From mschoene@math.rwth-aachen.de Tue Jan 31 08:58:26 1995 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08559; Tue, 31 Jan 95 08:58:26 EST Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0rZG32-000MP6C; Tue, 31 Jan 95 11:43 MET Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0rZG32-00025cC; Tue, 31 Jan 95 11:43 WET Message-Id: Date: Tue, 31 Jan 95 11:43 WET From: "Martin Schoenert" To: cube-lovers@life.ai.mit.edu In-Reply-To: Mark Longridge's message of Sun, 29 Jan 1995 23:40:00 -0500 <60.1021.5834.0C1CC7AE@canrem.com> Subject: Re: Skewb thoughts Mark Longridge wrote in his e-mail message of 1995/01/29 Extract from Martin's very detailed skewb analysis: >Then the group CG = < C, G > is the set of all positions a puzzler >could observe. There are 24 solved position in CG (solved up to >rotations). >The group CG has size 2 * 6!/2 * ((3^4*4!/2) * (3^4*4!/2) / 3^2) > |CG| = 75,582,720 Note that: |CG| /24 = 3,149,280 >The group G has size 6!/2 * ((3^4*4!/2) * (3^4*4!/2) / 3^2) > |G| = 37,791,360 Note that: |G| /12 = 3,149,280 The number of positions both David Singmaster and Tony Durham (the inventor) find for the skewb is 3,149,280. Right. The SKEWB has 75582720 basic states. Just as with the cube, we consider two basic states to be essential equal if the differ only by a rotation of the rigid cube. Since there are 24 rotations of the rigid cube, the SKEWB has 3149280 = 75582720/24 essential states. Mark continued If we use only one tetrad of the skewb then GAP also finds this number: ## corners centers ## (each turn permutes 4) (each turn permutes 3) skewb := Group( ( 1,11,17) ( 2,12,20)( 4,10,18)(22, 6,14) (25,27,29), ( 2,10,22) ( 1, 9,23)( 3,11,21)(17, 5,15) (25,27,30), ( 4,14,20) ( 1,15,19)( 3,13,17)( 7,11,23) (25,28,29), ( 6,12,18) ( 5,11,19)( 7, 9,17)(21, 1,13) (26,27,29) );; Size (skewb); > 3149280 In my message on the SKEWB I used the subgroup H generated by LUB, LUF, RUB, and RUF. As I noted, this subgroup has a nontrivial intersection with the subgroup C of rotations of the rigid cube. Thus it is *not* a model for the essential SKEWB. The subgroup that Mark uses, which is generated by RUF, RUB, LUF, and RDF is much better. It has trivial intersection with C and is a model for the essential SKEWB. Note however, that the corners corresponding to the four generators for this subgroups do *not* form a tetrad. They are the corner RUF and the three adjacent corners. In particular, those four generators do not fix the positions of the four corners; the generator RUF permutes the three corner cubies at RUB, LUF, and RDF. This subgroup has 7 other conjugated subgroups, corresponding to the 7 other possible choices of the first generator (the one that is adjacent to the other 3 generators). So allow me to use the subgroup H generated by RUF, LUB, RDB, and LDF. The corresponding four corners do form a tetrad. This H also has trivial intersection with C and also has size 3149280. Thus it also is a model for the essential SKEWB. Note that those four generators never change the positions of the four corner cubies. This subgroup is ``almost normal''; it has only 1 other conjugated subgroup, corresponding to the stabilizer of the other tetrad. Mark continued Mr. Singmaster had indicated in his last Cubic Circular that we may determine the skewb's orientation if only one of the tetrads are moved. I am not certain that I understand what this means. One possible interpretation is, that for each state g of the SKEWB we can easily find the rotation x of the rigid cube, such that (g x) is in the subgroup H. That means that for each state g we can easily find how to rotate the rigid cube, so that the rotated state can be solved using only the four generators above. But this is obvious. Since the four generators do not change the positions of the four corner cubies of the tetrad, the rotation x must be the one that puts those four corner cubies to their home positions. Mark continued By moving first one tetrad and then the other we can easily change the skewb's orientation in space. This comment I don't understand at all. Could you clarify it a bit? Mark continued Martin finds that the diameter of the skewb is 11 moves, with perhaps 90 antipodes. The idea that the skewb has 2 positions at 0 moves is rather odd, but I think if we divide Martin's table by 2 we should get the answer for visually distinguishable states for a skewb fixed in orientation. Right. The diameter of the SKEWB is 11 moves and there are 90 essential different antipodes. The essential SKEWB does *not* have 2 states at 0 moves, only the subgroup H which I used has 2 essentially solved states. This is not odder than the notion that the basic SKEWB has 24 essentially solved states. And yes, if you divide the numbers in my table by 2, you get the table for the essential SKEWB. I rerun the computation using the new subgroup H as a model for the essential SKEWB. Here is the output. 0 1 0 0 0 0 0 0 0 0 1 1 8 0 0 0 0 0 0 8 0 0 2 48 0 0 0 0 0 0 48 0 0 3 288 0 0 0 0 0 0 288 0 0 4 1728 0 0 0 0 0 120 1608 0 0 5 10248 0 0 0 36 377 1322 8513 0 0 6 59304 12 87 662 2217 7561 15698 33067 0 0 7 315198 4331 16897 37723 61161 76931 66997 51158 0 0 8 1225483 515249 311594 186221 115830 65096 25012 6481 0 0 9 1455856 1384909 61839 8280 708 95 25 0 0 0 10 81028 80938 90 0 0 0 0 0 0 0 11 90 90 0 0 0 0 0 0 0 0 Since the group is smaller it run faster and also used less memory. Using some additional tricks, I could cut down the time to 40 seconds and the memory needed to 2.5 megabytes. As you can see, the numbers in the first column are exactely half of the corresponding numbers in my previous message (as was expected). The numbers in the other columns are close to half of the corresponding numbers in my previous message but not exactely identical. I have to rethink what those numbers mean and how they relate to the corresponding numbers for the basic SKEWB. Have a nice day. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany