# Non trivial identities

As you know there is a recurrent formula based on the cube trivial identities that holds true up to level 5 in QTM. It breaks at level 6:

P = 1
P = 12
P = 114
P = 1,068
P = 10,011
P = 93,840
P < 879,624

The real number at level 6 is 878,880 which is 744 shorter than what the formula predicts.

This discrepancy is obviously caused by some identities other than the trivial ones.

Does anyone have the list of all those identities?

The only ones I know about are the ones listed in Jaap's website:

ab'=bc'=ca'=I where:

a= FU'R'FRF'
b= U'FRU'R'U
c= UL'U'LFU'

Are those the only ones? if so how do they account for the missing 744 positions?

Thanks

## Comment viewing options

### With my QTM otimal solver I g

With my QTM otimal solver I get these nontrivial identities with 12 moves

U U R U R' F' U U L' U' L F (12q*)
U U R B U' B' U U R' F' U F (12q*)
U U R' U' R B U U L U L' B' (12q*)
U U R' F' U F U U R B U' B' (12q*)
U R U U B L U' L' U U B' R' (12q*)
U R U R' F' U U L' U' L F U (12q*)
U R U' B U' L' B L B' U B' R' (12q*)
U R U' B' R R D' R' D B R R (12q*)
U R U' B' R B R' U' B U B' R' (12q*)
U R U' B' R B' D B D' R' B R' (12q*)
U R R F R F' U' R R B' R' B (12q*)
U R R F D R' D' R R F' U' R (12q*)
U R B U' B' U U R' F' U F U (12q*)
U R B B D B D' R' B B U' B' (12q*)
U R B' R D' R' D B R' B U' B' (12q*)
U R B' R' B U R' U' R B U' B' (12q*)
U R B' R' B U' B L U' L' U B' (12q*)
U R B' R' B B U' L' B L B B (12q*)
U R' U F R' F' R U' R B U' B' (12q*)
U R' U F' U' F R U' R B' R' B (12q*)
U R' U' R R B' D' R D R R B (12q*)
U R' U' R B U U L U L' B' U (12q*)
U R' U' R B U' B L' B' L U B' (12q*)
U R' U' R B U' B' U R B' R' B (12q*)
U R' U' R B' R D B' D' B R' B (12q*)
U R' F R F' U' R U' B U B' R' (12q*)
U R' F R' D' F D F' R F' U' R (12q*)
U R' F' U U L' U' L F U U R (12q*)
U R' F' U F U U R B U' B' U (12q*)
U R' F' U F U' R U' B' R B R' (12q*)
U R' F' U F U' R' F R F' U' R (12q*)
U R' F' U F' L F L' U' F U' R (12q*)

There are no other essentially different solutions. All other solutions you can get by applying any of the 48 symmetries of the cube.

### Conjugate relationships

Looking at Herbert's list there are some comminalities which might shed some light on how these identities arise. Nineteen of the identities share a common form: q [A] q' [B] or its inverse [A] q [B] q'

```U U	R U R'F' 	U U	L' U' L F
U U	R B U' B' 	U U	R' F' U F
U U	R' U' R B	U U	L U L' B'
U U	R' F' U F	U U	R B U' B'
U	R U U B L	U' 	L' U U B' R'
U R' 	U F R' F'	R U'	R B U' B'
U R'	U F' U' F	R U'	R B' R' B
U	R' F' U U L'	U'	L F U U R
U R'	F R F' U'	R U'	B U B' R'
U R'	F' U F U'	R U'	B' R B R'

U R U' B'	R R	D' R' D B	R R
U R U' B'	R B	R' U' B U	B' R'
U R U' B'	R B'	D B D' R' 	B R'
U R' U' R	B' R	D B' D' B	R' B
U R B B D	B	D' R' B B U'	B'
U R B' R' 	B U	R' U' R B	U' B'
U R B' R'	B U'	B L U' L' 	U B'
U R B' R'	B B	U' L' B L	B B
U R' U' R R	B'	D' R D R R	B

```

Seven of the identies share the form: p [A] p' q [B] q'

```U	R' U' R B	U'	B	L' B' L U	B'
U	R' U' R B	U'	B'	U R B' R'	B
U	R' F' U F	U'	R'	F R F' U'	R
U	R B U' B'	U'	U'	R' F' U F	U
U	R' U' R B	U'	U'	L U L' B'	U
U	R' F' U F	U'	U'	R B U' B'	U
U	R U R' F'	U'	U'	L' U' L F	U ```

Leaving six which follow neither pattern:

```U R' F' U F' L	F L' U' F U' R
U R U' B U' L' 	B L B' U B' R'
U R R F R F' 	U' R R B' R' B
U R R F D R'  	D' R R F' U' R
U R B' R D' R'	D B R' B U' B'
U R' F R' D' F	D F' R F' U' R ```

So many of the level 6 identities come from conjugate relationships at a lower level.

