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 11:46:33 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA19225; Thu, 20 Nov 1997 11:46:32 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 08:29:34 1997 From: bagleyd@americas.sun.sed.monmouth.army.mil (David Bagley x21081) Message-Id: <199711191329.IAA26271@java.sed.monmouth.army.mil> Subject: Re: A 4 Dimensional Rubik's Cube To: charlied@erols.com (Charlie Dickman), Cube-Lovers@ai.mit.edu Date: Wed, 19 Nov 1997 08:29:09 -0500 (EST) In-Reply-To: from "Charlie Dickman" at Nov 17, 97 08:01:03 pm Hi All I added Charlie Dickman's Tesseract (A 4 Dimensional Rubik's Cube) to my web pages ( http://www.tux.org/~bagleyd/ ). Its in two parts, the docs (mind twisting stuff) and the Mac Program. Charlie Dickman: If you make any updates I'll be happy to update the pages. By the way, I recently reorganized my web pages.... same old junk but its presented better. :) -- Cheers, /X\ David A. Bagley (( X bagleyd@bigfoot.com http://www.tux.org/~bagleyd/ \X/ xlockmore and more ftp://ftp.tux.org/pub/people/david-bagley From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 12:17:58 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA19324; Thu, 20 Nov 1997 12:17:58 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 16:10:23 1997 Sender: davidb@davidb.concentric.net Message-Id: <34735538.113A5129@iname.com> Date: Wed, 19 Nov 1997 13:08:08 -0800 From: David Barr Organization: Medweb To: Tenie Remmel , Cube-Lovers Subject: Re: Rubiks Revenge moves References: <19971118.204255.7126.1.tenie1@juno.com> Tenie Remmel wrote: > > Is there an easy way to cycle three adjacent top edges on the > Rubiks Revenge? I can't find one shorter than 62 moves, but if > there was a short one I could simplify my solution greatly. > > . b c . . a b . > a . . . => c . . . > . . . . . . . . > . . . . . . . . I hold the cube so the bottom looks like this: . . a . . . . b . . . c . . . . and do this sequence: F' b2 L2 / R' D r' D' R D r D' / L2 b2 F Capital letters are outer slices. Small letters are inner slices. The slashes are just to show the different parts of the sequence. The middle part, if performed alone, will cycle three edges. The first part of the sequence positions the cubies we want to move into the positions of the cubies that are cycled by the middle sequence. The last part of the sequence simply reverses the first part. Left view of cube: . . . . . . . a . . . . . c b . b2 R2 / L D' l D L' D' l' D / R2 b2 Bottom view of cube: . . a . c . . b . . . . . . . . F' / R' D r' D' R D r D' / F Bottom view of cube: . . . . a . . c . . . b . . . . b2 U' F / R' D r' D' R D r D' / F' U b2 Bottom view of cube: . . a . . . . b c . . . . . . . F' b' L2 / R' D r' D' R D r D' / L2 b F Here are some other three cycles you may find useful: R' D l D' R D l' D' R' D L D' R D L' D' r' D l D' r D l' D' -- mailto:davidbarr@iname.com http://www.concentric.net/~Davebarr/ From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 12:50:09 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA19471; Thu, 20 Nov 1997 12:50:07 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Nov 20 12:06:06 1997 Date: Thu, 20 Nov 1997 12:05:48 -0500 Message-Id: <20Nov1997.115617.Hoey@AIC.NRL.Navy.Mil> From: Dan Hoey Sender: Cube-Lovers-Request@ai.mit.edu To: cube-lovers@ai.mit.edu Subject: Auction on Rubik's Revenge (4x4x4) cubes (REPOST) Reply-To: Paul Hart whuang@ugcs.caltech.edu (Wei-Hwa Huang) has passed on a Usenet announcement of Paul Hart's auction of 6 unopened Rubik's Revenges, as mentioned previously in Cube-Lovers. The auction ends November 22. For details on the offer read http://www.enol.com/~hart, check a Usenet search engine, or inquire by e-mail to Paul Hart . - Dan Hoey Interim Cube-Lovers-Request operator From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 13:16:42 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA19547; Thu, 20 Nov 1997 13:16:42 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 09:32:05 1997 Date: Wed, 19 Nov 1997 14:30:34 GMT From: David Singmaster To: chrono@ibm.net Cc: cube-lovers@ai.mit.edu Message-Id: <009BD880.292EDB5C.279@ice.sbu.ac.uk> Subject: RE: Color schemes revisited In 1979(?) when I had my company David Singmaster Ltd which dealt in Cubes and cube-related items, we had a tee-shirt designed showing a jumbled cube with the caption Rubik's Cube Cures Sanity. Only one person ever wrote in pointing out that the cube was impossible! From the colouring of various visible pieces, one could tell that the white face was adjacent to all five other colours!! DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 13:47:50 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA19685; Thu, 20 Nov 1997 13:47:48 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 08:55:52 1997 Message-Id: In-Reply-To: <19971118.204255.7126.1.tenie1@juno.com> Date: Wed, 19 Nov 1997 08:56:30 -0500 To: Tenie Remmel , Cube-Lovers@ai.mit.edu From: Nichael Lynn Cramer Subject: Re: Rubiks Revenge moves Tenie Remmel wrote: >Is there an easy way to cycle three adjacent top edges on the >Rubiks Revenge? I can't find one shorter than 62 moves, but if >there was a short one I could simplify my solution greatly. > >. b c . . a b . >a . . . => c . . . >. . . . . . . . >. . . . . . . . > >Hopefully it won't mess up the corners, but it's ok if it does. The way I approach this is to begin with the following simple 3-cycle for edge cubies (note that this cycles only the cubies and leave the rest of the cube unaltered): 1] Imagine the involved cubies in the following configuration: Top face : . . c . Left face: . . . . . . . . a . . . . . . . . . . . . b . . . . . . 2] Perform the following sequence: - Rotate Front Face by 1/4 turn clockwise. - Rotate the slice just below the Top Layer by 180 dgs. - Rotate the Front Face by 1/4 turn counter-clockwise. - Rotate the Top Face by 180 dg. - Rotate Front Face by 1/4 turn clockwise. - Rotate the slice just below the Top Layer by 180 dgs. - Rotate the Front Face by 1/4 turn counter-clockwise. - Rotate the Top Face by 180 dg. This will result in: Top face : . . b . Left face: . . . . . . . . c . . . . . . . . . . . . a . . . . . . with all other cubies in their original locations. 3] Once this step is mastered, it is now only a question of moving the cubies that you want to swap into the approriate location for this operator to do its work. For example, in your example above this can be accomplished by (this assumes that the figure you have drawn above is your Top Face): - Rotating the Left-most two slices 1/4 turn clockwise (i.e. towards you) - Rotating the Top Face 1/4 turn counter-clockwise. If you now rotate the entire cube by 90dgs clockwise, you will see your three cubies are now in the proper location to use the above operator. (When you're done with the operator, repeat the steps just above in the reverse order to finish.) >I'd also like to see some short moves for the following 3-cycles: > >. * * . . . * . . . . . . . * . >. . . . * . . * * . . * . . . * >* . . . . . . . . . . * * . . . >. . . . . . . . . . . . . . . . These are just variations on the above. They will be left an exercise for the reader. ;-) Hope this helps. Nichael Nichael nichael@sover.net deep autumn my neighbor what does she do http://www.sover.net/~nichael/ --Basho From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 20:37:44 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA22609; Thu, 20 Nov 1997 20:37:43 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Nov 20 13:51:23 1997 Sender: davidb@davidb.concentric.net Message-Id: <3474866B.D1136871@iname.com> Date: Thu, 20 Nov 1997 10:50:19 -0800 From: David Barr Organization: Medweb To: Cube-Lovers Subject: Re: Rubiks Revenge moves References: <19971118.204255.7126.1.tenie1@juno.com> <34735538.113A5129@iname.com> David Barr wrote: > I hold the cube so the bottom looks like this: > > . . a . > . . . b > . . . c > . . . . > > and do this sequence: > > F' b2 L2 / R' D r' D' R D r D' / L2 b2 F Actually, you can save a couple moves by doing d2 L' R' D r' D' R D r D' L d2 but the pieces affected will be on the right side instead of the bottom. -- mailto:davidbarr@iname.com http://www.concentric.net/~Davebarr/ From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 20:54:43 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA22666; Thu, 20 Nov 1997 20:54:42 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 17:50:07 1997 Sender: mahoney@marlboro.edu Message-Id: <34736B2E.F9F4962@marlboro.edu> Date: Wed, 19 Nov 1997 17:41:50 -0500 From: Jim Mahoney Organization: Marlboro College To: Tenie Remmel Cc: Cube-Lovers@ai.mit.edu Subject: Re: Rubiks Revenge moves References: <19971118.204255.7126.1.tenie1@juno.com> Tenie Remmel wrote: > Is there an easy way to cycle three adjacent top edges on the > Rubiks Revenge? I can't find one shorter than 62 moves, but if > there was a short one I could simplify my solution greatly. > > . b c . . a b . > a . . . => c . . . > . . . . . . . . > . . . . . . . . You can cycle these three edges on the 4x4x4 in 14 quarter turns without disturbing the corners. With the "up" and "front" faces like this (in a kind of projection view; the corners are given by "*"; the "right" face is not shown), * . . * . . . . . . . C * A B * . . . . . . . . * . . * a procedure to cyle A,B,C is as follows: (1) 2 preparation moves which put C on "down" slice and B on "up/back" (2) 3 moves to get A off top slice and replace with C. (3) 2 moves (1/2 rotate) the top slice to put B where C (orginally A) was. (4) undo (2), restoring bottom layers and bring A back to top, in new spot. (5) undo (3) (6) undo (1), the prep moves. (In each case "undo" means to do the inverse of the same moves in the opposite order; that is, "undo" ABC means C'B'A' where C' is the inverse of move C.) The hardest part I have of describing the specifics of each of these is the notation; each of the 6 steps is only a couple of moves. Let me define U,R,F as the Up, Right, and Front faces, and number the slices by integers 1 to 4, so for example (F1,F2,F3,F4) are clockwise quarter turns on the 4 slices (front to back) parallel to the front face. Counterclockwise turns are indicated with either a ' (indicating "inverse") or lower case, so F1' = a counterclockwise quarter turn of the front face. In pictures, * - - * * - - * . 1 2 . . 3 1 . . 3 4 . . 4 2 . * - - * ==> U1 ==> * - - * | a b | | a b | | c d | | c d | * - - * * - - * * - - * * - - * . 1 2 . . 1 2 . . 3 4 . . 3 4 . * - - * ==> F1' ==> * - - * | a b | | b d | | c d | | a c | * - - * * - - * where the letters are on the "front" face and the numbers are on "up" face. Then with this notation, steps (1) through (6) of this procedure are (1) F2 R2 (2) F3' U4' F3 (3) U1 U1 (4) F3' U4 F3' (5) U1' U1' (6) R2' F2' which is 14 moves. [Moderator's note: Certainly (4) should be F3' U4 F3, but that still cycles the wrong edges. With (2)=R2 U4' R2', (4)=R2 U4 U2' we cycle the correct triple of edges, but in inverse order. ] I confess that I don't have a 4x4x4 anymore and so can't try this - I may have visualized one of the details wrong. Hope not. > I'd also like to see some short moves for the following 3-cycles: > > . * * . . . * . . . . . . . * . > . . . . * . . * * . . * . . . * > * . . . . . . . . . . * * . . . > . . . . . . . . . . . . . . . . You can do any of these as variations of the method I give above with different "preperation" moves to get the edges into the proper positions, namely two on the same slice which can be turned into one another (the top slice in these cases), and the third on another slice (usually the bottom slice) which can replace one of the edges from the top slice. I have a discussion of the NxNxN cube which includes in section (VI) this recipe for 3-cycles of any kind of edge, corner, or face piece; you can read it at http://www.marlboro.edu/~mahoney/cube/NxN.txt if you're interested. Regards, Jim Mahoney (mahoney@marlboro.edu) Marlboro College From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 21:52:54 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA22910; Thu, 20 Nov 1997 21:52:53 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 11:58:12 1997 Date: Wed, 19 Nov 1997 11:53:14 -0500 Message-Id: <000E9C40.001706@scudder.com> From: jdavenport@scudder.com (Jacob Davenport) Subject: Re: Rubiks Revenge moves To: Cube-Lovers@ai.mit.edu Forget three adjacent top edges, I just want to cycle two of them. I've been solving a 5x5x5, and finally figured out how to make it look like a 3x3x3 so that I could solve nearly all of it. However, the one place where I cannot do that is solving the second and fourth edges from any side, and have been using a short move that cycles three of them: .axx. .bxx. y.... z.... y.... => z.... b.... a.... .czz. .cyy. The move is 2L F' L F 2L' (where 2L means the second layer from the left) This works great for getting nearly all the edges in place, but I have two edges that are switched, and every time I use this move to put them in place, I either leave two other edges out of place or leave four edges out of place. That is, I have the following: .baa. .aaa. b.... c.... b.... which I can only make into c.... a.... b.... .ccc. .cbb. which does not help. I believe that my move works on 4x4x4 edges, and any move that helps a 4x4x4 cube will probably help me. From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 22 22:56:58 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA07129; Sat, 22 Nov 1997 22:56:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Nov 22 17:53:28 1997 From: roger.broadie@iclweb.com (Roger Broadie) To: Cc: "Tenie Remmel" Subject: Re: Rubiks Revenge moves Date: Sat, 22 Nov 1997 22:51:33 -0000 Message-Id: <19971122224927.AAA7296@home> Tenie Remmel wrote (19 November 1997 ) > Is there an easy way to cycle three adjacent top edges on the > Rubiks Revenge? I can't find one shorter than 62 moves, but if > there was a short one I could simplify my solution greatly. > > . b c . . a b . > a . . . => c . . . > . . . . . . . . > . . . . . . . . Rather than just throw a few more solutions into the pot, I'd like to start with some comments on the sort of process everyone, including me, seems to use to deliver 3-cycles of edge pieces in the 4x4x4. It is of the general form [P, TQT'] where the square brackets are used to show a commutator, that is, [A,B] means ABA'B'. In this process P and Q are turns of layers that are parallel to one another, and T is a turn of a layer transverse to P and Q. For instance, P and Q could