From mark.longridge@canrem.com Sat Oct 21 22:24:10 1995 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA20340; Sat, 21 Oct 95 22:24:10 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1FA20D; Sat, 21 Oct 95 22:16:38 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Spotty Megaminx From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1256.5834.0C1FA20D@canrem.com> Date: Sat, 21 Oct 95 22:12:00 -0500 Organization: CRS Online (Toronto, Ontario) Observations on the Magic Dodecahedron (Megaminx) ------------------------------------------------- I've never seen anything on patterns for the megaminx, with the sole exception of Kurt Endl's book "Megaminx". Unfortunately there are no detailed examples, only vague references to "many possible dot patterns" and "star patterns". A pattern similar to the 6 X order 3 of the cube is on the cover, but only part of the dodecahedron is visible. Using the solving skills I developed myself, I deliberately solved the megaminx with the centres not matching the surrounding face. Techniques like mono-twists and mono-flips carried over well from the cube. My conclusion: A 10-dot pattern is possible! Here is a description.... One pair of opposite faces is completely solid. The 5 faces adjacent to solid face A are spotted, also the 5 faces adjacent to solid face B (opposite to A) are spotted. If we look at one set of 5 faces we can observe that in this particular 10-spot that the 5 centres appear rotated to the left, or (since the centres don't really move in position) that the rest of the face is moved to the right. Similarly, in the lower tier of 5 faces, we can observe that 5 centres appear rotated to the left also. Let's try a small thought experiment. Imagine a skeleton, a disassembled megaminx. Grab the top and bottom with thumb and forefinger. Now, while keeping the top and bottom centres immobile, rotate the rest of the puzzle. What happens? The 10 other centres rotate in the same direction! If we do this on a cube skeleton the same thing happens, but on a fleshed out cube this would become a 4 cycle of centres, which is in the swap orbit and can't be reached by face turns. On the megaminx we have 2 five cycles of centres, and this is legal. There are 6 opposite pairs of faces on the megaminx. There are 4 ways to rotate the centres for each pair to generate a 10 spot. I'll speculate that there are 6*4 = 24 possible 10-spots. I suspect various 12-spots are possible. I have no idea how to easily permute centre pieces on the megaminx. -> Mark <- From dik@cwi.nl Sat Oct 21 22:52:22 1995 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA21513; Sat, 21 Oct 95 22:52:22 EDT Received: from bever.cwi.nl by charon.cwi.nl with SMTP id ; Sun, 22 Oct 1995 03:52:11 +0100 Received: by bever.cwi.nl id ; Sun, 22 Oct 1995 03:52:12 +0100 Date: Sun, 22 Oct 1995 03:52:12 +0100 From: Dik.Winter@cwi.nl Message-Id: <9510220252.AA04563=dik@bever.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Re: Spotty Megaminx Content-Length: 1116 > I've never seen anything on patterns for the megaminx, with the > sole exception of Kurt Endl's book "Megaminx". It is long ago I had it in my hands, and I have no books. What I say is from memory; probably correct. Note that a face turn induces an even permutation on both the corner and the edge "cubies". So odd permutations are not possible. On the other hand (if I remember well) *all* combinations of even permutations are possible. > There are 6 opposite pairs of faces on the megaminx. There are 4 ways > to rotate the centres for each pair to generate a 10 spot. I'll > speculate that there are 6*4 = 24 possible 10-spots. Right. > I suspect various 12-spots are possible. I have no idea how to > easily permute centre pieces on the megaminx. Indeed. Every rotation of the center skeleton is possible (if you consider the remainder fixed...). So there are 12 centers that can come out at top; for each center at top you have 5 possible positions of the remainder leading to 60 configurations. Of these 24 are 10-spots, 1 is the solved puzzle, so the remainder (35) is 12-spots. dik From mreid@ptc.com Mon Oct 23 11:20:02 1995 Return-Path: Received: from ptc.com (poster.ptc.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08395; Mon, 23 Oct 95 11:20:02 EDT Received: from ducie.ptc.com by ptc.com (5.x/SMI-SVR4-NN) id AA23979; Mon, 23 Oct 1995 11:15:45 -0400 Message-Id: <9510231515.AA23979@ptc.com> Received: by ducie.ptc.com (1.38.193.4/16.2.nn) id AA15699; Mon, 23 Oct 1995 11:42:32 -0400 Date: Mon, 23 Oct 1995 11:42:32 -0400 From: michael reid To: boland@sci.kun.nl, cube-lovers@ai.mit.edu Subject: Re: Embedding G in a symmetrical group michiel boland writes > It is clear that the group G of the cube (the one with > 4.3252x10^19 elements) can be embedded in a > symmetrical group, e.g. S_48, since each move of the cube can be > seen as a permutation of 48 objects. Hence, there is a smallest > number n such that G can be embedded in S_n. I'm curious to find > out what this number is. 48. first note that any homomorphism G --> S_n can be factored as G --> S_m_1 x S_m_2 x ... x S_m_k >--> S_n where m_1, m_2, ... , m_k are the sizes of the orbits of G acting on {1, 2, ... , n}, and thus m_1 + m_2 + ... + m_k = n. furthermore, the action of G on each {1, 2, ... , m_i} is transitive. transitive G-sets are easy to understand. for any subgroup H of G, G acts transitively on the cosets G/H by left multiplication. also, any transitive G-set is of this form. given a homomorphism G --> S_m with a transitive action, let H be the subgroup of G that fixes the element 1. then it's easy to see that the cosets G/H are in one-to-one correspondence with elements in the orbit of 1 (which by hypothesis are all of 1, 2, ... , m) and the action of G on G/H is isomorphic to the action of G on {1, 2, ... , m}. the kernel of the homomorphism G --> sym(G/H) is the largest normal subgroup of G contained in H , which is just the intersection of all G-conjugates of H. of course, in this case we have m = (G : H) (index of H in G). thus michiel's question can be settled by considering all subgroups of G with index less than 48. unless i've overlooked some, there are exactly 8 such, up to G-conjugacy. they are G itself G' = commutator subgroup of G = subgroup of positions an even number of quarter turns from start C_0 = subgroup where the corner UFR is in place, but may be twisted C'_0 = commutator subgroup of C_0 = intersection of C_0 and G' E_0 = subgroup where the edge UR is in place, but may be flipped E'_0 = commutator subgroup of E_0 = intersection of E_0 and G' C_1 = subgroup where the corner UFR is in place and is not twisted E_1 = subgroup where the edge UR is in place and is not flipped. for each of these, except the last two, the kernel of G --> sym(G/H) contains all elements that only flip edges in place and twist corners in place. number of subgroup index kernel conjugates G 1 G 1 G' 2 G' 1 C_0 8 {all corners in place, may be twisted} 8 C'_0 16 {all corners in place, may be twisted} 8 E_0 12 {all edges in place, may be flipped} 12 E'_0 24 {all edges in place, may be flipped} 12 C_1 24 {all corners in place, may not be twisted} 8 E_1 24 {all edges in place, may not be flipped} 12 thus the only way to get and embedding (i.e. injective homomorphism) G --> S_n using the subgroups above is G --> sym(G/C_1) x sym(G/E_1) >--> S_48 which in fact, is just the action of G on the 48 non-center facelets. i had previously stumbled across this exact same question, so now i'm curious: why are you interested in this? mike From bagleyd@source.asset.com Mon Oct 23 13:57:59 1995 Return-Path: Received: from source.asset.