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Discussions on the mathematics of the cube
Irreducible LoopsSubmitted by Peter on Wed, 04/18/2007 - 09:20.Some ideas for how to possibly proove that the diameter of the Cube-Group would be X.
Hi, I am new to the Cube problem, so probably the ideas are not new, or too naive, but I could not find them anywhere. This is possibly due to my lack of knowledge of terminology. (My background is theoretical solid states physics.) Thus I would like to share these ideas with you, which you hopefully find useful, or can tell me that these ideas are not new or possibly that they are useless. I kindly ask you to comment. I just go ahead... How could one calculate the diameter of the Group? Let A be a random permutation. I start by choosing a (not optimal) path from id to A, say in quarter turn metric, with A=prod_i(ai) (i=1,...,N) where ai in {U,U',D,...}. » 11 comments | read more
Welcome to my BlogSubmitted by Peter on Wed, 04/18/2007 - 07:21.Hello, I am Peter Jung, a physicist from Cologne University.
Only a couple of days ago I got very much interested in the Cube. At wikipedia I found a note that the diameter of the Cube group is not yet known, and a link to this site. Great work! Sniffing into the problem, it seems to be quite complex. But some ideas that came to me these days I could not find. That's the purpose of this visit: To ask whether attempts have been made along these lines of thought, and if so, what is the outcome. And if not, I would like to contribute some analysis. Antisymmetry and the 2x2x2 CubeSubmitted by Bruce Norskog on Sat, 03/17/2007 - 22:48.Someone on the Yahoo forum asked about how to do a 2x2x2 God's algorithm calculation and mentioned the "1152-fold" symmetry for the 2x2x2. I got to looking at some of the messages in the Cube-Lovers archives that Jerry Bryan had made about B-conjugation and the 1152-fold symmetry of the 2x2x2. He found that there were 77802 equivalence classes for the 2x2x2. I have used antisymmetry to further reduce the number of equivalence classes for the 2x2x2 to 40296. The following table shows the class sizes of these equivalence classes. class size class size/24 count ---------- ------------- ----- 24 1 1 48 2 1 72 3 3 96 4 1 144 6 14 192 8 11 288 12 49 384 16 22 576 24 337 768 32 6 1152 48 3353 2304 96 36498 ----- total 40296 I then performed God's algorithm calculations (HTM and QTM) to find the number of equivalence classes at each distance from the solved 2x2x2 cube. The results are given below. Because the 2x2x2 has no centers that provide a reference for the positions of the other cubies, the number of positions of the corners for the 2x2x2 (the only cubies it has) can be considered to be 1/24 the number of positions of the corners of the 3x3x3 (3674160 instead of 88179840). So in the tables below, I use the factor-of-24 reduced numbers for simplicity. The tables further break down the positions with respect to different class sizes. » 15 comments | read more
Odd Permutations of the Cube Shape of Square-1Submitted by Mike G on Fri, 03/02/2007 - 11:41.Results are presented from an exhaustive search on the odd
permutations of Square-1 in its solved shape. A maximum of 31
turns (U, D and /) are needed to solve these positions, which may be a new lower bound on
the length of God's Algorithm for Square-1, in this metric.
The method I used was suggested by Tom and Silviu's coset searches for the Rubik's Cube: Starting from a cube-shaped, odd-parity position of Square-1, an iterated depth-first search was made for all even-parity cube-shaped positions, with the search being pruned on [shape]x[parity]. » 3 comments | read more
God's Algorithm to 13q, Summarized by Symmetry ClassSubmitted by Jerry Bryan on Mon, 02/26/2007 - 12:25.God's Algorithm to 13q has been posted before, but it has not been summarized by symmetry class. The symmetry classes in the table below follow Dan Hoey's taxonomy.
|x| Symmetry Size Patterns Positions Class of xM 0 M 1 1 1 Total 1 1 1 CR 12 1 12 Total 1 12 2 I 48 2 96 Q 6 1 6 » 3 comments | read more
God's Algorithm to 11f, Summarized by Symmetry ClassSubmitted by Jerry Bryan on Mon, 02/26/2007 - 12:21.God's Algorithm to 11f has been posted before, but it has not been summarized by symmetry class. The symmetry classes in the table below follow Dan Hoey's taxonomy.
|x| Symmetry Size Patterns Positions Class of xM 0 M 1 1 1 Total 1 1 1 CR 12 1 12 Q 6 1 6 Total 2 18 2 I 48 4 192 » 5 comments | read more
Dan Hoey's Taxonomy of Symmetry Groups and Shoenflies SymbolsSubmitted by Jerry Bryan on Thu, 02/22/2007 - 23:40.I have received permission to post Dan Hoey's taxonomy of symmetry groups of Rubik's Cube. Also, I will relate Dan's taxonomy to Shoenflies symbols as implemented in Herbert Kociemba's Cube Explorer. (Go to http://kociemba.org/cube.htm and then click on Symmetric Patterns.) To that end, some preparatory comments are in order. In order to define any terminology for the symmetry groups of Rubik's Cube, it's necessary first to define some terminology for the symmetries of the cube. To the best of my knowledge, no standard terminology has been adopted by the Rubik's cube community for the symmetries of the cube. The terminology I'm going to use is very similar to some terminology I have seen before, but I can't remember the reference. It may have been Christoph Bandelow's book, Inside Rubik's Cube and Beyond. In any case, if I can find the reference I want to give proper credit. » 5 comments | read more
Solving the 4x4x4 in 68 turnsSubmitted by Bruce Norskog on Wed, 02/14/2007 - 01:03.I have completed a five-stage analysis of the 4x4x4 cube showing that
it can always be solved using at most 68 turns.
The analysis used the same five stages that were used in my prior posts where I claimed the 4x4x4 cube
can be solved in 79 single-slice turns, or alternatively in 85 twists.
The difference in this analysis is that it allows any single layer turn or double layer turn
(where the two layers are any two adjacent layers and moved together) to be counted as a "turn."
In some prior posts, I referred to these turns as "block turns."
So the set of turns about the U-D axis that count as one turn are the following: » 14 comments | read more
Lower bound using the Edges antipodeSubmitted by mdlazreg on Mon, 01/29/2007 - 15:01.One way of solving the cube is using two phases:
1) First solve all the edges [less than 18 moves as proved by Tom Rokicki ]. 2) Take it to the cube identity [less than 22 moves as proved by Silviu Radu]. Which gives a maximum of 40 moves. Is it possible to follow the same logic but use the Edges Antipode instead?. So the two phases become: 1) Solve all the edges by taking them to the Antipode edges instead of the identity edges. [this is guaranteed to be 18 moves]. 2) From the edges antipode position take it to the cube identity. [Does anyone know the maximum moves needed for this phase?]. To find this maximum it would take the same computational effort to prove the 22 moves I guess. Let's call this maximum X. » 15 comments | read more
A Little More on Odd and Even PermutationsSubmitted by Jerry Bryan on Thu, 01/11/2007 - 12:50.One of the basic tenents of cube theory (a highly specialized branch of group theory - grin!) is that odd permutations of the corners can only occur with odd permutations of the edges, and that even permutations of the corners can only occur with even permutations of the edges. This fact is one of the three factors that causes the order of the cube group to be smaller than the order of what is sometimes the illegal cube group.
If you take a cube apart and put it back together, you can put it back together in twelve times as many ways as there are positions in the cube group. Twelve is because 3 * 2 * 2 = 12. Three of the ways are because the twist of the last corner cubie you put back together can be set in one of three different ways, only one of which is legal. Two of the ways are because the flip of the last edge cubie you put back together can be set in one of two different ways, only one of which is legal. The last two of the ways are because the corners and edges can by put back together either with the same odd/even parity (legal) or with the opposite odd/even parity (illegal). » 3 comments | read more
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