be L and r and T could be U (capitals for the outer layers, lower case for the neighbouring inner layers, with sense parallel to the corresponding outer layer). That gives [L, UrU'] == L UrU'.L' Ur'U' which is a (not especially appealing) 3-cycle of edges. In fact any process of this form is a 3-cycle provided it takes one piece from the layer Q into the layer P. That will happen if T is a quarter turn in either sense - I haven't found anything useful with T as a half turn. But P and Q can be any power. The reason that processes of this form are 3-cycles is simple. If two permutations intersect at only one element, then their commutator is a 3-cycle. Thus if A = (...a1, x, a2...) and B = (...b1, x, b2...) then [A,B] -> (a1, x, b1) If you do just UrU' you will find there is a line of displaced pieces along the intersection Ub, but no other displaced piece in any of the layers parallel to r. Any of these pieces can be picked out to form part of a 3-cycle by selecting the layer that is parallel to r and contains the piece and using a turn of that layer as the component P of the commutator, with UrU' forming the component TQT'. In general, if all of P, Q and T are outer layers we will have a 3-cycle of corner pieces, if two are outer layers and one an inner layer we will have a 3-cycle of edge pieces, if one is an outer layer and the other two inner layers we will have a 3-cycle of centre pieces, and if all three are inner layers we will have done nothing visible to the cube, but in fact there will have been an invisible 3-cycle of the pieces of the imaginary internal 2x2x2. We can derive the last of these cases from the first quite neatly applying a fascinating concept called evisceration, which I recently met trawling through the archives. It was first quoted from David Singmaster's Cubic Circular by Stan Isaacs on 26 May 1983 and our present acting moderator also discussed it on 1 June 1983. If you turn a cube inside out by changing each outer layer in a process into an inner and vice-versa (i.e. capitals to lower case), then, in the effect of the process, you will interchange corner pieces with the pieces of the internal 2x2x2, and edge pieces with centre pieces. Making P, Q and T all to be outer layers gives just a 3-cycle of corner pieces; therefore applying evisceration takes that cycle into one on the pieces of the internal 2x2x2. Singmaster's Notes on Rubik's Magic Cube, the fifth edition, interprets processes of the type [P, TQT'] as [P,[T,Q]]. This expands to P TQT'Q'.P' QTQ'T', but the sequence Q'P'Q in the centre reduces to just P', giving the same expansion as before. Of course, the two components of the commutators TQT' and [T,Q] have different total effects, but what they have in common is that they put the same single piece into P. We can look at them both as being sort of like a mono-operation. Let's call it a "monopop": each process pops a piece into P; you then turn P, then reverse the pop operation, which extracts a different piece, and finally restore P. It's relatively straightforward to use this form to design specific processes. Say we want to move an edge piece from ULf to FLd and keep the third member of the 3-cycle in the top layer. Then we can take P to be L to achieve the required part of the cycle. We now know that Q must be in r or l. Let's take l. The transverse move in T has to take a piece from l into the point of intersection of the two components of the commutator, FLd. So it must be in F. Playing with F and l shows that the following does the job. [L, FlF'] == L FlF'. L' Fl'F' -> (ULf, FLd, UBl) If we'd taken Q to be in r we'd have needed a bit more care to keep the third piece of the 3-cycle in the top layer, but [L, F'r^2F] does, putting it at UBr. If we want to move a piece along a diagonal - from ULf to FLu, say - we need to use the other component of the commutator, TQT'. Thus we can build 3-cycles which include ULf to FLu around the component U'FU. For instance [f', U'FU] -> (ULf, LFu, RUf) With a clear head and a good following wind it's possible to work out these processes on the fly. They also transform nicely into another process of the same type by cycling the elements, which has the effect of conjugating the original process. Thus the last process can be dealt with as follows U[f', U'FU]U' = [Uf'U', F] -> (UFr, LFu, BUr) This cycling procedure comes from Singmaster. Let's now think about top-layer edge processes. I'll denote the pieces like this. X a1 a2 X d2 o o b1 d1 o o b2 X c2 c1 X The purpose of the numbering in pairs is to emphasize that the processes come in pairs. Each process has a twin created by changing each inner layer turn into its next-door inner neighbour. Thus the simplest U process of the general type we're using is of this form: [l, F'LF] -> (a1, c2, d1) Its twin is [r', F'LF] -> (a2, c1, d2) In the twin process, each edge piece is changed into its next door neighbour. We want to capture this regularity. I will therefore represent this pair by [M', F'LF] -> (a*, c', d) In this representation, M is either r or l', the asterisked piece defines the layer that contains M and primed letters denote a piece with the opposite suffix number to the asterisked piece. Obviously, these suffixes are closely related to flip in a 3x3x3 and the assignment of the numbers is arbitrary. Some assignments are more helpful than others in a particular context, and the method used in the diagram above is the obvious one of giving the same number ("flipperty"?) to the pieces in the positions that a single piece moves into during a complete U turn. Here, then is a complete set of top layer 3-cycles of edge pieces, to within a reflection. It comes from a fairly systematic search I did for processes of the type [P, TQT'] that can be conjugated by at most one turn into a top-layer process. They are oriented in a way I find easy to do. I will leave them as commutators, because it is very easy to perform the full set of turns from them. The T/T' sequence remains constant for both halves and the only adjustment needed is to invert P and Q the second time around. Inverses are also easy to perform, since all one has to do is read off the second component first. (a*, c', d) [M', F'LF] (a*, b, c) F2 [D R2 D', M2] F2 (c*, a, b') R2 [M' D' M, U2] R2 (d1, b2, b1) (Bb)' [U l U', R2] (Bb) (d2, b1, b2) (Ff) [U r' U', R2] (Ff)' (d2, c2, d1) (Bb) [L2, D l D'] (Bb)' (Tenie's question) (d1, c1, d2) f' [L2, D r' D'] f The last two pairs could have been left in the M form if say N was introduced to represent either f or b'. But keeping them separate lets us save a wrist movement for the first three by combining the inner and neighbouring outer layers for a turn relative to the central cut. That won't work for the final process, since the F layer is already included in the 3-cycle and can't be amalgamated with the f layer. All these process can be directly transferred to the 3x3x3, using the one single central slice as M (or l' or r). The primes then correspond to flipping the edge piece relative to the top surface. Roger Broadie 22 November 1997 From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 22 23:38:50 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA07249; Sat, 22 Nov 1997 23:38:49 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Nov 22 08:31:49 1997 Message-Id: <3.0.32.19971122082850.007b25c0@po9.mit.edu> Date: Sat, 22 Nov 1997 08:28:51 -0500 To: cube-lovers@ai.mit.edu From: Dennis Okon Subject: Large Cube I realize this is impossible to make, at least in the traditional way, but... I just saw an add for some computer conference which displayed a 9x9x9 cube (with some rather strange colors - at least 7 different ones too!). The text read something like (really paraphrased): "When you were young you could work a cube in 27 seconds, but now you're older and only have a week to solve this." So, my question is: Assuming the cube is solvable, can it be done? What kind of order of growth to the current algorithms, for human algorithms and computer algorithms, have in relation to the size of the cube? -Dennis [ Moderator's note: For solvability, see J.A. Eidswick's article "Cubelike Puzzles -- What Are They and How Do You Solve Them?" (American Mathematical Monthly', 93:3 (March 1986), pp. 157-176). There are some loose bounds in my two articles of 24 Jun 1987 in the archives at . ] From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 24 20:30:40 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA05807; Mon, 24 Nov 1997 20:30:40 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Nov 23 06:10:45 1997 Message-Id: <199711231109.LAA02433@GPO.iol.ie> From: "Goyra (David Byrden)" To: Subject: Re: Large Cube Date: Sun, 23 Nov 1997 11:01:33 -0000 > From: Dennis Okon > I just saw an add for some computer conference which displayed a 9x9x9 cube Java version can be played with at http://www.iol.ie/~goyra/Rubik.html a "back" button is under development, too. David From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 24 21:00:19 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA05989; Mon, 24 Nov 1997 21:00:19 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Nov 24 19:45:36 1997 To: Cube-Lovers@AI.MIT.Edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Rubiks Revenge moves Date: 25 Nov 1997 00:44:22 GMT Organization: California Institute of Technology, Pasadena Message-Id: <65d716$29p@gap.cco.caltech.edu> References: roger.broadie@iclweb.com (Roger Broadie) writes: >Tenie Remmel wrote (19 November 1997 ) >> Is there an easy way to cycle three adjacent top edges on the >> Rubiks Revenge? I can't find one shorter than 62 moves, but if >> there was a short one I could simplify my solution greatly. >> >> . b c . . a b . >> a . . . => c . . . >> . . . . . . . . >> . . . . . . . . >Rather than just throw a few more solutions into the pot, I'd like to start >with some comments on the sort of process everyone, including me, seems to >use to deliver 3-cycles of edge pieces in the 4x4x4. It is of the general >form > [P, TQT'] >where the square brackets are used to show a commutator, that is, [A,B] >means ABA'B'. >In this process P and Q are turns of layers that are parallel to one >another, and T is a turn of a layer transverse to P and Q. Count me among the few "self-taught" solvers who don't actually use this, then. The one I worked out for myself a long time ago turns out to be: [r, FUF'] which is of a similar form, but P and Q are not parallel. As a consequence of this, the permutation is not "clean": i.e., some other cubies get disturbed. As these are all face cubies anyway, I just modified my solution so that I do the face cubies last. :-) -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- "[Lucy's eyes] look like little round dots of India ink..." -- Charlie Brown From cube-lovers-errors@mc.lcs.mit.edu Thu Dec 4 21:21:56 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA26609; Thu, 4 Dec 1997 21:21:56 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Dec 4 11:42:31 1997 Message-Id: <19971204164104.18805.qmail@hotmail.com> From: "John Coffey" To: Cube-Lovers@ai.mit.edu Date: Thu, 04 Dec 1997 09:41:03 MST I have made a DOS program that solves the square-1 rubik's cube variant. If you would like to have this program then please contact me. Source code is also available. john2001@hotmail.com John Coffey. http://www.xmission.com/~jrcoffey/chess.htm http://www.xmission.com/~jrcoffey/play.htm [ Moderator's note: Soon to appear in ftp://ftp.ai.mit.edu/pub/cube-lovers/contrib/ ] From cube-lovers-errors@mc.lcs.mit.edu Mon Dec 22 20:20:10 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA00821; Mon, 22 Dec 1997 20:20:09 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Dec 22 19:05:02 1997 Message-Id: <199712230003.AAA21827@GPO.iol.ie> From: "David Byrden" To: Subject: Return of the Cube? Date: Tue, 23 Dec 1997 00:01:20 -0000 I have just seen a toy expert on UK tv say that the Rubik Cube is making a comeback this year. Any comment? David From cube-lovers-errors@mc.lcs.mit.edu Tue Dec 23 20:21:04 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA08651; Tue, 23 Dec 1997 20:21:03 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Dec 23 18:19:28 1997 Message-Id: <199712232318.BAA01255@mail2.dial-up.net> From: "Frederik Strauss" To: Subject: Re: Return of the cube? Date: Wed, 24 Dec 1997 01:18:17 +0200 Here in South Africa the Rubiks cube is now for sale in one of the biggest newsagents, where previously it was hard to find anywhere. From cube-lovers-errors@mc.lcs.mit.edu Tue Dec 23 21:04:46 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA08741; Tue, 23 Dec 1997 21:04:46 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Dec 23 03:03:42 1997 Message-Id: <349F637D.4DA23A12@ibm.net> Date: Mon, 22 Dec 1997 23:08:45 -0800 From: "Jin 'Time Traveler' Kim" Organization: The Fourth Dimension To: Cube-Lovers@ai.mit.edu Subject: Re: Return of the Cube? References: <199712230003.AAA21827@GPO.iol.ie> David Byrden wrote: > I have just seen a toy expert > on UK tv say that the Rubik Cube is > making a comeback this year. Any > comment? > > David How about, "yay?" I hope that by having the cube re-emerge and become once again a fairly popular item, many other unique pieces will make a return, like the RUBIKS REVENGE, as an example. And perhaps then they will become a stable commodity instead of just burning out after just an intense period of a couple of years. A Hula Hoop is a fad. A piece like the Cube is Timeless. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa / Puente Hills http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@mc.lcs.mit.edu Wed Dec 24 19:28:46 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA01798; Wed, 24 Dec 1997 19:28:45 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Dec 24 17:38:35 1997 Message-Id: <199712242237.AAA11999@mail2.dial-up.net> From: "Frederik Strauss" To: "Cube Lovers" Subject: 5x5x5 Date: Wed, 24 Dec 1997 00:20:56 +0200 Hi... I have the 5x5x5 cube and can solve it, but it takes me about 15 minutes. I've looked on the net but can find nothing on this cube. Does anyone where I can get a bit more info on it, like moves or patterns? If anyone else can solve it, I'd like to know how you do it. Cu Fred fstrauss@icon.co.za From cube-lovers-errors@mc.lcs.mit.edu Thu Dec 25 23:43:21 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA05224; Thu, 25 Dec 1997 23:43:20 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Dec 25 19:23:57 1997 Date: Thu, 25 Dec 1997 19:22:44 -0500 (EST) From: Nicholas Bodley To: "Jin 'Time Traveler' Kim" Cc: Cube Mailing List Subject: Re: Return of the Cube? In-Reply-To: <349F637D.4DA23A12@ibm.net> Message-Id: Jin Kim's comment, with which I heartily agree, leads me to wonder whether some entrepreneur might support small-scale manufacture of a very well-made 3^3, on the order of the Ideal deluxe Cube (I have forgotten what they called it). That Cube had a redesigned mechanism (it differed only in the details) that would tolerate much more misalignment, before making a maneuver, than the typical Cubes. It had attached plastic colored tiles instead of stickers, and was made of an excellent "engineering plastic". There are commercially-available materials that are self-lubricating, and these could be used for the wearing surfaces. The pivots could be true bearings, that is to say, like those in a typical piece of machinery. (The Ideal Cube might have had such.} Tests to 250,000 revolutions might be reasonable. Whether it makes sense to try to improve upon that already very-good design, I can't say. Consider chess, go, checkers or dominoes. I think the Cube quite rightly should take its place with them. Imho, the Cube and its direct derivatives are the most ingenious mechanisms ever invented. On Mon, 22 Dec 1997, Jin 'Time Traveler' Kim wrote: {Lots snipped} } A piece like the Cube is Timeless. -- }Jin "Time Traveler" Kim |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------Amateur musician-------------- |* nbodley@tiac.net <<<-- Possible change to nbodley@shore.net; will let oodles of folks know if I do. I'd try to overlap by a month or so to give time to change over. Apologies in advance if so! Hope I don't have to. From cube-lovers-errors@mc.lcs.mit.edu Fri Dec 26 22:50:16 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA08130; Fri, 26 Dec 1997 22:50:16 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Dec 26 04:04:16 1997 Date: Fri, 26 Dec 1997 10:03:05 +0100 (MET) Message-Id: <1.5.4.16.19971226100243.1137cba4@mailsvr.pt.lu> X-Sender: geohelm@mailsvr.pt.lu To: "Frederik Strauss" From: Georges Helm Subject: Re: 5x5x5 Cc: Cube-Lovers@ai.mit.edu I have a German book by Kurt ENDL on how to solve the whole bunch of 2x2x2, 3x3x3, 4x4x4 and 5x5x5 cubes. I have a xeroxed copy of a solution by myself. I do upper middles, edges, corners. Then 2d, 3d and 4th layer edges. Then 2d, 3d and 4th layer middles. Then last layer corners and finally last layer edges. Sometimes parity is uneven, i,e, there remain 2 edges to swap, and there is a move I use to resolve that problem without disturbing the rest of the cube by Helmut GEMBITZKY. Georges Helm geohelm@pt.lu http://ourworld.compuserve.com/homepages/Georges_Helm/ http://www.geocities.com/Athens/2715 [ Moderator's note: As has been mentioned previously, there is a general solution method in Eidswick, J. A., "Cubelike Puzzles -- What Are They and How Do You Solve Them?", 'American Mathematical Monthly', Vol. 93, #3, March 1986, pp. 157-176, though it's not optimized. ] From cube-lovers-errors@mc.lcs.mit.edu Sat Dec 27 20:24:59 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA10880; Sat, 27 Dec 1997 20:24:59 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Dec 27 11:45:01 1997 Message-Id: Date: Sat, 27 Dec 1997 11:43:58 -0500 To: cube-lovers From: Charlie Dickman Subject: Eidswick Reference In a recent message regarding a question about solving the 5x5x5 cube our moderator mentioned the reference Eidswick, J.A., "Cubelike Puzzles -- What Are They and How Do You Solve Them?", 'American Mathematical Monthly', Vol. 93, #3, March 1986, pp. 157-176 Does anyone know if this article/document is available on the web? Charlie Dickman charlied@erols.com From cube-lovers-errors@mc.lcs.mit.edu Tue Dec 30 18:20:01 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA24605; Tue, 30 Dec 1997 18:20:00 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Dec 29 08:34:36 1997 Message-Id: <3.0.3.32.19971229083148.0055d700@caddscan.com> Date: Mon, 29 Dec 1997 08:31:48 -0500 To: Cube-Lovers@ai.mit.edu From: "Bryan Main" Subject: Re: 5x5x5 In-Reply-To: <1.5.4.16.19971226100243.1137cba4@mailsvr.pt.lu> At 10:03 AM 12/26/97 +0100, Georges Helm wrote: >I have a German book by Kurt ENDL on how to solve the whole bunch of 2x2x2, >3x3x3, 4x4x4 and 5x5x5 cubes. >I have a xeroxed copy of a solution by myself. >I do upper middles, edges, corners. >Then 2d, 3d and 4th layer edges. >Then 2d, 3d and 4th layer middles. >Then last layer corners and finally last layer edges. > >Sometimes parity is uneven, i,e, there remain 2 edges to swap, and >there is a move I use to resolve that problem without disturbing the >rest of the cube by Helmut GEMBITZKY. > >Georges Helm I just got one of these for christmas and had a question or two. First is there a cube program so I can play with it and not destroy all the work I have done? And I have solved one side, and all of the edges without much problems. However, can I solve the middle pieces without destroying the edges? As of yet I haven't found a way to keep the one side I have finished and move one of the center pieces on another side. I don't want moves, I just want to know if it is possible to solve this way or if I need to start looking at another way to solve it. bryan From cube-lovers-errors@mc.lcs.mit.edu Tue Dec 30 18:50:07 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA24683; Tue, 30 Dec 1997 18:50:07 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Dec 29 10:22:12 1997 Message-Id: Date: Mon, 29 Dec 1997 10:21:01 -0500 To: cube-lovers@ai.mit.edu From: kristin@tsi-telsys.com (Kristin Looney) Subject: a Rubiks Xmas not only did I get 24 new cubes (as a gift to fill out my gameroom window which now holds 120 cubes poised and ready for new cube art) but my 9 year old niece brought me a cube to mix up... which she then solved! anyone know any kids younger than 9 that can solve the cube? I'm sure there are younger cubists out there... but I sure was impressed. -Kristin kristin@wunderland.com http://www.wunderland.com _________________________________________________________ Kristin Looney / Manager, Information Systems / TSI TelSys Inc. 7100 Columbia Gateway Drive * Columbia, MD 21046 * 410.872.3939 klooney@tsi-telsys.com * www.tsi-telsys.com * fax 410.872.3901 From cube-lovers-errors@mc.lcs.mit.edu Tue Dec 30 19:54:57 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA24798; Tue, 30 Dec 1997 19:54:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Dec 29 10:42:38 1997 Date: Mon, 29 Dec 1997 10:27:04 -0500 Message-Id: <00115225.001706@scudder.com> From: jdavenport@scudder.com (Jacob Davenport) Subject: Re: 5x5x5 To: "Cube Lovers" My way of solving the 5x5x5 has been to think about the cube in 3x3x3 terms. When I solve a 3x3x3, I do top corners, bottom corners, top and bottom edges at the same time, and then middle edges. When I do a 5x5x5, I think of the middle corners (those cubies directly diagonal from the center) as corners to the 3x3x3, ignoring completely the outside edges, and I solve them so that all the middle corners are aligned like a 3x3x3 would be aligned. Then I solve the middle edges (those cubies directly next to the center) like I would solve the edges from 3x3x3 cubes. This leaves the nine cubies in the center of each face sovled. I then use a move which many people use when solving a 4x4x4 to get all the edge pieces together without disturbing the center squares. This finally leaves me with messed up corners, solved center squares, and the three edges on each side together. I then view this as a 3x3x3 and solve that using my normal method. The only drawback is that I sometimes cannot get the edges to all work together, and the reason is that I have inadvertantly switched two middle corners, and it takes a long time for me to fix them. If anyone wants more information on this solution, I'll spell it out in detail on my web page. -Jacob Davenport http://wunderland.com/wts/jake From cube-lovers-errors@mc.lcs.mit.edu Thu Jan 1 22:43:23 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA06147; Thu, 1 Jan 1998 22:43:23 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Dec 31 19:31:26 1997 To: Cube-Lovers@ai.mit.edu Date: Wed, 31 Dec 1997 14:31:51 -0800 Subject: Where to get Dino Cube? Message-Id: <19971231.143151.14726.0.tenie1@juno.com> From: tenie1@juno.com (Tenie Remmel) Is there a current source for the Dino Cube? Both the vertex turning kind (with 12 pieces one for each edge) and the edge turning kind (with 24 pieces four on each face). Is Gametrends still around? The phone number given in a message in 1995 does not work. --Tenie Remmel (tenie1@juno.com) From cube-lovers-errors@mc.lcs.mit.edu Thu Jan 1 23:48:05 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA06283; Thu, 1 Jan 1998 23:48:04 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Dec 31 20:13:48 1997 Date: Wed, 31 Dec 1997 17:12:53 -0800 (PST) Message-Id: To: Cube-Lovers@ai.mit.edu From: lowfrqcy@west.net (Ryan Blum) Subject: Getting a 4x4x4 or a 5x5x5 As a relative newbie to the cube world, I only have a 3x3x3. Would there be any chance that I would find a used 4x4x4 or 5x5x5 cube around for less than an arm and a leg? The new 4's went for around $100 in that auction a little while ago, and that scared me.... Any Info would be greatly appreciated! Thanks, Ryan [Moderator's note: The 5x5x5 cubes may be more available. You can use one to practice 4x4x4 moves by ignoring the central slices (except in that you have to keep them aligned with an adjacent slice).] From cube-lovers-errors@mc.lcs.mit.edu Fri Jan 2 00:51:34 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id AAA06440; Fri, 2 Jan 1998 00:51:34 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jan 1 08:00:23 1998 Message-Id: <199801011258.OAA17338@mail2.dial-up.net> From: "Frederik Strauss" To: "Cube Lovers" Subject: Re: 5x5x5 Date: Wed, 31 Dec 1997 04:17:14 +0200 >From: Bryan Main >Subject: Re: 5x5x5 >problems. However, can I solve the middle pieces without destroying the >edges? As of yet I haven't found a way to keep the one side I have Yes, as a matter of fact that is the current method I use to solve it, e-mail me if you want the moves. Cu Fred fstrauss@icon.co.za From cube-lovers-errors@mc.lcs.mit.edu Fri Jan 2 21:37:09 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA11918; Fri, 2 Jan 1998 21:37:08 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jan 2 12:33:26 1998 Date: Fri, 2 Jan 1998 09:31:22 PST From: ccw@eql12.caltech.edu (Chris Worrell) Message-Id: <980102092825.23006d20@eql12.caltech.edu> Subject: Re: Where to get Dino Cube? In-Reply-To: Your message <19971231.143151.14726.0.tenie1@juno.com> dated 31-Dec-1997 To: tenie1@juno.com Cc: cube-lovers@ai.mit.edu > Is Gametrends still around? The phone number given in > a message in 1995 does not work. I think I was the one who posted that. Gametrends (in Pasadena, CA) has been gone for more than a year. Chris Worrell (ccw@EQL12.caltech.edu) From cube-lovers-errors@mc.lcs.mit.edu Sat Jan 3 02:01:09 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id CAA12748; Sat, 3 Jan 1998 02:01:09 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jan 2 19:50:46 1998 Message-Id: In-Reply-To: <199712230003.AAA21827@GPO.iol.ie> Content-Type: text/plain; charset="us-ascii" Date: Fri, 2 Jan 1998 15:15:49 -0500 To: Cube-Lovers@ai.mit.edu From: Nichael Lynn Cramer Subject: Re: Return of the Cube? Cc: David Byrden David Byrden wrote: > I have just seen a toy expert >on UK tv say that the Rubik Cube is >making a comeback this year. Any >comment? NWell, this is more a "cube-sighting" than an answer to the question whether cubes will soon become more easily available, but I notice that in the new (i.e. 4Jan98) New York Times Book Review, on the inside of the front cover is a small ad consisting of a (b&w) photo of a (3X) cube with the over-laying copy: NEVER A DULL WEEKEND Our enhanced, two-part Weekend section is full of ideas about ways to broaden your horizons. I couldn't quite figure out, though, whether the ad intended the cube to symbolize the "broaden[ed] horizons", or the "dull weekend"... From cube-lovers-errors@mc.lcs.mit.edu Sat Jan 3 02:35:28 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id CAA12913; Sat, 3 Jan 1998 02:35:27 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Jan 3 01:40:29 1998 Date: Sat, 3 Jan 1998 01:39:21 -0500 Message-Id: <3Jan1998.011449.Alan@LCS.MIT.EDU> From: Alan Bawden To: Cube-Lovers@ai.mit.edu Subject: "A Message from Professor Erno Rubik" I just accidentally tripped across http://www.rubiks.com/. Much to my amusement the home page has a little message from Erno Rubik that begins: It is 23 years since I created the Cube, some 17 years since this simple little six-coloured object attained its great world wide appeal. I often wondered what impact the Internet would have had if it had been around at the time. Cube awareness, for one thing, would have spread even faster, aggravating the already severe Cube shortages in the market place. Suggestions and disputes about different approaches to solving it would surely have filled the screens. Amusing, unusual, interesting tales to do with the Cube would have criss-crossed the globe on the Net and intrigued mathematicians would have proposed and discussed Cube related theories on-line. Those of you who have been on this mailing list for the last 17 years will recognize this as an uncannily accurate description of exactly what -did- happen! (Except, of course, the Net was so much smaller then that we had little effect on the market place.) So accurate is Prof. Rubik's description that I'd be surprised if he (or his ghost writer) hasn't actually read through some of our earliest archives. - Alan From cube-lovers-errors@mc.lcs.mit.edu Sat Jan 3 21:37:36 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA15379; Sat, 3 Jan 1998 21:37:35 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Jan 3 05:56:11 1998 From: roger.broadie@iclweb.com (Roger Broadie) To: , "Bryan Main" Subject: Re: 5x5x5 Date: Sat, 3 Jan 1998 10:55:50 -0000 Message-Id: <19980103105500.AAA11342@home> > From: Bryan Main > To: Cube-Lovers@ai.mit.edu > Subject: Re: 5x5x5 > Date: 29 December 1997 13:31 > I just got one of these for christmas and had a question or two. > First is there a cube program so I can play with it and not destroy > all the work I have done? And I have solved one side, and all of > the edges without much problems. However, can I solve the middle > pieces without destroying the edges? As of yet I haven't found a > way to keep