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17761; Mon, 23 Oct 95 13:57:59 EDT Received: by source.asset.com (AIX 3.2/UCB 5.64/4.03) id AA15823; Mon, 23 Oct 1995 13:33:08 -0400 Date: Mon, 23 Oct 1995 13:33:08 -0400 From: bagleyd@source.asset.com (David A. Bagley) Message-Id: <9510231733.AA15823@source.asset.com> To: cube-lovers@life.ai.mit.edu Subject: pyraminx-like puzzles (yet again) Hi > Recently I asked the question: > > I have a question, I hope this makes sence. ;) On a "nxnxn" > > tetrahedron with period 2 or period 3 turning or a "nxnxn" octahedron with > > period 3 or period 4 turning, can the orientation of any of the center > > triangles change when the puzzle is solved? If so, where does this > > start to happen. I know from "experience" that this is not true on > > a pyraminx. The reported to answer to * was incorrect > Well, if you believe proof by example on a simulated puzzle, then > Tetrahedron period 2 turning: never happens > Tetrahedron period 3 turning: starts when n=4 with center triangle > Octahedron period 3 turning: starts when n=4 with center triangle * > Octahedron period 4 turning: starts with n=4 with center triangle It should be: Octahedron period 3 turning: starts when n=2 with center triangle The case where n = 3 (here there is no one center triangle) was interesting because there seemed to be no easy repetition of moves where the colors of the puzzle would be solved but the orientation of the triangles would be changed. /\ /__\ /\C /\ /__\/__\ /\C /\C /\ /__\/__\/__\ After much experimentation, I found a way of rotating 3 center triangles on a face which involved 216 moves, (this can be bettered by one noting that 2 clockwise rotations = 1 counterclockwise rotation): Repeat 5 times { With reference to the top "C" in diagram, turn center to the right and then rotate face clockwise for a total of 42 moves. One will then get a pattern where only 3 center colors are out of place on 3 different faces. rotate this face clockwise } rotate this face clockwise New versions of my pyraminx and octahedron puzzles are now out. Cheers, clockwise New versions of my pyraminx and octahedron puzzles are now out. Cheers, --__--------------------------------------------------------------- / \ \ / David A. Bagley \ | \ \ / bagleyd@source.asset.com | | \//\ Some days are better than other days. | | / \ \ -- A short lived character of Blake's 7 | \ / \_\puzzles Available at: ftp.x.org/contrib/games/puzzles / ------------------------------------------------------------------- From hazard@niksula.hut.fi Mon Oct 23 14:42:32 1995 Return-Path: Received: from nukkekoti.cs.hut.fi by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA20954; Mon, 23 Oct 95 14:42:32 EDT Received: from ummagumma.tky.hut.fi (hazard@ummagumma.tky.hut.fi [130.233.33.120]) by nukkekoti.cs.hut.fi (8.6.12/8.6.11) with SMTP id UAA28308 for ; Mon, 23 Oct 1995 20:42:30 +0200 Date: Mon, 23 Oct 1995 20:42:30 +0200 Message-Id: <199510231842.UAA28308@nukkekoti.cs.hut.fi> X-Sender: hazard@pop.niksula.cs.hut.fi X-Mailer: Windows Eudora Pro Version 2.1.2 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: cube-lovers@ai.mit.edu From: Mikko Haapanen Subject: pull out the corner? Hello! I have a question (yes, again). This subject may be discussed here before, but i don't understand set theory or other high math, so i ask: If i had a 3x3x3 cube and i pull out a corner piece. I turn it and push back. Now the cube cannot be solved. I think the cube is now 'on the other orbit'. If i pull now an edge piece and flip it, the cube is again on some other orbit. Only one of those orbits are legal. How many different illegal orbits there are? -----Mikko Haapanen------hazard@niksula.hut.fi------ Another toy will help destroy The elder race of man Forget about your silly whim It doesn't fit the plan ---------------------------------------------------- From BRYAN@wvnvm.wvnet.edu Mon Oct 23 16:38:39 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29047; Mon, 23 Oct 95 16:38:39 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R3) with BSMTP id 8685; Mon, 23 Oct 95 16:38:12 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 7909; Mon, 23 Oct 1995 16:38:13 -0400 Message-Id: Date: Mon, 23 Oct 1995 16:38:12 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: pull out the corner? In-Reply-To: Message of 10/23/95 at 20:42:30 from hazard@niksula.hut.fi On 10/23/95 at 20:42:30 Mikko Haapanen said: >I have a question (yes, again). This subject may be discussed here before, >but i don't understand set theory or other high math, so i ask: >If i had a 3x3x3 cube and i pull out a corner piece. I turn it and push >back. Now the cube cannot be solved. I think the cube is now 'on the other >orbit'. If i pull now an edge piece and flip it, the cube is again on some >other orbit. >Only one of those orbits are legal. How many different illegal orbits there are ? In the terms you are using, there are 12 orbits. Of these, 1 is "legal" (contains Start), and 11 are "illegal" (do not contain Start). There is a factor of 3 from twisting the corners. Pull out a corner piece. There are 3 ways to put it back in. You can put it back in the way it came out, you can twist it right, or you can twist it left. There is a factor of 2 from flipping the edges. Pull out an edge piece. There are 2 ways to put it back in, flipped or unflipped. There is a factor of 2 from parity. The edges can be said to be in even parity or in odd parity, and the corners can be said to be in even parity or odd parity. Normally, the corners and edges are in the same parity. A quarter turn changes the parity both for the edges and for the corners. But pull out 2 edges pieces (or 2 corner pieces). Put them back where they came from, and their parity remains the same. Exchange them, and their parity changes. We therefore have 12=3x2x2. However (and draw a deep breath), for every expert there is an equal and opposite expert. This use of the term "orbit" agrees with some experts. However, other experts would say that the corners form an orbit, that the edges form an orbit, and that the face centers form an orbit. I don't know which use of the term orbit is correct (perhaps both are in the proper context). But in any case, if you take a cube apart, there are 12 disjoint sets of positions that you choose from when you put the cube back together. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From geohelm@pt.lu Tue Oct 24 11:31:19 1995 Return-Path: Received: from menvax.restena.lu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16179; Tue, 24 Oct 95 11:31:19 EDT Date: Tue, 24 Oct 95 11:31:18 EDT Received: from telinf1.pt.lu by menvax.restena.lu with SMTP; Tue, 24 Oct 1995 15:56:00 +0100 (MET) Received: from slip12.pt.lu by telinf1.pt.lu id aa24228; 24 Oct 95 13:49 CET X-Sender: geohelm@mailsvr.pt.lu (Unverified) X-Mailer: Windows Eudora Version 1.4.4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: cube-lovers@ai.mit.edu From: Georges Helm Subject: availability of cubes and other puzzles Message-Id: <9510241349.aa24228@telinf1.pt.lu> Here is the address of a German friend who is still selling the following puzzles (among others): 5x5x5, skewb, magic dodecahedron, German calender cube, pyraminx... Christoph Bandelow An der Wabeck 37 D-58456 Witten Germany Tel.: ++49-2302-71147 Fax : ++49-2302-77001 Books he is selling (among others): Bandelow: Inside Rubik's Cube and beyond (~$15) Singmaster: Notes (~$10) Georges Helm From boland@sci.kun.nl Tue Oct 24 17:31:16 1995 Return-Path: Received: from wn1.sci.kun.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10061; Tue, 24 Oct 95 17:31:16 EDT Received: from canteclaer.sci.kun.nl by wn1.sci.kun.nl via canteclaer.sci.kun.nl [131.174.132.34] with SMTP id WAA17657 (8.6.10/2.14); Tue, 24 Oct 1995 22:31:01 +0100 Message-Id: <199510242131.WAA17657@wn1.sci.kun.nl> To: Mikko Haapanen Cc: cube-lovers@ai.mit.edu Subject: Re: pull out the corner? In-Reply-To: Your message of "Mon, 23 Oct 95 20:42:30 +0200." <199510231842.UAA28308@nukkekoti.cs.hut.fi> Date: Tue, 24 Oct 95 22:31:00 +0100 From: Michiel Boland Mikkao Haapanen writes: >If i had a 3x3x3 cube and i pull out a corner piece. I turn it and push >back. Now the cube cannot be solved. [...] This has nothing to do with his question, but one of my old cubes has become so loose that it has become quite easy to twist a single corner piece - no doubt other people have expierenced this phenomenon. The last stage in my cube-solving algorithm used to be orienting the corners - this has now become trivial. :) -- Michiel Boland University of Nijmegen The Netherlands From boland@sci.kun.nl Mon Oct 30 07:48:01 1995 Return-Path: Received: from wn1.sci.kun.