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