the one side I have finished and move one of the center > pieces on another side. I don't want moves, I just want to know if > it is possible to solve this way or if I need to start looking at > another way to solve it. > bryan Yes, if the corners of the top layer are also in the right place. You can move them around by normal 3x3x3 moves, but in doing so you may find that the parity of the edge pieces is changed. If you can swap a pair of edge pieces on a 4x4x4, all will be well, and all the pieces in the ring of eight around the piece at the centre of each face can be dealt with by 3-cycles to move these pieces to a different face or around on the same face. There is a hidden complication. The new type of pieces introduced by the 5x5x5 are those at N, S, E and W in the central block of nine in each face. If the corner pieces of the cube are correctly placed, the parity of these new pieces is tied to that of the edge pieces introduced by the 4x4x4, i.e. those next to the corner pieces of the cube. So if a pair of these edge pieces is swapped, so will be a pair of the new 5x5x5 central pieces. But the swap of the edge pieces will cure them at the same time. Often the change to the centre pieces will not even show, because it will take place within the same face. Thus the sequence Georges Helm gave some time ago to swap the 4x4x4 edges also cures the 5x5x5 centre pieces. If it is applied to a cube in the start position, it swaps Bl and Br visibly, and interchanges FHl and FHr (where H is the central slice parallel to U) invisibly. It also makes an even-parity change to the 4x4x4 centre pieces on the front face - in fact it rotates by 180 degrees the (l, u+H+d) and (r, u+H+d) strips on this face. Roger Broadie From cube-lovers-errors@mc.lcs.mit.edu Mon Jan 5 21:48:30 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA25102; Mon, 5 Jan 1998 21:48:29 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Jan 4 10:49:11 1998 Message-Id: Date: Sun, 04 Jan 1998 10:23:51 -0500 To: cube-lovers@ai.mit.edu From: Aaron Weintraub Subject: Re: a Rubiks Xmas In-Reply-To: Kristin, Not to brag or anything, but I first solved the cube when I was 7 years old. I'm 22 now and still haven't lost interest. -Aaron At 10:21 AM 12/29/97, Kristin wrote: >anyone know any kids younger than 9 that can solve the >cube? I'm sure there are younger cubists out there... but >I sure was impressed. > >-Kristin >kristin@wunderland.com From cube-lovers-errors@mc.lcs.mit.edu Tue Jan 13 13:12:03 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA00779; Tue, 13 Jan 1998 13:12:02 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jan 12 23:26:33 1998 Date: Mon, 12 Jan 1998 23:25:18 -0400 (EDT) From: Jerry Bryan Subject: Face Turns Nine Moves from Start To: Cube-Lovers Message-Id: I have some new search results for the face turn metric. Here is a summary of the new search. Face Turns Patterns Positions Branching Positions/ from Start Factor Patterns 0 1 1 1.000 1 2 18 18.000 9.000 2 9 243 13.500 27.000 3 75 3240 13.333 43.200 4 934 43239 13.345 46.294 5 12077 574908 13.296 47.604 6 159131 7618438 13.252 47.875 7 2101575 100803036 13.231 47.965 8 27762103 1332343288 13.217 47.991 9 366611212 17596479795 13.207 47.998 The results at 8f and 9f from Start are new. Previously, the face turn metric had only been searched through 7f from Start. All the results in terms of patterns (M-conjugacy classes) are new. Previously, the face turn metric had been searched only in terms of positions. Note that the branching factor does not change very much. We already know (or strongly suspect by statistical arguments based on the results of Kociemba, Winter, Reid, and Korf) that it cannot change much this close to Start. Otherwise, the mode of the distribution would be greater than the 18f which is strongly suspected to be the case. I have not yet installed the logic to detect weak local maxima. The logic to detect strong local maxima is installed with an interesting result. Two patterns were detected at 9f from Start which are strong local maxima. Regrettably, I have no idea what they are. I will have to add something to the program to print out strong local maxima when they are detected. All I know is that the patterns are at least "somewhat symmetric" in that they collectively represent only 32 positions. I have begun to suspect that strong local maxima are fairly rare in the face turn metric. Recall that a strong local maximum is one where all 18 face turns carry the cube closer to Start. A weak local maximum, by contrast, is a local maximum where at least one face turn leaves the cube the same distance from Start. If I have not made a mistake in analyzing them (which is entirely possible), the only one of Mike Reid's "highly symmetric" positions which is a strong local maximum is superflip. Even Pons Asinorum is not a strong local maximum. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Wed Jan 14 19:00:40 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA05544; Wed, 14 Jan 1998 19:00:40 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Jan 14 13:43:31 1998 Date: Wed, 14 Jan 1998 13:42:20 -0500 (Eastern Standard Time) From: Jerry Bryan Subject: Performance Analyzers for Cube (and other) Programs To: cube-lovers@ai.mit.edu Reply-To: Jerry Bryan Message-Id: [Moderator's note: Please reply directly to Jerry.] This is a little off topic, but many cube searching programs run for dozens or hundreds of hours and we are always interested in speeding them up. The best speed ups usually come from algorithm improvements, but I am also interested in more mundane program improvements. Through the years, I have used various tools, usually for FORTRAN, usually on mainframes, which will analyze a running program, telling you where (which routines, which lines of source code) the program is spending its time. I am now running mostly C programs, mostly on a PC. I confess I am clueless as to what performance analysis tools might be available in this environment. (I use Borland C++ if it matters.) Any suggestions would be gratefully accepted. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Thu Jan 15 12:55:44 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA09053; Thu, 15 Jan 1998 12:55:44 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jan 15 09:34:15 1998 X-Sender: ddyer@10.0.2.1 Message-Id: Date: Wed, 14 Jan 1998 10:23:42 -0800 To: Cube-Lovers@ai.mit.edu From: Dave Dyer Reply-To: Dave Dyer Subject: save a cube for the price of a stamp My trusty 4x4x4 has died. I urgently need 2 replacement "1-sided" cubelets, preferably white and yellow. I'm hoping someone has a similarly defunct 4x4x4 that they saved for sentimental reasons (or to admire the amazing internal mechanism) and will send me some spare parts. From cube-lovers-errors@mc.lcs.mit.edu Fri Jan 16 16:29:01 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA14189; Fri, 16 Jan 1998 16:28:59 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jan 16 12:59:32 1998 Date: Fri, 16 Jan 1998 17:58:10 +0000 (GMT) From: Jonathan Tuliani To: Cube-Lovers Subject: MEGAMINX Message-Id: The following is based on an email I sent recently to Kurt Endl. He says that he does not have time to work on this at present, and, with a thesis to write, nor do I! I expect that someone has heard of this and the answer is known--I am new to this discussion group. Otherwise, hopefully someone will find it sufficiently interesting to think it through. I was delighted to be given a MEGAMINX this Christmas, together with Kurt Endl's instruction booklet. I resolved to attempt the puzzle without looking in the booklet, at least at first. My approach was similar--I built a solution in layers, starting at the bottom (which I shall call the north pole, consistent with the notation in the latter part of the book) and proceeding, layer by later towards the south pole. I was successful in my efforts until I reached the south cap, at which point I became stuck. My problem was to position and orient the south pole edges. Try as I might, the best I could do was to reach a position where two south pole edges needed to be exchanged. And try as I might, I could not find a way to do this. After a week, I gave up and turned to the instructions. I was delighted to see that their approach was similar to mine, and fascinated by the simple moves L_{**}, L^{**}, R_{**} and R^{**} used. My methods were, of course, far less elegant. I was able to start at section 8, `Setting the South Pole edges'. The procedure for setting the edges affects the southern equatorial corners, which you then arrange later. My layer-by-layer approach had, of course, already set these corners correctly before attempting the south pole edges (and indeed before setting the southern equatorial edges). Perhaps this was the key? Having the southern equatorial corners set should not affect the validity of the book's method, which should work with any arrangement of these corners. But following the instructions, I was unable to position the last two south pole edges correctly. The statement ``The remaining two South Pole edges will be correctly placed again at the same time'' on page 21, section 8 of the instructions must be incorrect--here after all was a counterexample! My last two south pole edges needed to be exchanged. I ignored the problem for the time being. I oriented the two edges correctly, with them still in the wrong positions. I was then able to complete the MEGAMINX, positioning and orienting the southern equatorial and south pole corners as in the instructions. The result was a complete MEGAMINX, except that it appeared that two little triangular stickers, each on the border of the southern cap, had been exchanged. After some thought, I have found a way out of this problem. I believe that this is a detail that may be required for solution in some circumstances that is not in this instruction booklet. I will try to describe what I think went wrong. The twelve faces of the MEGAMINX are coloured using only six colours, with opposite faces bearing the same colour. Thus, each edge piece has an `identical twin' on the opposite side of the completed MEGAMINX. When solving the MEGAMINX from a totally jumbled position, these twins are indistinguishable and may therefore be assigned either to their own original position, or to their twin's position at random. In this sense the solution of MEGAMINX is not unique. (Strictly, the solution may still be unique but we have not proved conclusively that it is so, and have demonstrated reason to believe that it may not be.) Now, suppose we return to the two edges I wished to exchange. Imagine them as south pole edges also in adjacent faces. Turn the MEGAMINX so that the two faces concerned are in the left and right positions (using your terminology). We need to exchange these edges, but the problem we have is that every sequence of moves seems to rotate 3 edges cyclicly, rather than exchange a pair! For example, the book uses L_{**} or R_{**} to bring one of the two edges concerned down into the front southern equatorial edge (``...the edge we have misused so brutally...''), and the same moves to move it back up again. But these moves, together with any I can find, cycle 3 edges. How can moves cycling 3 edges be used to exchange just 2 edges? I was stuck. The solution comes from the `identical twins' I talked about before. Suppose one of the two edges I'm interested in is, say, yellow/blue, and the other yellow/orange, so the southern pole is yellow. Tucked away on the opposite side of MEGAMINX is *another* yellow/blue edge. By a simple sequence of moves, this may be brought into the postion of the front southern equatorial edge. Now consider cycling these three edges. As two of the edges are identical, cycing these three edges *looks* like a swapping of just two edges! Now we return to the other side of MEGAMINX the twin of the piece that was originaly there. (Some fixing of the MEGAMINX is required to repair the damage caused by bringing an edge from the opposite side of the MEGAMINX to the front and sending it's twin back, but this isn't too hard.) Now, having apparently `swapped' two adjacent edges, we can proceed as per the instructions and complete the MEGAMINX. This I have done. Have any other people encountered this problem, or was I just extremely unlucky? The question arising is, of course, just how many solutions of MEGAMINX are there? There are 10 of these `identical twin' edge pairs. I reckon swapping just one twin pair is not possible in a complete solution, but that swapping any even number of twins may be (unproven), and so there are 512 solutions, each of which would be distinct if 12 different colours had been used on the original puzzle. Does anyone fancy having a stab at this conjecture? Jonathan Tuliani Mathematics Department Royal Holloway, University of London Egham Surrey TW20 0EX U.K. jont@dcs.rhbnc.