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06686; Mon, 30 Oct 95 07:48:01 EST Received: from canteclaer.sci.kun.nl by wn1.sci.kun.nl via canteclaer.sci.kun.nl [131.174.132.34] with SMTP id NAA19553 (8.6.10/2.14) for ; Mon, 30 Oct 1995 13:48:00 +0100 Message-Id: <199510301248.NAA19553@wn1.sci.kun.nl> To: cube-lovers@ai.mit.edu Subject: Exchanging just four edges in antislice impossible? Date: Mon, 30 Oct 95 13:47:59 +0100 From: Michiel Boland Hello all, can anyone provide an easy proof of the fact that it is impossible to exchange just four edges using just antislice moves, whilst leaving everything else fixed? (We're talking about the 3x3 cube of course.) Another way of putting it: why are the 2xH and 4-dot patterns not in the antislice group? I have thought about this a little, but not hard enough to find an answer. I looked it up in Singmaster's Notes but could not find a satisfying explanation either. Here is some more background. The antislice group is contained in the group of all positions that are symmetric under `cube half-turns' (the subgroup of M containing I,(FB)(LR),(FB)(UD) and (UD)(LR)). This group has (8*4*12*8*4*3*2^2)/2 = 73728 elements. It can be shown that in the antislice group, the orientation of the corners is determined by the edge positions [I am willing to explain this, but it is much easier visualized than written down], which means that the antislice group contains at most 73728/3=24576 elements. But apparently the antislice group contains just 6144 elements, which is a factor 4 below the abovementioned number. This factor 4 is explained by the fact above, which I am trying to prove. -- Michiel Boland University of Nijmegen The Netherlands From mark.longridge@canrem.com Tue Oct 31 01:27:02 1995 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09622; Tue, 31 Oct 95 01:27:02 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1FBDC8; Tue, 31 Oct 95 01:11:34 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Spotty Megaminx Revisited From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1257.5834.0C1FBDC8@canrem.com> Date: Tue, 31 Oct 95 01:02:00 -0500 Organization: CRS Online (Toronto, Ontario) Notes on the Spot Patterns on the Megaminx ------------------------------------------ Number the faces of the megaminx 1 through 12. Here are all the possible permutations of the 12 centres: dod := Group( (2,3,4,5,6) (7,8,9,10,11), (1,4,10,9,2)(5,11,12,8,6) );; Size (dod) = 60; NumberConjugacyClasses (dod) = 5; Elements (dod); [ (), 0 spot ( 2, 3, 4, 5, 6)( 7, 8, 9,10,11), 2 5-cycles = 10 ( 2, 4, 6, 3, 5)( 7, 9,11, 8,10), 2 5-cycles = 10 ( 2, 5, 3, 6, 4)( 7,10, 8,11, 9), 2 5-cycles = 10 ( 2, 6, 5, 4, 3)( 7,11,10, 9, 8), 2 5-cycles = 10 ( 1, 2)( 3, 6)( 4, 8)( 5, 9)( 7,10)(11,12), 6 2-cycles = 12 ( 1, 2, 3)( 4, 6, 9)( 5, 8,10)( 7,12,11), 4 3-cycles = 12 ( 1, 2, 6)( 3, 8, 5)( 4, 9, 7)(10,12,11), 4 3-cycles = 12 ( 1, 2, 8, 7, 5)( 3, 9,12,11, 4), 2 5-cycles = 10 ( 1, 2, 9,10, 4)( 5, 6, 8,12,11), 2 5-cycles = 10 ( 1, 3, 2)( 4, 9, 6)( 5,10, 8)( 7,11,12), 4 3-cycles = 12 ( 1, 3, 9, 8, 6)( 4,10,12, 7, 5), 2 5-cycles = 10 ( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7,12)( 8,11), 6 2-cycles = 12 ( 1, 3,10,11, 5)( 2, 9,12, 7, 6), 2 5-cycles = 10 ( 1, 3, 4)( 2,10, 5)( 6, 9,11)( 7, 8,12), 4 3-cycles = 12 ( 1, 4,10, 9, 2)( 5,11,12, 8, 6), 2 5-cycles = 10 ( 1, 4,11, 7, 6)( 2, 3,10,12, 8), 2 5-cycles = 10 ( 1, 4, 3)( 2, 5,10)( 6,11, 9)( 7,12, 8), 4 3-cycles = 12 ( 1, 4, 5)( 2,10, 7)( 3,11, 6)( 8, 9,12), 4 3-cycles = 12 ( 1, 4)( 2,11)( 3, 5)( 6,10)( 7, 9)( 8,12), 6 2-cycles = 12 ( 1, 5, 7, 8, 2)( 3, 4,11,12, 9), 2 5-cycles = 10 ( 1, 5, 6)( 2, 4, 7)( 3,11, 8)( 9,10,12), 4 3-cycles = 12 ( 1, 5,11,10, 3)( 2, 6, 7,12, 9), 2 5-cycles = 10 ( 1, 5, 4)( 2, 7,10)( 3, 6,11)( 8,12, 9), 4 3-cycles = 12 ( 1, 5)( 2,11)( 3, 7)( 4, 6)( 8,10)( 9,12), 6 2-cycles = 12 ( 1, 6, 2)( 3, 5, 8)( 4, 7, 9)(10,11,12), 4 3-cycles = 12 ( 1, 6, 8, 9, 3)( 4, 5, 7,12,10), 2 5-cycles = 10 ( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,11)(10,12), 6 2-cycles = 12 ( 1, 6, 5)( 2, 7, 4)( 3, 8,11)( 9,12,10), 4 3-cycles = 12 ( 1, 6, 7,11, 4)( 2, 8,12,10, 3), 2 5-cycles = 10 ( 1, 7, 2, 5, 8)( 3,11, 9, 4,12), 2 5-cycles = 10 ( 1, 7, 9)( 2, 6, 8)( 3, 5,12)( 4,11,10), 4 3-cycles = 12 ( 1, 7,10)( 2, 8, 9)( 3, 6,12)( 4, 5,11), 4 3-cycles = 12 ( 1, 7)( 2,11)( 3,12)( 4, 8)( 5, 6)( 9,10), 6 2-cycles = 12 ( 1, 7, 4, 6,11)( 2,12, 3, 8,10), 2 5-cycles = 10 ( 1, 8, 3, 6, 9)( 4, 7,10, 5,12), 2 5-cycles = 10 ( 1, 8)( 2, 6)( 3, 7)( 4,12)( 5, 9)(10,11), 6 2-cycles = 12 ( 1, 8, 5, 2, 7)( 3,12, 4, 9,11), 2 5-cycles = 10 ( 1, 8,10)( 2, 9, 3)( 4, 6,12)( 5, 7,11), 4 3-cycles = 12 ( 1, 8,11)( 2,12, 4)( 3, 9,10)( 5, 6, 7), 4 3-cycles = 12 ( 1, 9, 6, 3, 8)( 4,12, 5,10, 7), 2 5-cycles = 10 ( 1, 9)( 2, 3)( 4, 8)( 5,12)( 6,10)( 7,11), 6 2-cycles = 12 ( 1, 9, 7)( 2, 8, 6)( 3,12, 5)( 4,10,11), 4 3-cycles = 12 ( 1, 9, 4, 2,10)( 5, 8,11, 6,12), 2 5-cycles = 10 ( 1, 9,11)( 2,12, 5)( 3,10, 4)( 6, 8, 7), 4 3-cycles = 12 ( 1,10, 8)( 2, 3, 9)( 4,12, 6)( 5,11, 7), 4 3-cycles = 12 ( 1,10, 2, 4, 9)( 5,12, 6,11, 8), 2 5-cycles = 10 ( 1,10, 7)( 2, 9, 8)( 3,12, 6)( 4,11, 5), 4 3-cycles = 12 ( 1,10)( 2,11)( 3, 4)( 5, 9)( 6,12)( 7, 8), 6 2-cycles = 12 ( 1,10, 5, 3,11)( 2,12, 6, 9, 7), 2 5-cycles = 10 ( 1,11, 8)( 2, 4,12)( 3,10, 9)( 5, 7, 6), 4 3-cycles = 12 ( 1,11, 9)( 2, 5,12)( 3, 4,10)( 6, 7, 8), 4 3-cycles = 12 ( 1,11, 3, 5,10)( 2, 7, 9, 6,12), 2 5-cycles = 10 ( 1,11, 6, 4, 7)( 2,10, 8, 3,12), 2 5-cycles = 10 ( 1,11)( 2,12)( 3, 7)( 4, 5)( 6,10)( 8, 9), 6 2-cycles = 12 ( 1,12)( 2, 7)( 3,11)( 4,10)( 5, 9)( 6, 8), 6 2-cycles = 12 ( 1,12)( 2, 8)( 3, 7)( 4,11)( 5,10)( 6, 9), 6 2-cycles = 12 ( 1,12)( 2, 9)( 3, 8)( 4, 7)( 5,11)( 6,10), 6 2-cycles = 12 ( 1,12)( 2,10)( 3, 9)( 4, 8)( 5, 7)( 6,11), 6 2-cycles = 12 ( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7) 6 2-cycles = 12 Number Pattern ------ ------- 1 0 spots 24 2 five-cycles (10 spot) 15 6 two-cycles (12 spot) 20 4 three-cycles (12 spot) -- 60 orientations of the dodecahedron, 24 ten-spots, 35 twelve-spots >> I suspect various 12-spots are possible. I have no idea how to >> easily permute centre pieces on the megaminx. > > Indeed. Every rotation of the center skeleton is possible (if you > consider the remainder fixed...). So there are 12 centers that can > come out at top; for each center at top you have 5 possible positions > of the remainder leading to 60 configurations. Of these 24 are > 10-spots, 1 is the solved puzzle, so the remainder (35) is 12-spots. > dik Well, I was confused how there could be 35 twelve-spots (at first), but I am happy to confirm Dik's memory. -> Mark <- From alan@curry.epilogue.com Fri Nov 3 02:39:27 1995 Return-Path: Received: from curry.epilogue.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16466; Fri, 3 Nov 95 02:39:27 EST Received: (from alan@localhost) by curry.epilogue.com (8.6.8/8.6.6) id CAA18206; Fri, 3 Nov 1995 02:39:27 -0500 Date: Fri, 3 Nov 1995 02:39:27 -0500 Message-Id: <3Nov1995.011909.Alan@LCS.MIT.EDU> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu To: Cube-Lovers@ai.mit.edu In-Reply-To: Kathryn Kelly's message of Fri, 3 Nov 1995 01:00:07 -0500 (EST) Subject: Magazine Spam First off, let me apologize for sending a message that has nothing whatsoever to do with Rubik's Cube. But I know from private electronic mail that many of you were very annoyed by the six copies of the "magazine club" advertisement that were distributed through Cube-Lovers during the last month. I've been urging people to just sit tight and ignore the messages, because it wasn't entirely clear where the mail was actually coming from. All that appeared certain was that the ads were being gatewayed through an Internet Service Provider in New York named "American Network, Inc" (domain name IXC.NET). Recently I've been talking to the responsible folks at American Network. They are quite apologetic about the whole thing, but apparently the miscreant is working through accounts that they give away free for the asking. So they actually have no way to track the jerk down. They don't even know how many of their accounts might all belong to the same person! This whole experience has apparently woken them up to what a -bad- idea that is, and they're going to stop. Besides cleaning up their act about granting free anonymous accounts, they also tracked down the post office box number that the miscreant uses for his business. And they have a request: Date: Fri, 3 Nov 1995 01:00:07 -0500 (EST) From: Kathryn Kelly ... We wish you to do something for us. We have the address where the individual sending out this material receives his physical mail. Please send a letter of complaint to the postmaster at Postmaster Staten Island NY 10312 Regarding the person at this address: Magazine Club Inquiry Center Att. Internet Services Department P. O. Box 120990 Staten Island NY 10312 0990 So those of you who really want to complain to somebody, here's your chance. Compose a nice letter to the Postmaster at the address above explaining that the folks running the business at that P.O.Box are engaging in anti-social behavior on the Internet. Unfortunately, I don't believe there is anything -illegal- about what this jerk is doing (although it is closely analogous to some things that are illegal to do with a telephone), so it wont work to demand that the Postmaster actually -do- anything, but work up a good complaint anyway. Finally, I urge you all -not- to respond to this message in public. If you have further thoughts on Internet advertising, electronic mailing list administration, or