## Pattern Sub-Groups

Submitted by B MacKenzie on Sat, 01/12/2008 - 22:10.I've been fooling around with symmetric Rubik cube patterns: "Cross" patterns, involving corners only; "Check" patterns, involving edges only; and "Dots" patterns, involving edges and corners.

By rotating the eight corners a quarter turn about one of the cube axes a cube state with a cross on four faces is produced. A second such state may be produced by rotating about a different cube axes:

R U B R U R D R U' D B' U' B' D' B' R' D' B' R B D R B R F R B' F D' B' D' F' D' R' F' D'

By recursively forming binary products from these two generators a group of 24 cube states is produced composed of all the different pure cross patterns plus the identity element. This Cross group is composed of five conjugate classes:

## Hexadecimal Sudo-Kube puzzle

Submitted by crepeau on Sat, 01/12/2008 - 01:27.I have constructed the following puzzle based on a 4x4x4 Eastsheen cube. It is labelled with hexadecimal digits 0,1,2,...,9,A,B,C,D,E,F on each face of each cubbie (I have turned the labels at an angle of 45 degrees to avoid having positional information due to label orientations). The goal of the puzzle is to have each label eaxctly once per face and once per row (in all 3 dimensions). This is the only constraint that must be satisfied. It seems possible that a single puzzle has more than one valid solved configuration, but I have not fund more than one so far. I wrote a computer program which generates the instance to be solved "randomly" and outputs the labelling of a scrambled puzzle. This last step is necessary to avoid knowing the solved configuration which would reduce the puzzle to returning each cubbie to its correct place... The puzzle is really difficult because the solved configuration is not known !!!

## Hexadecimal Sudo-Kube puzzle

Submitted by crepeau on Wed, 01/09/2008 - 13:18.I have constructed the following puzzle based on a 4x4x4 Eastsheen cube. It is labelled with hexadecimal digits 0,1,2,...,9,A,B,C,D,E,F on each face of each cubbie (I have turned the labels at an angle of 45 degrees to avoid having positional information due to label orientations). The goal of the puzzle is to have each label eaxctly once per face and once per row (in all 3 dimensions). This is the only constraint that must be satisfied. It seems possible that a single puzzle has more than one valid solved configuration, but I have not fund more than one so far. I wrote a computer program which generates the instance to be solved "randomly" and outputs the labelling of a scrambled puzzle. This last step is necessary to avoid knowing the solved configuration which would reduce the puzzle to returning each cubbie to its correct place... The puzzle is really difficult because the solved configuration is not known !!!

## Using GAP to produce solutions to Pentultimate

Submitted by DougCube on Tue, 12/11/2007 - 16:14.You can play with this puzzle here (it doesn't physically exist yet):

http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/dodeca_f6.htm

## Independent proof of 26-move upper bound for half-turn metric

Submitted by rokicki on Thu, 12/06/2007 - 13:34.run on my personal machines on a bunch of the H cosets, starting with

the most symmetrical ones and moving down to the less symmetrical ones.

My H-coset solver uses the Kociemba two-phase algorithm, but for all

19B elements of an H-coset in parallel, rather than for a single cube

position. (It's just the Kociemba two-phase algorithm augmented with

a 19B-element bit table, roughly.)

As of today, I have completed 4,748 such cosets, using a depth of at

least 16 for phase 1 and proving that each element of most of these

## Symmetry of Cayley Graphs of Subgroups of the Cube Group G

Submitted by Jerry Bryan on Sun, 11/25/2007 - 10:35.
Normally, we think of Rubik's cube symmetry in terms of a particular
position x. The symmetry of x can be defined as Symm(x), where
Symm(x) is the set all symmetries m in M
such that x^{m}=x, and where M is the group
of 48 symmetries of the cube.
By the closure property, Symm(x) is a subgroup of M,
and there are 98 possible such
subgroups.
I wish to expand this definition of symmetry to encompass an entire
subgroup of the Cube group G.

The primary motivation for the expanded definition is to provide a standard mechanism to deal with the symmetry of subgroups of the Cube group G such as H=<U,R>. I'm probably going to make things more complicated than they need to be. However, I don't recall seeing a general discussion of group symmetries before, neither on Cube-Lovers nor on this forum.

## Optimal solver for QTM

Submitted by Herbert Kociemba on Sun, 10/28/2007 - 12:41.I have written an optimal solver console program for the quarter turn metric in C which compiles under Linux and Windows. The target subgroup for the pruning table is the permutations of <U,D,R2,F2,L2,B2> which have an even corner parity, lets call it H'. I experimented with another type of pruning table which does not store the distance to H' for each coset element but the moves which reduce the distance to H'. Because there are 12 possible moves in QTM, a 16bit word is enough for each coset element - a move reduces the distance to H' by one or increases it by one. The table fits in about 583 MB of RAM.

## 4x4x4 Supercube Squares Group

Submitted by Bruce Norskog on Mon, 10/01/2007 - 22:40.At last, I have completed my God's algorithm calculation for the 4x4x4 supercube squares group, in the single-slice half-turns only metric. It was found to have 47,968,768 antipodes, many more elements than are in the entire 3x3x3 supercube squares group. (OK, it's only 11,992,192 antipodes after reducing the group by four cube orientations.) The analysis showed that the elements in the group can always be solved using no more than 20 half-turns, only one more half-turn than the number required if the four centers for each face are considered indistinguishable. (The 4x4x4 supercube differs from the usual 4x4x4 cube in that all 24 centers are considered to be distinguishable from one another.)

## Supercube Squares Group

Submitted by Bruce Norskog on Sat, 09/15/2007 - 12:56.Before I finish and report the results of a huge analysis that I'm working on, I thought it only fitting that I first perform this much smaller analysis, an analysis of the 3x3x3 supercube squares group (using half-turns only). Since this group has a mere 5,308,416 elements, it wouldn't be surprising to me if others have already done this analysis. However, I did not find any such analysis searching on the internet. I know that Mark had noted the size of this group in a message in the "Cube-lovers" archives: http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Mark_Longridge__Super_Groups.html as well as a God's algorithm calculation of the ordinary 3x3x3 squares group (a group 8 times smaller). Or at least he discussed the antipodes of that group: http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Mark_Longridge__SQUARE'S_GROUP_ANALYS IS.html

## the new magic number is 26...

Submitted by crepeau on Thu, 08/16/2007 - 11:09.http://www.ccs.neu.edu/home/gene/papers/rubik.pdf