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Sun Jan 18 15:03:51 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA05245; Sun, 18 Jan 1998 15:03:50 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Jan 17 00:05:36 1998 Message-Id: In-Reply-To: <199801170435.UAA02625@eve.speakeasy.org> References: Date: Sat, 17 Jan 1998 00:04:00 -0500 To: cube-lovers From: Charlie Dickman Subject: Re: save a cube for the price of a stamp Dave... The other day you wrote... >>>My trusty 4x4x4 has died. I urgently need 2 replacement "1-sided" >>>cubelets, preferably white and yellow. I'm hoping someone has a >>>similarly defunct 4x4x4 that they saved for sentimental reasons (or to >>>admire the amazing internal mechanism) and will send me some spare >>>parts. I took the liberty of forwarding your message to Mike Green at Puzzletts. Here is his response. >I still have parts for the 4x4x4 @ $2.50 each plus postage. I believe I >can handle your request. ( - : > >MG Hope this helps. Charlie Dickman charlied@erols.com From cube-lovers-errors@mc.lcs.mit.edu Sun Jan 18 15:40:29 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA05358; Sun, 18 Jan 1998 15:40:28 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Cube-Lovers-Request@ai.mit.edu Mon Oct 6 23:26:19 1997 Date: Sun, 18 Jan 1998 15:39:22 -0500 (EST) Message-Id: <18Jan1998.153922.Cube-Lovers@AI.MIT.EDU> From: Cube Lovers Moderator To: Cube-Lovers@AI.MIT.EDU Subject: Megaminx -- [Digest v23 #257] Cube-Lovers Digest Sun, 18 Jan 1998 Volume 23 : Issue 257 Today's Topic: Megaminx [5 messages] ---------------------------------------------------------------------- Date: Fri, 16 Jan 1998 21:59:11 -0500 To: Cube-Lovers From: Charlie Dickman Subject: Re: MEGAMINX Where can one obtain Kurt Endl's instruction booklet for the megaminx? Charlie Dickman charlied@erols.com ------------------------------ From: roger.broadie@iclweb.com (Roger Broadie) To: "Cube-Lovers" Cc: "Jonathan Tuliani" Subject: Re: MEGAMINX Date: Sat, 17 Jan 1998 22:54:40 -0000 In my opinion Jonathan is quite right in his analysis of the problem and its solution. When cubes or similar puzzles are coloured ambiguously, it is always possible that the puzzle will be in an apparently impossible configuration which must be cured by a move which changes identically coloured pieces in an invisible way. I am lucky enough to have a dodecahedron puzzle from the 80s called the Supernova, which was made in Hungary and sold in the UK by Pentangle. That used twelve different colours, and the problem did not arise. Perhaps the instructions for the Megaminx were originally written for this form. Other puzzles can show similar effects - the variant of the cube with the vertical edges bevelled so that it is octagonal in horizontal cross-section has edge-pieces in the middle horizontal slice that have only one colour, so their orientation is ambiguous and the top and bottom edge-pieces can appear to have impossible flip states, such a single edge-piece flipped. Jonathan remarks that every sequence of moves seems to rotate 3 edges cyclically. The reason (as usual) is to be found in parity considerations. A single turn of a face of the dodecahedron yields two 5-cycles, one of the edge-pieces and one of the corners. Both these are even permutations, and it is therefore impossible to create a permutation of odd parity by any combination of turns. Hence a 3-cycle is the minimum possible, and a single 2-cycle is impossible. Obviously any 3-cycle of edges can in principle be conjugated into a form to solve Jonathan's problem, but I thought I'd look for a process for a 3-cycle of edge-pieces which would take one piece from the bottom into the top and preserve orientation (in the sense that the faces in the top and bottom remain so). I don't have the instruction booklet that goes with the Megaminx, so I don't know what notation it used, or indeed what anyone else may have used for this puzzle, so with the usual apologies if I'm ignoring a standard notation: Position the puzzle on a table, so there is a horizontal top plane and bottom plane, and turn it so that one of the faces in the top band directly faces you. Call that face F, the top T, the two faces on either side of F respectively Ru on the right and Lu on the left, and the two faces in the lower band that join in the centre Rl on the right and Ll on the left (i.e. u=upper and l=lower). The arrangement of faces you see is thus (nb use a non-proportionally spaced typeface); T Lu F Ru Ll Rl D Then Rl Ll' T Ru Lu' F^2 Ru' Lu T' Ru Lu' F^-2 Ru' Lu Ll Rl' does (TRu, TF, Drl). Yes, you can swap pairs of edges - this is an even permutation which can be composed out of 3-cycles, and in general any even number of swaps is possible. Roger Broadie ------------------------------ Date: Sun, 18 Jan 1998 02:28:59 +0100 (MET) From: Dik.Winter@cwi.nl To: Cube-Lovers@ai.mit.edu Subject: Re: MEGAMINX > I reckon swapping just one twin pair is not possible in > a complete solution, but that swapping any even number of twins may be > (unproven), and so there are 512 solutions, each of which would be > distinct if 12 different colours had been used on the original puzzle. It is not so difficult to prove. Just as with the cube, also for the dodecahedron it is easy to see that whenever you turn a face, the parity of the edge and corner permutations remain the same. So a single swap of two edges is not possible, that is an odd permutation and would also require an odd permutation of the corner. However, interchanging two pairs is possible. Actually any even permutation of the edges is possible with the corners in place. This is because there are simple procedures that rotate a triple of edges, leaving the corners in place. Actually these procedures can be extremely similar to those used for the cube. Anyhow, this proves it. dik [ Moderator's note: Lest a reader misunderstand, let me note that the parity situation is different between the cube and megaminx. On the cube an odd permutation of edges is achievable provided the corner permutation is also odd. On the megaminx, neither the corner permutation nor the edge permutation can ever be odd. ] ------------------------------ From: "Philip Knudsen" To: cube-lovers@ai.mit.edu Subject: Re: MEGAMINX Date: Sun, 18 Jan 1998 00:36:43 PST Jonathan Tuliani wrote: >I was delighted to be given a MEGAMINX this Christmas, together with >Kurt Endl's instruction booklet. This is a recently re-issued version of the Megaminx, using 6 colors instead of 12. This puzzle is available at Spielkiste The original Meffert Megaminx, however, used 12 colors, and the same goes for a slightly different hungarian version called Supernova. Christoph Bandelow still has a few of the latter for sale. >The result was a complete MEGAMINX, except that it appeared that two >little triangular stickers, each on the border of the southern cap, >had been exchanged. This indeed is caused by the fact that in this version of the puzzle there exist 15 (not 10) pairs of edges that are alike. Thus solving the puzzle is somewhat similar to solving Alexander's Star and Impossiball at the same time! ____________________________________ Philip Knudsen Recording Artist Vendersgade 15, 3th DK - 1363 Copenhagen K Denmark Phone : +45 3393 2787 E-mail : philipknudsen@hotmail.com ------------------------------ Date: Sun, 18 Jan 1998 14:08:10 -0500 From: Walter Smith To: cube-lovers@ai.mit.edu Subject: Megaminx On 1/6/98 Jonathan Tuliani described a Megaminx with 6 colors. I have a Megaminx purchased when they first came out. It has 10 colors. They are positioned so that there is only one "twin" pair on the puzzle. There are two red/yellow edge pieces. It was a major personal discovery when I found that the puzzle was only solvable if the red/yellow pieces were put in the "right" places. Only having one twin pair made the puzzle particularly difficult because it took so long before I noticed their existence. I am sure that many readers will want to know if Megaminx is back in production. Jonathan, were did yours come from? Walt Smith walsmith@erols.com Germantown Md. ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Mon Jan 19 23:18:59 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA09746; Mon, 19 Jan 1998 23:18:58 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jan 19 00:02:09 1998 Message-Id: <199801190500.AAA29703@life.ai.mit.edu> Date: Sun, 18 Jan 1998 23:59:30 -0500 From: michael reid To: cube-lovers@ai.mit.edu Subject: Re: Face Turns Nine Moves from Start Cc: jbryan@pstcc.cc.tn.us jerry writes about strong local maxima in the face turn metric. he says that superflip is such a position, but pons asinorum is not. there are some other positions with a high degree of symmetry that are also strong local maxima, for example pons asinorum composed with superflip superfliptwist supertwist and some of the T-symmetric positions that may not have standard names #1. B U L' F' U R U2 D2 F' L U' B' L D R2 L2 B2 #4. D' R' U B' D' R' L F L B' R' F B' U L D' F U' D2 #5. B2 L U' L D R' L' D2 R U L' B2 U R2 U2 F2 U #6. D' L F' B' L F2 B2 U R L' U D' L F' R2 L2 F2 U' D2 #9. U F2 D B' U' B2 R B2 D' F2 U' D2 B2 L' U2 B D2 #11. D' F2 U2 B2 R F' L U' F2 B R' F' D L2 D R2 F2 U' F2 and there might be more among the H-symmetric and T-symmetric positions (i can't tell right now without doing more searching). yes, strong local maxima in the face turn metric are probably quite rare. in the quarter turn metric, any global maximum is necessarily a local maximum, because of parity considerations. however, in the face turn metric, a global maximum is a local maximum, but it may not be a strong one! so the only strong local maxima we have here are found as a result of (lots of) computer searching. i look forward to seeing your 2 strong local maxima at distance 9f from start. mike From cube-lovers-errors@mc.lcs.mit.edu Mon Jan 19 23:57:43 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA09847; Mon, 19 Jan 1998 23:57:42 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jan 19 05:27:26 1998 Date: Mon, 19 Jan 1998 10:25:32 +0000 (GMT) From: Jonathan Tuliani To: Cube-Lovers Subject: MEGAMINX Message-Id: I'm glad my comments regarding MEGAMINX seemed to have sparked interest. As someone points out, and as I realised over the weekend, there are of course 15 pairs of `twin' edges, not 10 as I stated before. Of course the conjecture I made (suggested by one correspondent to be true, but well beyond my group theory) would mean there are then 2^{15-1} = 16384 `distinct' complete solutions to this version of the puzzle! I'll ask my friend where he got the MEGAMINX and the instruction booklet from. He said there were a fair few other similar puzzles there as well. Jonathan Tuliani Date: Mon, 19 Jan 1998 18:14:56 +0000 (GMT) From: Jonathan Tuliani To: Cube-Lovers Subject: Source for my Megaminx Since somebody asked...apparently, my Megaminx came from Toys'R'Us, a large toy retailer here in the UK. The instruction booklet came with it. Jonathan From cube-lovers-errors@mc.lcs.mit.edu Tue Jan 20 00:25:19 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id AAA09944; Tue, 20 Jan 1998 00:25:15 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jan 19 11:26:05 1998 Sender: hainesd@ai.mit.edu Message-Id: <34C37E50.41C67EA6@kentrox.com> Date: Mon, 19 Jan 1998 08:24:48 -0800 From: Darin Haines To: Cube-Lovers@ai.mit.edu Subject: re: save a cube for the price of a stamp My situation was similar to Dave's. I sent him a response, and forgot to cc it to the list. I thought it would be beneficial [for me to send it] for everyone on the list who may be in the same, or similar boat. Here's what I told Dave... The guy to talk to is Christoph Bandelow. His email address is Christoph.Bandelow@ruhr-uni-bochum.de. (I'm pretty sure he monitors the list.) He is located in Germany. He also has a bunch of other puzzles that you will be interested in. He is very prompt with delivering orders, and is very easy to work with. Hope this helps. -Darin Dave Dyer wrote: > > My trusty 4x4x4 has died. I urgently need 2 replacement "1-sided" > cubelets, preferably white and yellow. I'm hoping someone has a > similarly defunct 4x4x4 that they saved for sentimental reasons (or to > admire the amazing internal mechanism) and will send me some spare > parts. From cube-lovers-errors@mc.lcs.mit.edu Tue Jan 20 01:11:57 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id BAA10057; Tue, 20 Jan 1998 01:11:56 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jan 19 17:29:03 1998 Message-Id: <19980119222713.19893.qmail@hotmail.com> From: "HADER MESA" To: cube-lovers@ai.mit.edu Subject: deseo cubos Date: Mon, 19 Jan 1998 14:27:11 PST [Moderator's note: As Cube-lovers is conducted in ASCII, the ISO accent characters in the original message are replaced by two-character sequences here. E.g., "informaci'on" refers to an acute accent over the "o".] Hola, yo soy un aficionado al cubo de rubik, y me gustar'ia tener en mi poder el cubo de rubik 3x3x3 y sus variantes (4x4x4, 5x5x5,2x2x2, etc) ya que el que yo ten'ia, lo perd'i, y en mi pa'is no lo he podido conseguir, adem'as yo recibo muchas noticias de los cube-lovers porque estoy inscrito en su grupo de noticias, que con un gran esfuerzo logro traducir. El ingl'es se me dificulta y por eso escribo en espa~nol. Si me pueden dar alguna informaci'on sobre donde los puedo adquirir, o comprar a trav'es de Internet, estar'ia muy agradecido. Cordialmente: Hader Mesa Pareja hamepa@hotmail.com From cube-lovers-errors@mc.lcs.mit.edu Tue Jan 20 12:14:46 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA11032; Tue, 20 Jan 1998 12:14:45 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jan 20 11:57:15 1998 Date: Tue, 20 Jan 98 10:55:25 CST Message-Id: <9801201655.AA23859@dvorak.amd.com> Sender: clive1@dvorak.amd.com From: "HADER MESA" To: cube-lovers@ai.mit.edu Subject: Translation: [hamepa@hotmail.com: deseo cubos] Reply-To: "HADER MESA" Here's a translation of Hader's message for those who are interested: Hi, I am a fan of Rubik's cube, and I would like to have in my posession the 3x3x3 cube and its variants (4x4x4, 5,x,5, 2x2x2, etc.) since the one I had, I lost, and in my country I haven't been able to obtain it. Also, I receive many items from the cube-lovers because I am already subscribed to your news group, that I manage to translate with great effort. English is hard for me, and that is why I write in Spanish. If you can give me any information about where I can obtain them, or buy them over the Internet, I would be very grateful. Cordially, Hader Mesa Pareja hamepa@hotmail.com ------ Original message: [Moderator's note: As Cube-lovers is conducted in ASCII, the ISO accent characters in the original message are replaced by two-character sequences here. E.g., "informaci'on" refers to an acute accent over the "o".] Hola, yo soy un aficionado al cubo de rubik, y me gustar'ia tener en mi poder el cubo de rubik 3x3x3 y sus variantes (4x4x4, 5x5x5,2x2x2, etc) ya que el que yo ten'ia, lo perd'i, y en mi pa'is no lo he podido conseguir, adem'as yo recibo muchas noticias de los cube-lovers porque estoy inscrito en su grupo de noticias, que con un gran esfuerzo logro traducir. El ingl'es se me dificulta y por eso escribo en espa~nol. Si me pueden dar alguna informaci'on sobre donde los puedo adquirir, o comprar a trav'es de Internet, estar'ia muy agradecido. Cordialmente: Hader Mesa Pareja hamepa@hotmail.com From cube-lovers-errors@mc.lcs.mit.edu Thu Jan 22 12:46:32 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA17608; Thu, 22 Jan 1998 12:46:31 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jan 22 00:44:32 1998 Date: Thu, 22 Jan 1998 00:43:14 -0400 (EDT) From: Jerry Bryan Subject: Re: Face Turns Nine Moves from Start In-Reply-To: To: Cube-Lovers Message-Id: On Mon, 12 Jan 1998, Jerry Bryan wrote: > I have not yet installed the logic to detect weak local maxima. The logic > to detect strong local maxima is installed with an interesting result. Two > patterns were detected at 9f from Start which are strong local maxima. > Regrettably, I have no idea what they are. I will have to add something > to the program to print out strong local maxima when they are detected. > All I know is that the patterns are at least "somewhat symmetric" in that > they collectively represent only 32 positions. I have not yet added the logic for weak local maxima, but further perusing of my printout from the run which has already been made does yield a bit of confirmation to some previous results reported by others. At each distance from Start, my program summarizes the number of patterns and positions by the symmetry class (one of the 33 symmetry classes of M, the group of 48 symmetries of the cube). Hence, I can easily look for "highly symmetric" positions based on the symmetry class. Of the 72 positions defined as q-transitive by Jim Saxe and Dan Hoey in Symmetry and Local Maxima, only 4 of them show up in the search through 9f. One of them is at 0f (Start), one of them is at 6f (Pons Asinorum, a weak local maximum, only the six half turns move closer to Start), and two of them are at 8f (the two conjugate 6-H positions, weak local maxima with only the six half turns moving closer to Start). We therefore know that the two patterns at 9f which are strong local maxima are not q-transitive, and cannot be shown to be local maxima by symmetry considerations alone. Strictly speaking, we already knew all this based on Dan's study of Pons Asinorum from many years ago, and based on Mike Reid's recent studies of highly symmetric positions with his optimal cube solver. Jim Saxe found the 8f processes for the 6-H positions many years ago, but I do not believe that they were shown to be minimal until Mike's recent work. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Wed Jan 28 13:49:36 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA06018; Wed, 28 Jan 1998 13:49:35 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Jan 28 13:18:22 1998 Date: Wed, 28 Jan 1998 18:14:52 GMT From: David Singmaster Computing To: cube-lovers@ai.mit.edu Message-Id: <009C0FA1.17365270.46@ice.sbu.ac.uk> Subject: Megaminx The UK stores of Toys R Us have been selling a number of Meffert's products, including the Megaminx, Skewb, Impossiball, etc. I had assumed these were also available in the USA. Is this not the case?? DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Wed Jan 28 17:37:14 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA06624; Wed, 28 Jan 1998 17:37:13 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Jan 28 13:32:58 1998 Date: Wed, 28 Jan 1998 10:31:35 -0800 From: mrhip@sgi.com (Jason Werner) Message-Id: <9801281031.ZM27368@neuhelp.corp.sgi.com> To: Cube-Lovers@ai.mit.edu Subject: 9X9X9 cube (fictional) If you can get your hands on the February 1998 edition of "Sys Admin: The Journal For UNIX Systems Administrators", check out the ad on page 56. Or, catch a glimpse at: http://www.sd98.com/ Enjoy! -Jason -- Jason K. Werner Email: mrhip@sgi.com Systems Administrator Phone: 650-933-9393 USFO I/S Technical Support Fax: 650-932-9393 Silicon Graphics, Inc./Cray Research Pager: 888-491-2906, mrhip_p@sgi.com "Winning is a habit"-Vince Lombardi "These go to eleven"-Nigel Tufnel From cube-lovers-errors@mc.lcs.mit.edu Thu Jan 29 13:56:49 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA09304; Thu, 29 Jan 1998 13:56:49 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Jan 28 15:39:47 1998 Message-Id: From: "joyner.david" To: "'David Singmaster Computing'" , "'cube-lovers@ai.mit.edu'" Subject: RE: Megaminx Date: Wed, 28 Jan 1998 15:38:09 -0500 >From: David Singmaster Computing[SMTP:david.singmaster@sbu.ac.uk] >Sent: Wednesday, January 28, 1998 1:14 PM >To: cube-lovers@ai.mit.edu > > The UK stores of Toys R Us have been selling a number of >Meffert's products, including the Megaminx, Skewb, Impossiball, etc. >I had assumed these were also available in the USA. Is this not the >case?? No this is not the case. The Toys R Us don't even sell Rubik's cubes around were I live (in the Washington DC area). - David Joyner [ Moderator's note: Michael Swart notes that the Toys Ya us stores in Kitchener-Waterloo, Ontario Canada don't have them either. ] From cube-lovers-errors@mc.lcs.mit.edu Thu Jan 29 15:45:06 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA09604; Thu, 29 Jan 1998 15:45:05 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jan 29 02:24:42 1998 Date: Thu, 29 Jan 1998 02:22:17 -0500 From: Edwin Saesen Subject: Re: 9X9X9 cube (fictional) To: CUBE Message-Id: <199801290222_MC2-3110-7043@compuserve.com> >If you can get your hands on the February 1998 edition of "Sys Admin: >The Journal For UNIX Systems Administrators", check out the ad on >page 56. That will probably be similar to the one I saw in the February edition of Visual Basic Programmer's Journal on page 90. I wanted to post about this one, but somehow forgot... Although, I think a deadline of one week to solve it should be enough time...after I worked out the 5x5x5, the 9x9x9 should be just more of the same :-) Michael Ehrt From cube-lovers-errors@mc.lcs.mit.edu Fri Jan 30 13:08:04 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA13119; Fri, 30 Jan 1998 13:08:03 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jan 29 05:57:13 1998 Message-Id: <34D05FEC.B6FE6B14@ibm.net> Date: Thu, 29 Jan 1998 02:54:37 -0800 From: "Jin 'Time Traveler' Kim" To: Cube-Lovers@ai.mit.edu Subject: Rubik's Cube FAQ References: <9801281031.ZM27368@neuhelp.corp.sgi.com> Is there such a thing as a Rubik's-type puzzle FAQ? There is interest among several people who wish to create one (specifically for www.rubiks.com) but if there's already another that exists, there's no reason to duplicate effort if it's not necessary. If anybody knows of such a FAQ, please let me know. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa / Puente Hills http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@mc.lcs.mit.edu Fri Jan 30 13:43:05 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA13218; Fri, 30 Jan 1998 13:43:05 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jan 29 15:22:57 1998 Message-Id: <3.0.3.32.19980129151831.005569e0@caddscan.com> Date: Thu, 29 Jan 1998 15:18:31 -0500 To: cube-lovers@ai.mit.edu From: "Bryan Main" Subject: RE: Megaminx In-Reply-To: >No this is not the case. The Toys R Us don't even sell Rubik's cubes >around were I live (in the Washington DC area). - David Joyner It's kind of difficult to find the Cubes in Toy's R Us but they are there. They also have some new game out for two people, but I don't remember what it's called. I know that the cubes are not where you would expect them to be, with other games and puzzles, they are normally on an end cap near the front of the stores on the bottom shelf. Also a lot of the other toy stores have them and they all have them in strange places around the store. I don't think that I have seen anything new besides the new game. They normally sell the cube, snake, magic rings, and a pyramid that comes apart and you put back together. bryan From cube-lovers-errors@mc.lcs.mit.edu Mon Feb 2 14:45:11 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA21883; Mon, 2 Feb 1998 14:45:10 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Feb 1 04:11:00 1998 From: peter@mold.demon.co.uk (Pete Thomas) To: Cube-Lovers@ai.mit.edu Subject: Centre Turns - A simple solution? Date: Sun, 01 Feb 1998 09:07:10 GMT Organization: Virtual Mold Reply-To: peter@mold.demon.co.uk Message-Id: <34db3a3e.3154932@post.eng.demon.net> I've a 3 x 3 cube with a rather abstract pattern made up of three colours. It does require the centered to be orientated correctly. If I solve the cube, but fail to get the centers correct; is there an easy solution to rotating them about their axis (I would guess in opposite pairs? Singmaster Notation if poss.. Regards Pete (Cube dabbler over the past 20 years). Pete --------------------------------------------- Virtual Mold.... Better than the real thing! http://www.mold.demon.co.uk From cube-lovers-errors@mc.lcs.mit.edu Mon Feb 2 15:25:57 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA22024; Mon, 2 Feb 1998 15:25:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Feb 1 12:38:55 1998 Message-Id: <199802011737.MAA08347@life.ai.mit.edu> From: "John Coffey" To: , "Bryan Main" Subject: Re: Megaminx Date: Sun, 1 Feb 1998 10:30:04 -0700 Just as a side note, I was able to find a Square 1 puzzle at KayBee toys on closeout. I had been looking for one for about 2 months. Thanks, John coffey From cube-lovers-errors@mc.lcs.mit.edu Mon Feb 2 18:07:51 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA22436; Mon, 2 Feb 1998 18:07:49 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jan 30 17:00:36 1998 Date: Fri, 30 Jan 1998 22:59:06 +0100 (MET) Message-Id: <199801302159.WAA02024@relay.euronet.nl> To: Cube-Lovers@ai.mit.edu From: Sytse de Maat <4xs2fs@euronet.nl> Subject: Re: Rubik's Cube FAQ At 02:54 29-1-98 -0800, chrono@ibm.net (Jin "Time Traveler" Kim) wrote: >Is there such a thing as a Rubik's-type puzzle FAQ?... At least a universal notation for 4x4x4 and 5x5x5 would be very welcome to me. Sytse de Maat <4xs2fs@euronet.nl> Designer of a 5x5x5 cube in 1982 From cube-lovers-errors@mc.lcs.mit.edu Tue Feb 3 14:15:18 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA25095; Tue, 3 Feb 1998 14:15:18 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Feb 3 04:22:43 1998 Message-Id: <3.0.5.16.19980203101216.2f57e01a@vip.cybercity.dk> Date: Tue, 03 Feb 1998 10:12:16 To: cube-lovers@ai.mit.edu From: Maria Skou & Philip Knudsen Subject: Lights Out Cube I just heard there's a new puzzle out in the U.S. called "Lights Out Cube". My first thought was that it's probably from the same company (Tiger Electronics) who introduced the "Lights Out" and "Deluxe Lights Out". Anybody who can confirm or knows some more?? It would also be nice if anyone knew a store in the San Francisco Area where these puzzles are available (i have a friend who can buy them for me). I haven't seen them in Europe. Thanks in advance, Philip K recording and performing artist Vendersgade 15, 3th DK - 1363 Copenhagen K Phone: +45 33932787 Mobile: +45 21706731 E-mail: skouknudsen@email.dk E-mail: philipknudsen@hotmail.com E-mail: 4521706731@sms.tdk.dk (leave subject blank!) From cube-lovers-errors@mc.lcs.mit.edu Tue Feb 3 16:10:34 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA25347; Tue, 3 Feb 1998 16:10:34 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Feb 3 15:49:29 1998 Message-Id: <01a501bd30e5$5e06f440$da460318@CC623255-A.srst1.fl.home.com> From: "Chris and Kori Pelley" To: Subject: Re: Lights Out Cube Date: Tue, 3 Feb 1998 15:50:28 -0500 >I just heard there's a new puzzle out in the U.S. called "Lights Out Cube". Yes the cube is by the makers of Lights Out. It is available here in the U.S. at places like Target and Wal-Mart. It's quite a conversation piece because people often ask if it's an electronic Rubik's Cube! Of course it's just a 3-D version of the Lights Out game. It's fun because the puzzles get progressively more difficult, plus there is a multi-player mode. Chris Pelley ck1@home.com From cube-lovers-errors@mc.lcs.mit.edu Thu Feb 5 13:30:56 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA02690; Thu, 5 Feb 1998 13:30:55 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From hoey@AIC.NRL.Navy.Mil Thu Feb 5 12:17:12 1998 Date: Thu, 5 Feb 98 12:16:55 EST Message-Id: <9802051716.AA02692@sun33.aic.nrl.navy.mil> From: Dan Hoey To: cube-lovers@mc.lcs.mit.edu Subject: Test This is a test of cube-lovers forwarding. It shouldn't go to anyone but the administrator, but if it does, that's a mistake. Dan Hoey Interim Cube-lovers administrator From cube-lovers-errors@mc.lcs.mit.edu Mon Feb 9 17:53:53 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA15144; Mon, 9 Feb 1998 17:53:53 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Feb 9 15:36:15 1998 Message-Id: Date: Mon, 9 Feb 1998 15:36:13 -0500 To: Cube-Lovers@ai.mit.edu From: kristin@wunderland.com (Kristin Looney) Subject: looking for a phone number... Reply-To: kristin@wunderland.com (Kristin Looney) Does anyone on this list have contact information for Oddz-On? -K. kristin@wunderland.com http://www.wunderland.com/wts/kristin From cube-lovers-errors@mc.lcs.mit.edu Sat Feb 14 15:02:57 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA02713; Sat, 14 Feb 1998 15:02:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Feb 14 09:38:36 1998 Message-Id: <199802141438.JAA02445@life.ai.mit.edu> From: "David Byrden" To: Subject: Rubik lawyers up in arms over website Date: Sat, 14 Feb 1998 14:37:00 -0000 I have a website at http://Byrden.com/puzzles/ where I keep playable versions of many varieties of cube, pyramid, dodecahedron, etc etc. All of them are built in Java and have an "undo" button allowing you to explore your moves. The site was titled "The Rubik Gallery" in honour of Erno Rubik. It explicitly said that there was no further connection with him. I got a letter from a Washington firm of lawyers about a week ago, saying that they were the advisers to Seven Towns Limited, holders of the 'Rubik' trademark. The site was "diluting" the value of the trademark and causing "customer confusion". I was engaging in "unfair competition" (despite not selling or distributing anything or taking any money or having any advertising on the site). Not only did they want the word 'Rubik' removed from the website, they wanted one of the Java puzzles removed as well. They called it an "electronic version of the RUBIK'S CUBE". Fair enough, being a hexahedron sliced into 26 equal parts it bore a certian visual resemblance, but obviously there was none of their mechanism involved. It was all brand-new software. Anyway, I took it off, and they seem happy enough now. But...does anyone know what company owns the rights to the 4x4 cube? David Byrden From cube-lovers-errors@mc.lcs.mit.edu Sun Feb 15 23:45:18 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA05620; Sun, 15 Feb 1998 23:45:17 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Feb 15 19:06:46 1998 Date: Sun, 15 Feb 1998 19:06:35 -0400 (EDT) From: Jerry Bryan Subject: Strong Local Maxima 9f from Start To: Cube-Lovers Message-Id: #1. D2 F2 L2 D' U L2 F2 D' U' #2. U D B2 R2 U D' L2 B2 U2 These positions "look" very symmetric, especially #2, but I have not yet examined their symmetry characteristics in detail. They are certainly not Q-transitive. I do not know if either position has been reported before, has a name, etc. The corners are identical between the two positions, but the edges are a good bit different. I find both positions to be very pretty. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Mon Feb 16 00:16:58 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id AAA05697; Mon, 16 Feb 1998 00:16:57 -0500 (EST) From: cube-lovers-errors@mc.lcs.mit.edu Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Feb 14 16:39:52 1998 Message-Id: <15Feb1988.235225.Cube-Lovers@AI.MIT.EDU> To: Cube-Lovers@AI.MIT.EDU Subject: Rubik lawyers up in arms over website -- Digest v23 #279] Date: Sun, 15 Feb 1998 23:52:25 EST Cube-Lovers Digest Sun, 15 Feb 1998 Volume 23 : Issue 279 Today's Topic: Rubik lawyers up in arms over website [3 messages] ---------------------------------------------------------------------- From: "Philip Knudsen" To: cube-lovers@ai.mit.edu Subject: Re: Rubik lawyers up in arms over website Date: Sat, 14 Feb 1998 13:38:35 PST This confirms a common (european?) prejudice about the U.S. and their tendency to file lawsuits over just about anything. I feel certain Rubik himself would have nothing against Byrden's online cube. On the other hand it's good to know his business is well taken care of... Don't know about the 4x4x4 copyright, but it's pretty well known Rubik did not design the actual mechanism for it. Ideal just used his name to market the puzzle. By the way, I did visit the www.Byrden.com site some time ago and really liked it, not the least the "special octahedron". ____________________________________ Philip K [ Moderator's note: I don't think the archive has anything about the origin of the 4^3 (or any other) design. Can you give a source for this well-known information? ] ------------------------------ Date: Mon, 16 Feb 98 08:59:30 +0900 From: Norman Diamond 16-Feb-1998 0859 To: cube-lovers@ai.mit.edu Subject: Re: Rubik lawyers up in arms over website David Byrden wrote: >I got a letter from a Washington >firm of lawyers about a week ago, saying >that they were the advisers to Seven Towns >Limited, holders of the 'Rubik' trademark. >The site was "diluting" the value of the >trademark and causing "customer confusion". Fine. Please restore your web site, but say: "Please enjoy these puzzles yourself. However, we cannot honor the famous Dr. Rubik because his lawyers won't let us honor him." >I was engaging in "unfair competition" >(despite not selling or distributing anything >or taking any money or having any advertising >on the site). It doesn't matter if you take money or not. I thought it didn't hurt if you had advertisements honoring Dr. Rubik, but ... Well, I guess everyone had better remove Rubik's signature from our instances of his merchandise, because whenever we play with one of his products, we're advertising his name illegally. [Note: This is the only sarcastic sentence in this message. Please take the rest seriously.] > Not only did they want the word 'Rubik' >removed from the website, they wanted one of >the Java puzzles removed as well. They called it >an "electronic version of the RUBIK'S CUBE". >Fair enough, being a hexahedron sliced into 26 >equal parts it bore a certian visual resemblance, >but obviously there was none of their mechanism >involved. It was all brand-new software. It is true that none of their mechanism is involved. Therefore I believe their patent doesn't apply. That is, if they actually still have a pattent, after Ishige and some American who preceded all of them (whose name I've forgotten) ... but wait, it's been more than 20 years (or 17 in the US), so ALL their patents have expired. So don't honor Dr. Rubik. Please restore all of your mathematical puzzles. -- Norman Diamond diamond@jrdv04.enet.dec-j.co.jp [Speaking for Norman Diamond not for Digital.] ------------------------------ Date: Sun, 15 Feb 1998 20:23:38 -0500 From: Alan Bawden To: David@Byrden.com Cc: Cube-Lovers@ai.mit.edu From: "David Byrden" Date: Sat, 14 Feb 1998 14:37:00 -0000 ... Not only did they want the word 'Rubik' removed from the website, they wanted one of the Java puzzles removed as well. They called it an "electronic version of the RUBIK'S CUBE". Fair enough, being a hexahedron sliced into 26 equal parts it bore a certian visual resemblance, but obviously there was none of their mechanism involved. It was all brand-new software. This latter seems totally outrageous to me. If I were in your shoes, I would consider contacting the EFF to see if they were interested in making a case out of this. The request that you remove Rubik's name from your site is the kind of petty stupidity we're seeing all to often these days, and is probably pretty mundane to the cyberlawyers at EFF, but the notion that they can torpedo your software if it merely duplicates the user interface (the "look-and-feel") of their physical puzzle might be something genuinely new. Heck, do these guys claim that they own the underlying mathematical group? ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Tue Feb 17 18:08:36 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA10207; Tue, 17 Feb 1998 18:08:35 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Feb 16 19:02:35 1998 Date: Mon, 16 Feb 1998 19:02:17 -0400 (EDT) From: Jerry Bryan Subject: Re: Strong Local Maxima 9f from Start In-Reply-To: To: Cube-Lovers Message-Id: On Sun, 15 Feb 1998, Jerry Bryan wrote: > > #1. D2 F2 L2 D' U L2 F2 D' U' > #2. U D B2 R2 U D' L2 B2 U2 > It ocurred to me that because these positions are strong local maxima (and the shortest ones, at that), maybe I should show maneuvers of length 9f ending with each of the 18 possible face turns. Here they are. No uniqueness is claimed for the maneuvers. #1 F2 L2 U2 B F' D2 L2 B' F' F2 L2 U2 B F' D2 L2 F' B' R2 D2 F2 L R' F2 D2 L' R' R2 D2 F2 L R' F2 D2 R' L' D2 F2 L2 D' U L2 F2 D' U' D2 F2 L2 D' U L2 F2 U' D' B2 R2 D2 B F' U2 R2 B F B2 R2 D2 B F' U2 R2 F B L2 U2 B2 L R' B2 U2 L R L2 U2 B2 L R' B2 U2 R L U2 B2 R2 D' U R2 B2 D U U2 B2 R2 D' U R2 B2 U D B' F' U2 R2 B F' L2 U2 F2 B F D2 L2 B F' R2 D2 B2 L' R' F2 U2 L R' U2 F2 R2 L R B2 D2 L R' D2 B2 L2 D U L2 B2 D' U B2 L2 U2 D' U' R2 F2 D' U F2 R2 D2 #2 B2 U2 R2 B' F L2 U2 B' F' B2 U2 R2 B' F L2 U2 F' B' R2 F2 D2 L R' U2 F2 L' R' R2 F2 D2 L R' U2 F2 R' L' D2 L2 B2 D' U F2 L2 D' U' D2 L2 B2 D' U F2 L2 U' D' F2 D2 L2 B' F R2 D2 B F F2 D2 L2 B' F R2 D2 F B L2 B2 U2 L R' D2 B2 L R L2 B2 U2 L R' D2 B2 R L U2 R2 F2 D' U B2 R2 D U U2 R2 F2 D' U B2 R2 U D B F R2 D2 B' F U2 R2 F2 B' F' L2 U2 B' F D2 L2 B2 L' R' U2 F2 L R' B2 U2 R2 L R D2 B2 L R' F2 D2 L2 D U B2 R2 D' U L2 B2 U2 D' U' F2 L2 D' U R2 F2 D2 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Tue Feb 17 19:44:21 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA10408; Tue, 17 Feb 1998 19:44:21 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Feb 16 07:57:12 1998 Message-Id: <17Feb1998.181625.Cube-Lovers@AI.MIT.EDU> Date: Tue, 17 Feb 1998 08:16:25 -0500 Subject: Rubik lawyers up in arms over website -- Digest v23 #281] To: Cube-Lovers@ai.mit.edu From: Cube-Lovers@ai.mit.edu Cube-Lovers Digest Tue, 17 Feb 1998 Volume 23 : Issue 281 Today's Topic: Rubik lawyers up in arms over website [3 messages] ---------------------------------------------------------------------- Date: Mon, 16 Feb 1998 07:58:28 -0500 To: Cube-Lovers@ai.mit.edu From: Nichael Lynn Cramer Subject: Laches Message-Id: >From: "Philip Knudsen" > >This confirms a common (european?) prejudice about the U.S. and their >tendency to file lawsuits over just about anything. >I feel certain Rubik himself would have nothing against Byrden's >online cube. On the other hand it's good to know his business is well >taken care of... [Apologies for taking this even farther off topic. But in an attempt to clear up one point...] Indeed it's quite possible that Rubik knows nothing about it. The issue here is the legal concept of "Laches", which says --effectively-- that if it can be shown that you, the copyright owner, did not pursue all incidents of copyright infringement of which you were aware, then the copyrighted item is in real danger of being declared as in the public domain. (This is also what's behind those silly stories of, say, some vet in the wilds outside Buccolia, Maine with a picture of Snoopy painted on his barn who one day gets a letter from Charles Schulz's lawyers requesting that he either remove the picture or else sign a license arrangement at a zillion dollars a year.) So, in short, a copyright owner is legally "required" to go after _any_ known or perceived abuse of the copyright or face the very real danger of losing it. Nichael Cramer nichael@sover.net Gather the folks, tell the stories, http://www.sover.net/~nichael/ break the bread. -- John Shea ------------------------------ To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Date: 16 Feb 1998 17:38:31 GMT Organization: California Institute of Technology, Pasadena Message-Id: <6c9tin$2ni@gap.cco.caltech.edu> "David Byrden" writes: > I got a letter from a Washington >firm of lawyers about a week ago, saying >that they were the advisers to Seven Towns >Limited, holders of the 'Rubik' trademark. >The site was "diluting" the value of the >trademark and causing "customer confusion". >I was engaging in "unfair competition" >(despite not selling or distributing anything >or taking any money or having any advertising >on the site). They have to say stuff like this to demonstrate that they've protected their trademark. Apparently the word "Rubik", when applied to puzzles, is trademarked. In US law, if one doesn't protect a trademark by this manner, one may lose it. Of course, since Rubik is also the name of a person, you should be able to use "Rubik" when referring to the person. The specific thing they're worried about is phrases like "The Rubik Page" or "Rubik Puzzles." Change the wording to "Puzzles based on those invented by Erno Rubik," and I don't think they can touch you. > Not only did they want the word 'Rubik' >removed from the website, they wanted one of >the Java puzzles removed as well. They called it >an "electronic version of the RUBIK'S CUBE". >Fair enough, being a hexahedron sliced into 26 >equal parts it bore a certian visual resemblance, >but obviously there was none of their mechanism >involved. It was all brand-new software. This is bunk. No one can trademark that stuff. At most, there's a patent (which you can't have violated). The lawyers are just asking for extra. - -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ - --------------------------------------------------------------------------- "...he put a wire in his cap and called himself Marconi." ------------------------------ Date: Mon, 16 Feb 1998 00:45:33 -0500 From: mark longridge Subject: Re: Rubik lawyers up in arms over website -- Digest v23 #279] Message-Id: <34E7D27D.6083@idirect.com> > From: "Philip Knudsen" > ...Don't know about the 4x4x4 copyright, but it's pretty well known > Rubik did not design the actual mechanism for it. Ideal just used > his name to market the puzzle. Probably (I'm not totally certain) it was Udo Krell, an inventor whose design was used by Uwe Meffert to make the 5x5x5. Norman Diamond wrote: > ... > It is true that none of their mechanism is involved. > Therefore I believe their patent doesn't apply. That is, > if they actually still have a pattent, after Ishige and > some American who preceded all of them (whose name I've > forgotten) ... but wait, it's been more than 20 years > (or 17 in the US), so ALL their patents have expired.... What about Karl Hornell's Java Applet "Rubik Unbound"?? It's all over the internet on hundreds of sites including my own!! I don't think the name Rubik itself can expire since that is his name... so the name of the product is always "Rubik's Cube"... ummmm right? :-) [Moderator's note: _Patents_ expire. _Trademarks_ don't necessarily expire. _Names_ are not protected by law. ] Alan Bawden wrote: > ... The request that you remove Rubik's name from your > site is the kind of petty stupidity we're seeing all to often these days, > and is probably pretty mundane to the cyberlawyers at EFF, but the notion > that they can torpedo your software if it merely duplicates the user > interface (the "look-and-feel") of their physical puzzle might be something > genuinely new. Heck, do these guys claim that they own the underlying > mathematical group? I don't think you can prevent people from making java applets and the like of cubes... but I think they (the lawyers that be) can protect Rubik's name. I don't think anyone can say "Don't show a rubik's cube-like construction on your web page". The other guy who history has forgotten was Larry Nichols who made a 2x2x2 cube called twizzle which routinely came apart and was rejected by Ideal Toy! I'd restore the 3x3x3 java applet if it was my web page. - -> Mark <- ------------------------------ End of Cube-Lovers Digest ************************* From cube-lovers-errors@mc.lcs.mit.edu Wed Feb 18 13:29:22 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA12405; Wed, 18 Feb 1998 13:29:22 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Feb 18 11:08:58 1998 Message-Id: <3.0.5.16.19980218165604.29af224c@vip.cybercity.dk> X-Sender: ccc10207@vip.cybercity.dk Date: Wed, 18 Feb 1998 16:56:04 To: cube-lovers@ai.mit.edu From: Philip Knudsen Subject: Game designers [was Re: Rubik lawyers...] Dan wrote: >I don't think the archive has anything about the origin of the 4^3 (or any >other) design. Can you give a source for this well-known information? Maybe i was a little fast to state that it is "generally known Rubik did not design the 4x4x4 mechanism". I've also digged through the entire list archives as well as my own stuff, and have found nothing which directly indicated that Rubik DID design it. Now the earliest mention of 4x4x4 is in Hofstadter's article in S.A. from march '81, page 26. Quote: "Rest assured, it's being developed in the Netherlands, and it may be ready soon..." >From: mark longridge - >Probably (I'm not totally certain) it was Udo Krell, an inventor whose >design was used by Uwe Meffert to make the 5x5x5. Don't you think it would be known that Udo Krell also invented the 4x4x4 if this was indeed the case? All of this is getting a little vague, it could be nice to have this matter cleared up by someone who has some REAL info! Maybe someone who worked for Ideal at the time. Yours Truly, Philip K From cube-lovers-errors@mc.lcs.mit.edu Wed Feb 18 16:13:49 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA13064; Wed, 18 Feb 1998 16:13:48 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Feb 18 15:12:55 1998 Message-Id: Date: Wed, 18 Feb 1998 15:13:03 -0500 To: cube-lovers@ai.mit.edu