clever acts of revenge, you can send them to -me-, but don't CC your message to Cube-Lovers as a whole. The whole point here is to keep Cube-Lovers relatively free of off-topic mail. As the list administrator I get to send out occasional administrivia such as this message because I do actual -work- to keep the list running. And yes, I am still working on some new list management technology that should eliminate problems like this in the future. (And yes, I know all about MajorDomo, listserv, and their relatives -- you don't need to enlighten me about them.) - Alan (Cube-Lovers-Request@AI.MIT.EDU) (I never imagined when Dave Plummer and I started this list in 1980 that one day I'd be spending significant time trying to prevent people from using it to sell magazines. Amazing.) From mark.longridge@canrem.com Sun Nov 5 02:14:26 1995 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08624; Sun, 5 Nov 95 02:14:26 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1FCC4E; Sun, 5 Nov 95 02:09:17 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Halpern's Tetrahedron From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1258.5834.0C1FCC4E@canrem.com> Date: Sun, 5 Nov 95 01:59:00 -0500 Organization: CRS Online (Toronto, Ontario) # Ben Halpern's Tetrahedron # 4 faces rotate tetra := Group( (1,3,5)(2,4,6)(7,13,24)(12,18,23)(11,17,22), (7,9,11)(8,10,12)(22,15,3)(2,21,14)(1,20,13), (13,15,17)(14,16,18)(11,20,5)(10,19,4)(9,24,3), (22,20,24)(21,19,23)(7,15,5)(8,16,6)(9,17,1) );; # Size (tetra) = 3,732,480; # # Centre (tetra) = (2,12)(4,18)(6,23)(8,21)(10,14)(16,19); # # Tetrahedron has 6 edges, 4 corners # 6! /2 * 2^5 * 4!/2 * 3^3 Just a few more combinations than a 2x2x2 pocket cube... -> Mark <- From rjh@on-ramp.ior.com Mon Nov 6 22:09:28 1995 Return-Path: Received: from on-ramp.ior.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26893; Mon, 6 Nov 95 22:09:28 EST Received: from cs2-07.ior.com by on-ramp.ior.com with smtp (Smail3.1.28.1 #10) id m0tCePb-000RomC; Mon, 6 Nov 95 19:09 PST Message-Id: Date: Mon, 6 Nov 95 19:09 PST X-Sender: rjh@on-ramp.ior.com X-Mailer: Windows Eudora Version 1.4.4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: Cube-Lovers@ai.mit.edu From: rjh@on-ramp.ior.com (RonH) Subject: Rubik's stuff Saw your message in rec.puzzles. Please add me to your mailing list for Rubik's Cube info. My address is rjh@on-ramp.ior.com Thanks in advance! RON From joemcg@catch22.com Thu Nov 9 13:26:02 1995 Return-Path: Received: from B17.Catch22.COM by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA22560; Thu, 9 Nov 95 13:26:02 EST Received: (from joemcg@localhost) by B17.Catch22.COM (8.6.9/8.6.12) id KAA06094; Thu, 9 Nov 1995 10:29:44 -0800 X-Url: http://www.Catch22.COM/ Date: Thu, 9 Nov 1995 10:29:43 -0800 (PST) From: Joe McGarity To: "Rubik's Cube Mailing List" Subject: Flowers in you hair Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Hello, I just joined this group and I thought that I would throw out a place here in San Francisco where one can find some interesting things (such as the elusive 5x5x5 cube). Game Gallery One Embarcadero Center Street Level San Francisco, CA 94102 (415) 433-4263 Also, there is a second-hand store on the corner of 17th and Mission which seems to have some type of mix-up-and-fix puzzle every time I go in. Let me know what people are looking for and I'll keep an eye out for it. Later, Joe ------------------------------------------------------------------------------ Joe McGarity "Mufasa, Mufasa, Mufasa!" 418 Fair Oaks San Francisco, CA 94110 joemcg@catch22.com ------------------------------------------------------------------------------ From joemcg@catch22.com Thu Nov 9 14:10:43 1995 Return-Path: Received: from B17.Catch22.COM by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA01378; Thu, 9 Nov 95 14:10:43 EST Received: (from joemcg@localhost) by B17.Catch22.COM (8.6.9/8.6.12) id KAA06094; Thu, 9 Nov 1995 10:29:44 -0800 X-Url: http://www.Catch22.COM/ Date: Thu, 9 Nov 1995 10:29:43 -0800 (PST) From: Joe McGarity To: "Rubik's Cube Mailing List" Subject: Flowers in you hair Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII From mark.longridge@canrem.com Sun Nov 12 01:51:15 1995 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26404; Sun, 12 Nov 95 01:51:15 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1FDFC0; Sun, 12 Nov 95 01:35:04 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Magic Platonic Solids From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1260.5834.0C1FDFC0@canrem.com> Date: Sun, 12 Nov 95 01:33:00 -0500 Organization: CRS Online (Toronto, Ontario) First a correction (sorry Dave!) > # Perhaps David Badley could confirm the following orders: The above should be "David Bagley". I have some further comments on the "Magic Platonic Solids". One can stretch (abuse?) the concept of the slice and anti-slice groups of the cube to include the Megaminx (Magic Dodecahedron). In the case of the Megaminx we can consider one-fifth turns of opposite faces. Unfortunately my experiments with "slice" turns on the Megaminx has not generated any spot patterns as yet. Ben Halpern was not the only one to make a prototype of a tetrahedron with rotating faces, as Kersten Meier made one as well. Only 3 of the 4 generators of the Halpern-Meier Tetrahedron are necessary to generate the 3,732,480 possible states. If we use only 2 generators we only get 19,440 possible states. It is not possible to swap just 1 pair of corners and 1 pair of edges, as is possible with the standard Rubik's cube. The number of possible states of the Halpern-Meier Tetrahedron break down like this: 6! /2 * 2^5 * 4!/2 * 3^3 = 3,732,480 The number of pairs of exchanges of the 6 edges must be even. The number of pairs of exchanges of the 4 corners must be even. 5 of the 6 edges may have any flip, the last edge is forced. 3 of the 4 corners may have any twist, the last corner is forced. The H-M Tetrahedron is roughly comparable to the 2x2x2 cube and the standard Skewb in terms of the number of combinations. Halpern's Tetrahedron 3.7*10^6 Ben Halpern, Kersten Meier Pocket Cube (2x2x2) 3.6*10^6 Erno Rubik Skewb 3.1*10^6 Tony Durham -> Mark <- From coumes@issy.cnet.fr Mon Nov 13 04:23:07 1995 Return-Path: Received: from xr3.atlas.fr by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17534; Mon, 13 Nov 95 04:23:07 EST X400-Received: by /PRMD=INTERNET/ADMD=ATLAS/C=FR/; Relayed; Mon, 13 Nov 1995 09:13:08 +0100 X400-Received: by mta xr3.atlas.fr in /PRMD=INTERNET/ADMD=ATLAS/C=FR/; Relayed; Mon, 13 Nov 1995 09:13:08 +0100 X400-Received: by /ADMD=ATLAS/C=FR/; Relayed; Mon, 13 Nov 1995 09:13:04 +0100 X400-Received: by /PRMD=cnet/ADMD=atlas/C=FR/; Relayed; Mon, 13 Nov 1995 09:13:10 +0100 Date: Mon, 13 Nov 1995 09:13:10 +0100 X400-Originator: coumes@issy.cnet.fr X400-Recipients: non-disclosure:; X400-Mts-Identifier: [/PRMD=cnet/ADMD=atlas/C=FR/;816250393@x400.issy.cnet.fr] X400-Content-Type: P2-1984 (2) Content-Identifier: unsubscribe Alternate-Recipient: Allowed From: Jean-Philippe COUMES CNET PAA/RGE/TSR Message-Id: <9511130914.AA15168@peyoti.i...> To: cube-lovers@life.ai.mit.edu Subject: unsubscribe X-Mailer: InterCon TCP/Connect II 2.2.1 Mime-Version: 1.0 Content-Type: Text/Plain; charset=US-ASCII Content-Disposition: Inline unsubscribe From JBRYAN@pstcc.cc.tn.us Tue Nov 14 09:12:58 1995 Return-Path: Received: from VAX1.PSTCC.CC.TN.US (PSTCC4.PSTCC.CC.TN.US) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05060; Tue, 14 Nov 95 09:12:58 EST Resent-From: JBRYAN@pstcc.cc.tn.us Resent-Message-Id: <9511141412.AA05060@life.ai.mit.edu> Received: from pstcc.cc.tn.us by pstcc.cc.tn.us (PMDF V5.0-3 #11457) id <01HXMMHE4MCQ8X60LN@pstcc.cc.tn.us> for cube-lovers@ai.mit.edu; Tue, 14 Nov 1995 09:13:44 -0400 (EDT) Resent-Date: Tue, 14 Nov 1995 09:13:43 -0400 (EDT) Date: Tue, 14 Nov 1995 09:13:41 -0400 (EDT) From: Jerry Bryan Subject: God's Algorithm for the 1x1x1 Rubik's Cube Sender: JBRYAN@pstcc.cc.tn.us To: Cube-Lovers Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: 7BIT Solving the 1x1x1 Rubik's cube is probably a bit silly and whimsical, but let's look at it anyway. I was led in this direction by rereading some articles in the archives from Dan Hoey and others concerning NxNxN Rubik's cubes. For example, consider Dan's discussion "Cutism, Slabism, and Eccentric Slabism" from 1 June 83 19:39:00. Sometimes degenerate cases are slightly interesting. I guess the 1x1x1 case is the most degenerate we have, unless you want to consider the 0x0x0. It seems to me that either cutism or slabism, as Dan calls them, reduce to whole cube rotations for the 1x1x1 case. For example, a quarter turn face turn or a quarter turn slice would be interpreted as a whole cube quarter turn for the 1x1x1. Hence, the cube group for the 1x1x1 is simply C, the group of 24 rotations of the cube. By analogy with some of our previous work, I can think of essentially three different ways to model the 1x1x1. 