From: kristin@wunderland.com (Kristin Looney) Subject: working for Ideal... At 4:56 PM 2/18/98, Philip Knudsen wrote: > All of this is getting a little vague, it could be nice to have this > matter cleared up by someone who has some REAL info! Maybe > someone who worked for Ideal at the time. I worked for Ideal at the time... but only as a 16 year old kid demonstrating the cube and giving away free T-Shirts and posters in shopping malls in the Chicago area. Sorry, I have no idea who invented the 4x4 mechanism. I do remember anxiously waiting for the mail every day for a couple of weeks when they had said they were sending me one hot off the assembly line... -K. kristin@wunderland.com http://www.wunderland.com/wts/kristin http://www.wunderland.com/Home/Rubik.html From cube-lovers-errors@mc.lcs.mit.edu Wed Feb 18 19:38:23 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA13638; Wed, 18 Feb 1998 19:38:22 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Feb 18 15:29:06 1998 Message-Id: Date: Wed, 18 Feb 1998 15:29:15 -0500 To: cube-lovers@ai.mit.edu From: kristin@wunderland.com (Kristin Looney) Subject: custom cubes / awesome cube art Two other URL's to point the members of this list to... First, I have perfected custom cube sticker technology! Check out: http://www.wunderland.com/WTS/Kristin/CustomCubes.html If anyone has a custom cube they have been itching to make for years... as long as you provide the cube sans stickers and artwork in the size and format that I need it in... and probably postage to send it back to you if very many people take me up on this... talk to me... together we can make you the cube of your dreams. A word of warning: Odz On has done an excellent job of solving the stickers-always-falling-off problem on the new cubes. You will have a MUCH easier time getting the stickers off an old cube than you will one of the new ones. Zarf designed a REALLY cool cube last week... a picture of it should go up on this page with this weeks update on Thursday. Also, Jake has been continuing to evolve his cube art... Check out: http://www.wunderland.com/WTS/Jake/CubeInfo/ I have a window in my gameroom with 120 cubes arranged within it... and every couple of weeks or so Jake comes over and solves them into some sort of a cool 36 x 30 pixel picture. Jake and Zarf and Andy and myself have all designed images, and we are not out of ideas yet... but I'm guessing Jake would take outside design submissions if you asked him. -K. kristin@wunderland.com http://www.wunderland.com/wts/kristin To all the fishies in the deep blue sea, Joy. From cube-lovers-errors@mc.lcs.mit.edu Thu Feb 19 15:32:08 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA16096; Thu, 19 Feb 1998 15:32:08 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From zingmast@sbu.ac.uk Wed Feb 18 13:38:54 1998 Sender: zingmast@sbu.ac.uk Date: Wed, 18 Feb 1998 18:36:07 +0000 From: David Singmaster To: cube-lovers@ai.mit.edu Message-Id: <009C2024.8A2B6B63.8@ice.sbu.ac.uk> Subject: RE: Rubik lawyers up in arms over website -- Digest v23 #279] Norman Diamond refers to the patents expiring. However, Rubik only had a Hungarian patent. As a result, the various Rubik companies took legal action under copyright law, and copyright lasts much longer. Perhaps I should hassle Seven Towns about the use of my notation! DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Thu Feb 19 16:53:14 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA16296; Thu, 19 Feb 1998 16:53:14 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Feb 18 21:36:39 1998 Date: Thu, 19 Feb 1998 02:03:04 +0000 From: David Singmaster To: skouknudsen@email.dk Cc: cube-lovers@ai.mit.edu Message-Id: <009C2062.FA899020.3@ice.sbu.ac.uk> Subject: RE: Game designers [was Re: Rubik lawyers...] In my Cubic Circular 1 (Autumn 1981), I recorded that Wim Osterholt, of the Netherlands, had made and patented a 4^3 which he showed me. I don't remember it and I'm not sure when he brought it to London - perhaps Summer 1981? I also recorded that Rainier Seitz (product manager of Arxon which was Ideal's German agent) showed me some German patents and applications for the 4^3 and 5^3. In Cubic Circular 2 (Spring 1982), I record talking with another person who had devised a 4^3 mechanism. In Cubic Circular 3/4 (Spring/Summer 1982), I describe playing with examples. However, I don't recall ever knowing who devised the mechanism that was produced for Ideal. It was common knowledge that it was not Rubik's mechanism. One may be able to get details from the web site that Oddz On (sp??) has set up. Tom Kremer (of Seven Towns, who is Rubik's agent) is supervising this site and he would be one of the most likely people to know. I have a huge file of material comprising all US puzzle patents and I'll look there, but I think I would have known about the patent for the 4^3 already. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Mon Feb 23 17:21:17 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA27553; Mon, 23 Feb 1998 17:21:14 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Feb 23 11:00:45 1998 Date: Mon, 23 Feb 1998 10:59:54 -0500 (Eastern Standard Time) From: Jerry Bryan Subject: Re: Strong Local Maxima 9f from Start In-Reply-To: To: Cube-Lovers Message-Id: On Sun, 15 Feb 1998, Jerry Bryan wrote: > #1. D2 F2 L2 D' U L2 F2 D' U' > #2. U D B2 R2 U D' L2 B2 U2 > > These positions "look" very symmetric, especially #2, .... The reason #2 "looks" so symmetric is that it is an isoglyph. That is, each face contains only two colors and the pattern of colors is the same on all six faces. If I am reading Dan Hoey's glyph table correctly, the glyph is of type 20. The glyph looks like the following, and the glyph itself is fairly symmetric. XOX OXO OOO On a lark, I asked Herbert Kociemba's Cube Explorer 1.5 program to find all isoglyphs which can be built with this glyph. Any such isoglyph is likely to be pretty. Up to symmetry, it found four isoglyphs (one of which is #2, which is a strong local maximum in the face turn metric). The other three are as follows: F2 U2 L' R D2 F2 L' R 8f F2 U2 B2 L2 U' B2 U' B2 L2 D2 L2 U R2 U' 14f U' L2 D' L2 D B2 F2 L2 R2 D F2 U' F2 U' 14f Cube Explorer 1.5 was able to show that the 8f maneuver is minimal. This position is not a strong local maximum, because the shortest strong local maxima are 9f. Cube Explorer 1.5 was not able to show that the 14f maneuvers are minimal in the time I gave it (six hours each on a Pentium 166). But I suspect that 14f is in fact minimal. Also, I do not know if the 14f maneuvers are strong local maxima because my search for strong local maxima extended only through 9f from Start. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Mon Feb 23 18:59:06 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA27870; Mon, 23 Feb 1998 18:59:05 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Feb 23 18:53:55 1998 Date: Mon, 23 Feb 1998 15:53:21 -0800 (PST) Message-Id: <199802232353.PAA05251@denali.cs.ucla.edu> From: Richard E Korf To: jbryan@pstcc.cc.tn.us Cc: cube-lovers@ai.mit.edu In-Reply-To: (message from Jerry Bryan on Mon, 23 Feb 1998 10:59:54 -0500 (Eastern Standard Time)) Subject: Re: Strong Local Maxima 9f from Start The two 14f move isoglyphs reported by Jerry Bryan in his last message do indeed require 14f moves. -rich From cube-lovers-errors@mc.lcs.mit.edu Tue Feb 24 17:38:27 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA01098; Tue, 24 Feb 1998 17:38:26 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Feb 24 05:49:14 1998 Date: Tue, 24 Feb 1998 11:48:55 +0100 (MET) From: Christian X-Sender: ceggermo@hengstdal.nijmegen.inter.nl.net Reply-To: Christian To: cube-lovers@ai.mit.edu Subject: RE: Rubik lawyers up in arms over website -- Digest v23 #279] In-Reply-To: <009C2024.8A2B6B63.8@ice.sbu.ac.uk> Message-Id: DAVID SINGMASTER, Professor of Mathematics and Metagrobologist wrote: > Norman Diamond refers to the patents expiring. However, Rubik > only had a Hungarian patent. As a result, the various Rubik companies > took legal action under copyright law, and copyright lasts much > longer. I think the following URL will answer most questions and resolves the issue: http://www.csun.edu/~hcmth014/comicfiles/copyright.html > Perhaps I should hassle Seven Towns about the use of my notation! Mmm interesting Idea... (-: Christian --------------------------------------------------- E-Mail: C.Eggermont@inter.NL.net Homepage: http://www.inter.nl.net/users/C.Eggermont --------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Wed Feb 25 12:25:52 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA03419; Wed, 25 Feb 1998 12:25:52 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Feb 25 00:39:17 1998 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Taiwanese Invention of the Cube? Date: 25 Feb 1998 05:38:19 GMT Organization: California Institute of Technology, Pasadena Message-Id: <6d0aob$6vq@gap.cco.caltech.edu> I have heard through multi-generation, unreliable sources that the Cube was first invented and patented by a Taiwanese person. This story strikes me as strongly false, but perhaps may have some basis somewhere. Any guesses? Perhaps a particularly different sort of mechanism was patented? A trademark? -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- "...he put a wire in his cap and called himself Marconi." From cube-lovers-errors@mc.lcs.mit.edu Wed Feb 25 19:33:53 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA04469; Wed, 25 Feb 1998 19:33:52 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Feb 25 16:47:43 1998 From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Message-Id: <199802252147.NAA14815@liquefy.ugcs.caltech.edu> Subject: RE: Taiwanese Invention of the Cube? (fwd) To: zingmast@sbu.ac.uk, cube-lovers@ai.mit.edu Date: Wed, 25 Feb 1998 13:47:07 -0800 (PST) Reply-To: whuang@ugcs.caltech.edu David Singmaster Computing & Maths South Bank Univ typed something like this in a previous message: >From zingmast@sbu.ac.uk Wed Feb 25 13:17:35 1998 Sender: zingmast@sbu.ac.uk Date: Wed, 25 Feb 1998 21:14:57 +0000 From: David Singmaster Computing & Maths South Bank Univ To: whuang@ugcs.caltech.edu Message-ID: <009C25BA.E326A9B3.4@ice.sbu.ac.uk> Subject: RE: Taiwanese Invention of the Cube? The earliest idea was due to someone in California, named William O. Gustafson (US patent 3,081,089 of 12 Mar 1963). He had a 2^3 in the shape of a sphere, but he had the problem of keeping the interior parts in synch with the outer parts and so he left gaps between the pieces. Basically he had an interior sphere with grooves and the pieces had lips. There are two versions - the first seems like it wouldn't work well, if at all, but the second seems fairly feasible. Unfortunately, Gustafson let his patent lapse, so the patent of Larry Nichols was the next, with US patent 3,655,201 (applied 4 Mar 1970, issued 11 Apr 1972). This had only the idea of a cubical puzzle and no practical mechanism, so I don't consider it very significant, but Nichols sued Rubik, more or less successfully - I never heard the conclusion of the story. Frank Fox (UK patent 1,344,259, applied for on 9 Apr 1970 and issued on 16 Jan 1974) seems to be next. He had a 3^3 sphere with tongue and grooves holding the pieces together, with a hollow center. He had let his patent lapse also. In 1976-1980, Terutoshi Ishige devised and patented two mechanisms for a 2^3., similar to Rubik's. This may be the source of the story you were asking about. However, there is another odd story. The first French writers on the Cube record that a old friend said he had played with such a cube (in wood) in Istanbul in 1920 and in Marseilles about 1935. However, no further evidence of such an early version has appeared. I forgot to set this to send myself a copy. Could you forward it back to me. Also you might like to send it to cube-lovers@ai.mit.edu Regards DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Have you heard the one about the guy Jean who visited Japan? From cube-lovers-errors@mc.lcs.mit.edu Wed Feb 25 21:29:52 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA04737; Wed, 25 Feb 1998 21:29:51 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Feb 25 18:29:45 1998 Message-Id: <9802252330.AA19883@jrdmax.jrd.dec.com> Date: Thu, 26 Feb 98 08:30:27 +0900 From: Norman Diamond 26-Feb-1998 0830 To: cube-lovers@ai.mit.edu Subject: Re: Taiwanese Invention of the Cube? Wei-Hwa Huang wote: >I have heard through multi-generation, unreliable sources that the Cube >was first invented and patented by a Taiwanese person. Invention is more or less possible. Surely someone like the famous Mr. Wu (whose given names I've forgotten) would be able to invent it. But if it happened, surely it would be hard to say who came first. As for patenting, somehow the mixture of "patent" and "Taiwan" in the same sentence strikes me as an oxymoron. >A trademark? Somehow the mixture of "trademark" and "Taiwan" strikes me as an oxymoron too, even though they're not in the same sentence. Want to try "copyright" next? :-) -- Norman Diamond diamond@jrdv04.enet.dec-j.co.jp [Speaking for Norman Diamond not for Digital.] From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 6 19:16:00 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA29248; Fri, 6 Mar 1998 19:15:59 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Mar 4 16:17:05 1998 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Taiwanese Invention of the Cube? Date: 4 Mar 1998 21:16:09 GMT Organization: California Institute of Technology, Pasadena Message-Id: <6dkgap$rsv@gap.cco.caltech.edu> References: Norman Diamond 26-Feb-1998 0830 writes: >As for patenting, somehow the mixture of "patent" and "Taiwan" in the >same sentence strikes me as an oxymoron. >Somehow the mixture of "trademark" and "Taiwan" strikes me as an >oxymoron too, even though they're not in the same sentence. >Want to try "copyright" next? :-) Is it possible to copyright the Cube? That's why I didn't try it. In any case, stop sneering -- Taiwan has local copyright, trademark, and patent laws, and has had them for decades. Sure, they haven't honored international copyright laws, but then again, most other countries don't think Taiwan exists as an independent country. When it became economically viable to honor international copyright, they did so -- such legislation was passed in 1994. Perhaps you are getting a biased view from living in Japan? -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Have you heard the one about the guy Jean who visited Japan?