1) With the 2x2x2, we normally wish to consider the puzzle solved if each face is all of one color. That is, whole cube rotations are to be considered equivalent. With the Singmaster fixed face center view of the 3x3x3, the issue of whole cube rotations does not arise. But with the 2x2x2 we would normally consider (for example) RL' equivalent to I. The most common way to accomplish this type of equivalence is to fix one of the corners. If we fix one of the corners of the 1x1x1, then we have a most remarkable puzzle. There is only one state, nothing can ever move, and the puzzle is always solved. 2) A second way to model the 2x2x2 such that whole cube rotations are considered to be equivalent is to consider the set of states to be the set of cosets of C, that is, the set of all xC. If we take this approach with the 1x1x1, then there is only one coset, namely iC (or just C, if you prefer). The cube can rotate, but all 24 states are considered to be equivalent and the puzzle is always solved. 3) Finally, if you model the 2x2x2 in such a way that whole cube rotations are considered to be distinct, then you are really modelling the corners of the 3x3x3. Indeed, a naive program that simply modelled the permutations of the 2x2x2 facelets would in fact unwittingly be modelling the corners of the 3x3x3. If you take the same approach of modelling the permutations of the 1x1x1 facelets, then you in effect are considering whole cube rotations to be distinct. You have a very easy problem, but the problem is not totally trivial as it is with approach #1 or approach #2. The rest of this note will therefore consider the problem of the 1x1x1 cube where whole cube rotations are considered to be distinct. Since we need to deal with whole cube rotations, I will use lower case letters as our standard E-mail simulation of Frey and Singmaster's script notation for whole cube quarter turns -- t for Top, r for Right, etc. We need only three of the six letters because, for example, we have l=r', d=t', b=f', etc. I will use t, r, and f. We know before we start that there are 24 states. We also know before we start that these 24 states form 5 M-conjugacy classes, where M is the set of 48 rotations and reflections of the cube. (There are 10 M-conjugacy classes of M, of which 5 are rotations and 5 are reflections.) Hence, any discussion of God's algorithm will involve 5 conjugacy classes and 24 states. The obvious searches to look at are for qturns only, and for qturns plus hturns. We may generate the qturn case as C=. We may generate the qturn plus hturn case as C=. Qturns Only Distance Conjugacy Positions from Classes Start 0 1 1 {i} 1 1 6 {t,t',r,r',f,f'} 2 2 11 {tt,rr,ff},{tr,tr',tf,tf',t'r,t'r',t'f,t'f'} 3 1 6 {ttf,ttf',ffr,ffr',rrt,rrt'} --- ---- ---- Total 5 24 Qturns Plus Hturns Distance Conjugacy Positions from Classes Start 0 1 1 {i} 1 2 9 {t,t',r,r',f,f'},{t2,r2,f2} 2 2 14 {tr,tr',tf,tf',t'r,t'r',t'f,t'f'}, {t2f,t2f',f2r,f2r',r2t,r2t'} --- ---- ---- Total 5 24 There are some additional problems we can look at. For an example, an interesting problem on the 3x3x3 is variously called the stuck axle problem or the five generator problem. In the case of the 1x1x1, we have the "two generator problem" because we certainly can generate C as C= (Proof: r=tft'). But can we generate C with only one generator? The answer is no. (Proof: Order(i)=1, Order(t)=4, Order(tt)=2, Order(tf)=3, and Order(ttf)=2. All the orders are less than 24. Note that it suffices to calculate the order for one representative of each conjugacy class.) I will leave it as an exercise for the reader to determine the lengths of each of the 24 positions if we generate C as , and to determine the appropriate conjugacy classes to take into account the symmetry of C generated as . By the way, do we know the minimum number of generators required to generate the 3x3x3? Here I do not mean the minimum number of quarter turns. I am asking the question if we are permitted to use as generators any elements of G. Here is one final item about the 1x1x1. We do not know how many subgroups of G there are for the 3x3x3. But we do know how many subgroups of C there are. There has been much discussion of the 98 subgroups of M which can be arranged in 33 conjugacy classes. The subgroups of C are simply those subgroups of M which consist entirely of rotations. There are 30 such subgroups, and they may be arranged in 11 conjugacy classes. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us From boland@sci.kun.nl Wed Nov 15 19:34:28 1995 Return-Path: Received: from wn1.sci.kun.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16127; Wed, 15 Nov 95 19:34:28 EST Received: from canteclaer.sci.kun.nl by wn1.sci.kun.nl via canteclaer.sci.kun.nl [131.174.132.34] with SMTP id BAA05998 (8.6.10/2.14); Thu, 16 Nov 1995 01:33:07 +0100 Message-Id: <199511160033.BAA05998@wn1.sci.kun.nl> To: Jerry Bryan Cc: Cube-Lovers Subject: Re: God's Algorithm for the 1x1x1 Rubik's Cube In-Reply-To: Your message of "Tue, 14 Nov 95 09:13:41 -0400." Date: Thu, 16 Nov 95 01:33:05 +0100 From: Michiel Boland Jerry wrote: >But can we generate C with only one generator? The >answer is no. (Proof: Order(i)=1, Order(t)=4, Order(tt)=2, Order(tf)=3, >and Order(ttf)=2. All the orders are less than 24. Note that it suffices >to calculate the order for one representative of each conjugacy class.) Another way to see that C cannot be generated by one generator is to note that C is not abelian. Singmaster mentions in his Notes that the cube group G itself can also be generated by two elements. -- Michiel Boland University of Nijmegen The Netherlands From hoey@aic.nrl.navy.mil Wed Nov 29 12:18:52 1995 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23715; Wed, 29 Nov 95 12:18:52 EST Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA26702; Wed, 29 Nov 95 12:14:41 EST Return-Path: Received: by sun13.aic.nrl.navy.mil; Wed, 29 Nov 95 12:14:40 EST Date: Wed, 29 Nov 95 12:14:40 EST From: hoey@aic.nrl.navy.mil To: mschoene@math.rwth-aachen.de (Martin Schoenert), frb6006@cs.rit.edu (Frank R Bernhart), Cube-Lovers@life.ai.mit.edu Newsgroups: sci.math Subject: Generating Rubik's Cube Message-Id: <9511291210.Hoey@AIC.NRL.Navy.Mil> References: <1995Nov29.054118.9651@cs.rit.edu> Distribution: About generating the cube's group with arbitrary elements of that group, mschoene@Math.RWTH-Aachen.DE (Martin Schoenert) writes: > ... Rubik's cube can be generated by 2 elements. > Moreover almost any random pair of elements will do the trick.... Actually, I think it's more accurate to say that a random pair of elements has nearly a 75% probability of generating the cube. At least, I'm pretty sure that's an upper bound, and I don't see any reason why it shouldn't be fairly tight. That's for the group where the whole cube's spatial orientation is irrelevant. I think it's more like 56% (9/16) if you also need to generate the 24 possible permutations of face centers. About the minimal presentation of the cube group on the usual generators, frb6006@cs.rit.edu (Frank R Bernhart) writes: > The answers may be in SINGMASTER, et.al. > "Handbook of Cubic Math" or BANDEMEISTER (sp?) "Beyond R. Cube" I recall Singmaster wanted to know if anyone found a reasonably-sized presentation; I don't know if any have been found in the intervening fifteen years. The best I know of is a few thousand relations, some of them several thousand letters long. I've been meaning to try chopping that down a bit. Dan posted and e-mailed Hoey@AIC.NRL.Navy.Mil From mbparker@share.ai.mit.edu Fri Dec 1 13:31:02 1995 Return-Path: Received: from share ([199.171.190.200]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19968; Fri, 1 Dec 95 13:31:02 EST Received: by share (NX5.67e/NX3.0M) id AA00728; Fri, 1 Dec 95 10:24:24 -0800 Date: Fri, 1 Dec 95 10:24:24 -0800 From: Michael B. Parker Message-Id: <9512011824.AA00728@share> To: PuzzleParty@cytex.com, Cube-Lovers@ai.mit.edu, 506maple-residents@cytex.com, www-designers@cytex.com Subject: PUZZLE PARTY 4 -- SATURDAY (Dec. 2), 7pm, Orange! Reply-To: mbparker@cytex.com PUZZLE PARTY IV! The 3 Puzzle Parties this year have been a big success! The last party brought puzzle collectors from as far as Australia, plus the world-famous Jerry Slocum,... and didn't quit 'til 3:30am! So if you missed the one before, you absolutely don't want to miss Puzzle Party 4!... Bring your brain teasers, mechanical puzzles and mental games, and prepare yourself to have an incredibly good time. Join us to dine on a tasty ``puzzle potluck'' along with drinks and leisurely conversation with friends by the fireside. Plenty of snacks & refreshments and good spirits provided, so grab that brain and some puzzles, and see you there! WHEN: Saturday, December 2nd 7:00 PM until... WHERE: Mike's house, 506 N. Maplewood St., Orange, CA From the 5 fwy S, exit 22E, to the end, then 55N, take 2nd exit Chapman West, at 1st light right/north on Tustin, 2nd light left/west onto Walnut, 3rd left is Maplewood. We are the big yellow house on the NW corner of Walnut and Maplewood. COST: $4 MITCSC Members & Guests with puzzles $6 MITCSC Non-Members & Guests with puzzles $8 MITCSC Members & Guests w/o puzzles $10 MITCSC Non-Members & Guests w/o puzzles RSVP: You may pay at the door, but please try to contact me beforehand so I can put you on the list. Please email, fax, or phone the following info: Your NAME, ADDRESS, PHONE, FAX, EMAIL, and what you're bringing: ___ puzzle-bearing members at $ 4 each: $___ ___ puzzle-bearing non-members at $ 6 each: $___ ___ puzzle-less members at $ 8 each: $___ ___ puzzle-less non-members at $10 each: $___ ___ <- total persons total cost -> $___ total number of puzzles being brought ___ SPONSOR: Michael B. Parker, MIT '89 email mbparker@cytex.com, 1-800-MBPARKER xLIVE, xPAGE, xFAXX (This info is online! See http://www.cytex.com/~mitcsc/w96/w96-pzl4.htm) From hoey@aic.nrl.navy.mil Sun Dec 3 14:46:32 1995 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA12457; Sun, 3 Dec 95 14:46:32 EST Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA01617; Sun, 3 Dec 95 14:46:31 EST Return-Path: Received: by sun13.aic.nrl.navy.mil; Sun, 3 Dec 95 14:46:30 EST Date: Sun, 3 Dec 95 14:46:30 EST From: hoey@aic.nrl.navy.mil Message-Id: <9512031946.AA24122@sun13.aic.nrl.navy.mil> To: Cube-Lovers@life.ai.mit.edu Subject: Re: Generating Rubik's Cube On the probability that two random elements will generate the entire cube group, I wrote: > ... a random pair of elements has nearly a 75% probability of > generating the cube. At least, I'm pretty sure that's an upper > bound, and I don't see any reason why it shouldn't be fairly tight. > That's for the group where the whole cube's spatial orientation is > irrelevant. I think it's more like 56% (9/16) if you also need to > generate the 24 possible permutations of face centers. I can now answer the spatial orientation part of the question, and it's much lower. We're talking about C, the 24-element group of proper motions of the whole cube. If we select two elements at random with replacement, the probability is only 3/8 that they will generate the whole group. The kinds of motions that can take part in a generating pair are a 90-degree rotation about an axis, a 120-degree rotation about a major diagonal, and a 180-degree rotation about a minor diagonal. Note that the last kind fixes two major diagonals and an axis. Two motions generate C iff they are (48 ways) a 120 and a 180, unless they fix the same major diagonal, (48 ways) a 180 and a 90, unless they fix the same axis, (24 ways) two 90s at right angles, or (96 ways) a 90 and a 120. The number comes out so even I suspect there's something deeper going on than the exhaustive analysis I used. As for generating the (fixed-face) Rubik's group, I still suspect that two elements almost always generate the entire group unless they are both even. Anyone who has a Sims's-algorithm implementation handy could help out with a Monte-carlo approximation to see if this is approximately right. Or, I wonder, is there a way of getting an exact number, perhaps with the help of GAP? Dan posted and e-mailed Hoey@AIC.NRL.Navy.Mil From mark.longridge@canrem.com Sun Dec 3 20:32:36 1995 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27214; Sun, 3 Dec 95 20:32:36 EST Received: by canrem.com (PCB-UUCP 1.1f) id 201705; Sun, 3 Dec 95 20:17:41 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: & G From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1261.5834.0C201705@canrem.com> Date: Sun, 3 Dec 95 20:09:00 -0500 Organization: CRS Online (Toronto, Ontario) A while back Jerry asked.... > Finally, pick any cube X in . We know > |X| in G <= |X| in . Can anybody find a cube X such that > |X| in G < |X| in ? Well, we basically know the answer is yes. There are elements in which require less moves if we use all the generators of G. To be more specific, look the 6 twist pattern in which requires 22 q turns: ^^^^^^^^^^ >> Equivalent to (U1 R1)^35= (R1 U1)^35 & Shift Invariant >> UR11 = U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 After a bit of computer cubing I found: p183 6 Twist R1 U3 R2 U3 R1 D3 U3 R1 U3 R3 D2 R3 U3 R1 D3 U3 (18 q, 16 q+h moves) ^^^^^ I'll spare everyone all the gory details. I'm certain there are all sorts of other examples, but here is one case where we can save 4 q turns. It may be of some small interest to see which of the two processes can be executed more rapidly by the human hand. -> Mark <- From mschoene@math.rwth-aachen.de Mon Dec 4 09:10:17 1995 Return-Path: Received: from hurin.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23100; Mon, 4 Dec 95 09:10:17 EST Received: from samson.math.rwth-aachen.de by hurin.math.rwth-aachen.de with smtp (Smail3.1.28.1 #30) id m0tMZOr-0009KuC.951204.150809; Mon, 4 Dec 95 12:49 MET Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0tMZOr-000I7wC; Mon, 4 Dec 95 12:49 MET Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #26) id m0tMZOq-0009ejC.951204.124936; Mon, 4 Dec 95 12:49 MET Message-Id: Date: Mon, 4 Dec 95 12:49 MET From: Martin Schoenert To: hoey@aic.nrl.navy.mil Cc: Cube-Lovers@life.ai.mit.edu In-Reply-To: hoey@aic.nrl.navy.mil's message of Sun, 3 Dec 95 14:46:30 EST <9512031946.AA24122@sun13.aic.nrl.navy.mil> Subject: Re: Re: Generating Rubik's Cube I have used GAP to compute the subgroup generated by 300 random pairs of elements of G. 151 of those pairs generated the entire group, so the probability is about 50%. I don't think we can figure out the exact number, since we don't know the maximal subgroups of G. One maximal subgroup we know is the derived subgroup (on which the upper bound of 75% is based). Then there are the 8 stabilizers of the corners (of index 8) and the 12 stabilizers of the edges (of index 12). Using those it should be possible to push the upper bound down to something about 60%. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From csstto@alpcom.it Wed Dec 6 06:16:17 1995 Return-Path: Received: from nic.alpcom.it by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14301; Wed, 6 Dec 95 06:16:17 EST Received: from monviso.alpcom.it by ALPcom.it (PMDF V4.3-10 #4712) id <01HYHH3ABBHS0017KF@ALPcom.it>; Wed, 06 Dec 1995 11:14:08 +0000 (GMT) Received: by monviso.alpcom.it (950911.SGI.8.6.12.PATCH825/940406.SGI) id LAA09416; Wed, 6 Dec 1995 11:14:06 GMT Date: Wed, 06 Dec 1995 11:14:06 +0001 (GMT) From: "C.S.S.T. Torino" Subject: Information request To: cube-lovers@life.ai.mit.edu Message-Id: X-Envelope-To: cube-lovers@life.ai.mit.edu Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: 7BIT How it is possible to access to "Cube lovers at MIT" ? Do we need a password ? Thank You and best regards Domenico Inaudi From alan@curry.epilogue.com Thu Dec 7 02:44:55 1995 Return-Path: Received: from curry.epilogue.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23735; Thu, 7 Dec 95 02:44:55 EST Received: (from alan@localhost) by curry.epilogue.com (8.6.12/8.6.12) id CAA01816; Thu, 7 Dec 1995 02:44:49 -0500 Date: Thu, 7 Dec 1995 02:44:49 -0500 Message-Id: <7Dec1995.013844.Alan@LCS.MIT.EDU> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu Subject: Magazine Spam To: Cube-Lovers@ai.mit.edu So I thought it was time to send you all an update on unwanted magazine advertisements that have been broadcast over Cube-Lovers about once a week for the last few months. Here's the final story. First off, there is absolutely -nothing- that I can do IN THE SHORT TERM to stop these advertisements. Internet electronic mail was not designed to prevent unwanted advertising. As things are set up now, Cube-Lovers is a simple mailing list, so anybody, anywhere, can send mail to Cube-Lovers and you all get it. It turns out that the source of the advertising we've been getting is a fellow named Kevin Jay Lipsitz . I've written directly to Mr. Lipsitz politely asking him to remove Cube-Lovers from his list of advertising targets (it was hard to be polite, but I was) -- but Mr. Lipsitz apparently doesn't answer his electronic mail. Actually, I doubt he even -reads- his electronic mail, because he is a well-known Spammer, and probably gets hundreds of complaints a day delivered to his address. (For those of you new to the Internet, "Spamming" is the technical term for the kind of advertising Mr. Lipsitz engages in.) I really doubt that Mr. Lipsitz's technique has sold any magazines to any of -you-, but I suppose he gets enough suckers to make it pay, and he's got no motivation to bother removing Cube-Lovers, since MIT is paying for the resources that he's using to reach you all. So we're stuck with him. At least we're stuck with him until I can get the filtering technology in place to cut him off. Which I wanted to avoid, because I have better things to do with my time, but now I have no choice. So relief from Mr. Lipsitz's magazines is on its way eventually, but probably not until you've seen several more copies of his advertisement -- sorry. By the way, here's more information on Mr. Lipsitz. You'll notice that he has his own domain name: KJL.COM. They don't give you a domain name unless you provide a mailing address and a phone number, so the following information is publicly available from the NIC: Kevin Jay Lipsitz (KJL-DOM) PO Box 120990 Staten Island NY 10312-0990 Domain Name: KJL.COM Administrative Contact, Technical Contact, Zone Contact: Lipsitz, Kevin Jay (KJL2) krazykev@KJL.COM 718-967-1234 Record last updated on 25-Aug-95. Record created on 20-Apr-95. Domain servers in listed order: NS1.ABS.NET 206.42.80.130 NS2.ABS.NET 206.42.80.131 NS1.NET99.NET 204.157.3.2 If you want proof that this is the guy, you need only note that the address given here is the -same- as the address for ordering magazines given in all those advertisements. The phone number is in the same area code and exchange as the Fax number he sometimes gives. (Although the fact that the phone number ends in "1234" makes me suspect it is bogus -- I don't think think the NIC tries to -verify- any of this information.) Notice that ABS.NET provides the domain service for KJL.COM. You will find that ABS.NET is no more interested in answering your mail than Mr. Lipsitz is. Finally, I urge you all -not- to respond to this message in public. If you have further thoughts on Internet advertising, electronic mailing list administration, or clever acts of revenge, you can send them to -me-, but don't CC your message to Cube-Lovers as a whole. The whole point here is to keep Cube-Lovers relatively free of off-topic mail. As the list administrator I get to send out occasional administrivia such as this message because I do actual -work- to keep the list running. - Alan (Cube-Lovers-Request@AI.MIT.EDU) From walts@federal.unisys.com Thu Dec 7 09:04:57 1995 Return-Path: Received: from www.han.federal.unisys.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04683; Thu, 7 Dec 95 09:04:57 EST Received: from homer.MCLN.Federal.Unisys.COM by www.han.federal.unisys.com (8.6.12/mls/8.0) id JAA00555; Thu, 7 Dec 1995 09:04:55 -0500 Received: from h3-91.MCLN.Federal.Unisys.COM by homer.MCLN.Federal.Unisys.COM (8.6.12/mls/4.1) id JAA05663; Thu, 7 Dec 1995 09:07:10 -0500 Message-Id: <199512071407.JAA05663@homer.MCLN.Federal.Unisys.COM> Date: Thu, 07 Dec 95 09:06:45 -0800 From: "Walter P. Smith" Organization: Installation Services X-Mailer: Mozilla 1.22 (Windows; I; 16bit) Mime-Version: 1.0 To: Cube-Lovers@ai.mit.edu Subject: Mini Cube & Revenge Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=us-ascii My first message. Oh boy! It's back. A new version of the 2x2x2 cube previously called the Pocket Cube is in production. The new version is called the Mini Cube. They are available from GameKeepers I got mine in Tysons Corner Shopping Mall Virginia, but GameKeepers is a chain and should be in most large cities. If readers can't locate one, let me know and I'll get a list of locations. They also carry a wide line of puzzles including Triamid, Snake, regular Rubik's cubes, Master Balls etc. The stickers are glossy paper. I don't think they will be very durable. Also the red and orange sides are very hard to tell apart. How could they be so stupid? The mechanism seems to work better than the old ones. I can't tell if the inner workings are the same. The original Pocket Cube had a ball in the center with six cap like pieces screwed to it (with springs under the screw head) to form a series of tracks. Each piece had a shaft that extended down into the grove with a triangular foot on it. This design requires a lot of pieces and drives the price up. I always thought they could be made by making the ball, three of the caps and one corner piece, all into one piece. The Mini Cube uses cubies that are solid on all sides. This may account for the smoother action. They include a complete solution sheet. One comes with the Master Ball also. I personally think manufacturers should't do this. Many people will turn to the solution sheet before giving it a good effort and will miss the pleasure of solving it for themselves. My Mini Cube cost over six dollars. A little pricey but a collector should never pass up an opportunity. Walt "The Puzzler" From serge@nexen.com Tue Dec 12 17:07:38 1995 Return-Path: Received: from guelah.nexen.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03846; Tue, 12 Dec 95 17:07:38 EST Received: from maelstrom.nexen.com (maelstrom.nexen.com [204.249.98.5]) by guelah.nexen.com (8.6.12/8.6.12) with ESMTP id QAA09465 for ; Tue, 12 Dec 1995 16:51:39 -0500 Received: from spank.nexen.com (spank.nexen.com [204.249.98.79]) by maelstrom.nexen.com (8.6.12/8.6.12) with ESMTP id RAA01960 for ; Tue, 12 Dec 1995 17:07:46 -0500 Received: (from serge@localhost) by spank.nexen.com (8.6.12/8.6.12) id RAA26101; Tue, 12 Dec 1995 17:07:19 -0500 Date: Tue, 12 Dec 1995 17:07:19 -0500 From: Serge Kornfeld Message-Id: <199512122207.RAA26101@spank.nexen.com> To: cube-lovers@ai.mit.edu Subject: subscribe Please subbscribe Serge serge@nexen.com From nichael@sover.net Wed Dec 13 21:09:22 1995 Return-Path: Received: from maple.sover.net by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17249; Wed, 13 Dec 95 21:09:22 EST Received: from [204.71.18.82] (st32.bratt.sover.net [204.71.18.82]) by maple.sover.net (8.6.12/8.6.12) with SMTP id VAA08585 for ; Wed, 13 Dec 1995 21:09:18 -0500 Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 13 Dec 1995 21:16:38 -0400 To: Cube-Lovers@ai.mit.edu From: nichael@sover.net (Nichael Lynn Cramer) Subject: Pocket Stuff [Possibly minimal relevance; take it in the sense of cool stocking-stuffer hacks.] Someone asked a couple of weeks back for pocket/key-ring cubes. Can't help with that, but this afternoon in Sandy's and Son's (Inman Square (Cambridge (Ma))) beside the cash register they had a basket of EtchASketch keyrings. Seemed pretty solidly built for the the $3.50. N From walts@federal.unisys.com Thu Dec 14 10:23:10 1995 Return-Path: Received: from www.han.federal.unisys.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA15946; Thu, 14 Dec 95 10:23:10 EST Received: from homer.MCLN.Federal.Unisys.COM by www.han.federal.unisys.com (8.6.12/mls/8.0) id KAA05464; Thu, 14 Dec 1995 10:23:08 -0500 Received: from h3-91.MCLN.Federal.Unisys.COM by homer.MCLN.Federal.Unisys.COM (8.6.12/mls/4.1) id KAA05516; Thu, 14 Dec 1995 10:25:26 -0500 Message-Id: <199512141525.KAA05516@homer.MCLN.Federal.Unisys.COM> Date: Thu, 14 Dec 95 10:25:01 -0800 From: "Walter P. Smith" Organization: Installation Services X-Mailer: Mozilla 1.22 (Windows; I; 16bit) Mime-Version: 1.0 To: Cube-Lovers@ai.mit.edu Subject: Twist Torus Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset=iso-8859-1 In Sept. 1992 Mark Longridge described an idea he had for a puzzle. He called it a Twist Torus. Well I bought a puzzle that fits his description very closely. I would have sent this in sooner except that I am new to Cube Lovers. I bought mine several years ago. Did he get his design into production or was it independently invented or did someone implement Mark's idea? Will we ever know? I bought mine in a department store (cant remember which) toy department. It was not in any packaging and cost less than $2 US.. They only had one. It was quite by chance that I determined that it is a puzzle. It will test my ability to describe it in words but here goes. It is torus shaped (dough nut shaped). At first glance it looks like a bracelet. It has one slice made the same way a bagel is sliced. The puzzle can turn along this cut. There are eight differently colored sections separated by fixed black sections around the circumference of the torus. Each colored section is subdivided into 4 sub-segments that can turn at right angles to the main circumference. As a segment is turned, different parts of the segment are brought to the other side of the main cut. It operates smoothly and is brightly colored. It is fairly easy to solve but the geometry is novel and interesting. Does anyone else have one of these? Does anyone know who manufactured this? Does anyone know what it is called? Walt "The Puzzler" Smith From devo@vnet.ibm.com Thu Dec 14 12:30:51 1995 Return-Path: Received: from VNET.IBM.COM by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA24743; Thu, 14 Dec 95 12:30:51 EST Message-Id: <9512141730.AA24743@life.ai.mit.edu> Received: from GDLVM7 by VNET.IBM.COM (IBM VM SMTP V2R3) with BSMTP id 3486; Thu, 14 Dec 95 12:07:06 EST Date: Thu, 14 Dec 95 12:03:41 EST From: "Dave Eaton" To: Cube-Lovers@ai.mit.edu Subject: Lumination Has anyone seen a puzzle called Lumination by Parker Bros. A guy at work says he got one from his wife about five years ago. It is a tetrahedron (like the Pyraminx) except that instead of having any moving parts, it has lights in the four points that change color when you rotate the whole puzzle in space. It sounds really neat. Does anyone know where I can get one? Dave Eaton From serge@nexen.com Thu Dec 14 13:17:04 1995 Return-Path: Received: from guelah.nexen.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27230; Thu, 14 Dec 95 13:17:04 EST Received: from maelstrom.nexen.com (maelstrom.nexen.com [204.249.98.5]) by guelah.nexen.com (8.6.12/8.6.12) with ESMTP id NAA20551 for ; Thu, 14 Dec 1995 13:00:45 -0500 Received: from spank.nexen.com (spank.nexen.com [204.249.98.79]) by maelstrom.nexen.com (8.6.12/8.6.12) with ESMTP id NAA02709 for ; Thu, 14 Dec 1995 13:17:05 -0500 Received: (from serge@localhost) by spank.nexen.com (8.6.12/8.6.12) id NAA28270; Thu, 14 Dec 1995 13:16:04 -0500 Date: Thu, 14 Dec 1995 13:16:04 -0500 From: Serge Kornfeld Message-Id: <199512141816.NAA28270@spank.nexen.com> To: Cube-Lovers@ai.mit.edu Subject: Re Twist Torus > It will test my ability to describe it in words but here goes. > It is torus shaped (dough nut shaped). At first glance it looks > like a bracelet. It has one slice made the same way a bagel is > sliced. The puzzle can turn along this cut. There are eight > differently colored sections separated by fixed black sections > around the circumference of the torus. Each colored section is > subdivided into 4 sub-segments that can turn at right angles to > the main circumference. As a segment is turned, different parts > of the segment are brought to the other side of the main cut. I came to US 4 years ago from Russia. Living in Russia I use to collect mechanical puzzles. I remember the article in magazine and a picture of the puzzle you described. I think it was 1987 ????. Article was saying that there are some problems to actually make this type of puzzle and ..... I cant remember the end of the article and I never saw this toy in real. Serge serge@nexen.com From SCHMIDTG@beast.cle.ab.com Thu Dec 14 21:36:02 1995 Return-Path: Received: from beast.cle.ab.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27654; Thu, 14 Dec 95 21:36:02 EST Date: Thu, 14 Dec 1995 21:35:58 -0500 (EST) From: SCHMIDTG@beast.cle.ab.com To: Cube-Lovers@ai.mit.edu Message-Id: <951214213558.20212e52@iccgcc.cle.ab.com> Subject: Re: Twist Torus Walter P. Smith wrote, >[stuff about a "Twist Torus" puzzle deleted...] > >Does anyone else have one of these? Does anyone know who >manufactured this? Does anyone know what it is called? I purchased one of these back in 1992 at a toystore in a Florida mall. I paid $4.99 for mine. The tags says: WrisTwist (tm) Puzzle & Bracelet WACO Riverdale, NJ 07457 Made in Indonesia Item #20003 The puzzle is still in its pristine state and the color progression is: Red-Blue-Green-Yellow-OrangeRed-Violet-Orange-LightGreen -- Greg From rh@thi.informatik.uni-frankfurt.de Fri Dec 15 03:39:59 1995 Return-Path: Received: from riese.informatik.uni-frankfurt.de (riese.thi.informatik.uni-frankfurt.de) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09295; Fri, 15 Dec 95 03:39:59 EST Received: from kassandra.thi.informatik.uni-frankfurt.de by riese.informatik.uni-frankfurt.de (4.1/THI-PeLeuck2.2a) id AA07726; Fri, 15 Dec 95 09:40:50 +0100 Date: Fri, 15 Dec 95 09:40:50 +0100 From: rh@thi.informatik.uni-frankfurt.de (Roger Haschke) Mime-Version: 1.0 Content-Transfer-Encoding: binary Content-Type: text/plain; charset=ISO-8859-1 Message-Id: <9512150840.AA07726@riese.informatik.uni-frankfurt.de> To: cube-lovers@ai.mit.edu Subject: unsubscribing ... can anybody please tell me the correct email-adresse for sending an unsubscribe-command? thanks - Roger From alan@curry.epilogue.com Fri Dec 15 04:07:12 1995 Return-Path: Received: from curry.epilogue.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09752; Fri, 15 Dec 95 04:07:12 EST Received: (from alan@localhost) by curry.epilogue.com (8.6.12/8.6.12) id EAA03212; Fri, 15 Dec 1995 04:06:23 -0500 Date: Fri, 15 Dec 1995 04:06:23 -0500 Message-Id: <15Dec1995.035416.Alan@LCS.MIT.EDU> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu To: rh@thi.informatik.uni-frankfurt.de Cc: cube-lovers@ai.mit.edu In-Reply-To: Roger Haschke's message of Fri, 15 Dec 95 09:40:50 +0100 <9512150840.AA07726@riese.informatik.uni-frankfurt.de> Subject: unsubscribing ... Date: Fri, 15 Dec 95 09:40:50 +0100 From: Roger Haschke can anybody please tell me the correct email-adresse for sending an unsubscribe-command? thanks - Roger As I'm certain you've been told -multiple- times, the address is: CUBE-LOVERS-REQUEST@AI.MIT.EDU For crying out loud, why can't people remember that? Let me give everybody a little bit of advice. For every mailing list you subscribe to, keep a file that contains the information you will need in order to cancel or update your subscription. This isn't hard to do. I do it myself. It's a great way to avoid looking foolish in front of hundreds of people. If you don't have such a file for Cube-Lovers already, start one RIGHT NOW and put this message in it. -- Alan Bawden From walts@federal.unisys.com Fri Dec 15 08:37:52 1995 Return-Path: Received: from www.han.federal.unisys.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16543; Fri, 15 Dec 95 08:37:52 EST Received: from homer.MCLN.Federal.Unisys.COM by www.han.federal.unisys.com (8.6.12/mls/8.0) id IAA12694; Fri, 15 Dec 1995 08:37:49 -0500 Received: from h3-91.MCLN.Federal.Unisys.COM by homer.MCLN.Federal.Unisys.COM (8.6.12/mls/4.1) id IAA12161; Fri, 15 Dec 1995 08:40:10 -0500 Message-Id: <199512151340.IAA12161@homer.MCLN.Federal.Unisys.COM> Date: Fri, 15 Dec 95 08:39:46 -0800 From: "Walter P. Smith" Organization: Installation Services X-Mailer: Mozilla 1.22 (Windows; I; 16bit) Mime-Version: 1.0 To: Cube-Lovers@ai.mit.edu Subject: WrisTwist Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=us-ascii I received the following mail from KINSMAN. I did not reply to him or make note of his address. I am retyping his note to get comment from others. He is refering to the WrisTwist puzzle. I have just such a puzzle too. It came from my local toy store. I also have a digital camera sitting next to me. Should I bring mine in and post a low resolution copy in GIF format to the group? -AAK Does anyone want him to do this? Walter P. Smith walts@federal.unisys.com From kinsman@ycc.kodak.com Fri Dec 15 08:51:42 1995 Return-Path: Received: from doolittle.ycc.Kodak.COM ([129.126.74.2]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17526; Fri, 15 Dec 95 08:51:42 EST Received: from crestone.ycc.Kodak.COM by doolittle.ycc.Kodak.COM with SMTP id AA12918 (5.67b/IDA-1.5 for cube-lovers@ai.mit.edu); Fri, 15 Dec 1995 08:50:55 -0500 Received: from newt.PCD1 (newt.ycc.Kodak.COM) by crestone.ycc.kodak.com with SMTP id AA13182 (5.65c/IDA-1.5 for ); Fri, 15 Dec 1995 08:50:52 -0500 Received: by newt.PCD1 (5.0/SMI-SVR4) id AA20112; Fri, 15 Dec 1995 08:50:51 +0500 Date: Fri, 15 Dec 1995 08:50:51 +0500 From: kinsman@ycc.kodak.com (Andy Kinsman 66672) Message-Id: <9512151350.AA20112@newt.PCD1> To: cube-lovers@ai.mit.edu Subject: Re: Twist Torus [small picture] Content-Length: 61490 Since I have both the puzzle in hand and a camera attached to my computer... here is a picture of the torus puzzle, slightly missaligned for effect. to decode save this note into a file, possibly trim stuff before begin and after end line. type 'uudecode the-file' find ringpuz.tif in your directory after this operation. view it with your favorite tif viewer. 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