### Actually all identities at 12

Actually all identities at 12q* can be generated using only the following identities:

I12-1 FR' F'R UF' U'F RU' R'U
I12-2 FR' F'R UF' F'L FL' U'F
I12-3 FR' F'R UF' UL' U'L FU'

http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Dan_Hoey__Results_of_an_exhaustive_search_to_six_quarter-twists.html

It was later proven that I12-3 can be generated using I12-1 and I12-3.

It is amazing, to me at least, that they are only 2 identities essentially at level 12. All the rest can be deduced using symmetry, antisymmetry, shifting and replacement by lower identities...

Unfortunately I know of no code that will generate only "generators identities" at all levels.

This idea of generator identities can be understood by looking at 4q* identities which are up to symmetry:

I4-1 UUUU
I4-2 UDU'D'
I4-3 UD'U'D

The third one however can be deduced from the first two as follow:

UD'U'D = U DDD U' D [thanks to DDDD=I which is I4-1 up to symmetry]
= U DD DU' D
= UD D U'D D [thanks to UDU'D'=I which is I4-2]
= UD DU' D D
= UD U'D D D [thanks to UDU'D'=I which is I4-2]
= UD U'D DD
= UD U' DDD
= UD U' D' [thanks to DDDD=I which is I4-1 up to symmetry]
= I [thanks to UDU'D'=I which is I4-2]

This means that at level 4q* they are only two identities that are essentially different. All the rest can be deduced from them. It is amazing that at level 12q* also they are only two identities that are essentially different.

I am trying to reduce the 14q* identities to their generators only without success so far...

### I was unaware that analysis.

I was unaware that analysis. That is amazing. All of the different identites can be put in one of these three basic forms, constructed of nested conjugations?

I have a hazy conception of these turn sequences as zig-zag paths through 20D parameter space forming a complex web. Complex but highly ordered and symmetric. Every node is like every other. The web looks exactly the same from every state. All the different 12 turn identities and only three basic shapes.

### I am not sure what you mean b

I am not sure what you mean by nested conjugations but yes it is indeed very amazing that all identities at 12q* are derived from these three basic forms. Thats why I believe God can make miracles:).

If that was not enough, even the third one can be derived from the first two!. This means that if someone tells us only about these two identities at 12q* then he has told us everything. Talk about information compression.

So given these two identities, you can derive all the rest using symmetry, anti-symmetry, shifting and replacement by lower identities.

shifting is when you take the identity: ABCDEFGHIJKL and you deduce out of it identities like:

LABCDEFGHIJK
KLABCDEFGHIJ
JKLABCDEFGHI

The question of what is the smallest dimension of the space where it would be possible to arrange all nodes in such a way that every node is like every other, is yet another unanswered interesting question.

If we imagine that each node is a charged particle, then throw all of them on the surface of a sphere, would they arrange themselves in such a way that every node looks like every other? If so then their parameter space is simply a 2D space. This however is highly unlikely...

I do not know what is the basis of your 20D.

### Parameter Space

There are twenty cubies which may be in any of 24 states. With twenty degrees of freedom, any state may be mapped to a point in 20D space. It would be a highly non-euclidian space with the dimensions looping back on themselves, UUUU and you're back where you started.

Actually(and I'm going way beyond my level of competence here)I think that you need the same number of dimensions as the order of the group in order to have a completely accurate map, every element being a distance of 1 from every other. With fewer dimensions there will be distortion like mapping the spherical earth onto a plane.

Ah, glad you pointed out what you mean by shifting. Actually, I would describe this as transposing a product: [A][B] to [B][A]. This is a conjugation, really. Since identity is identity's only conjugate one gets another identity. Neat trick.

### re. Parameter Space

On reflection, only elements of order two generate a dimension per element. A higher order element and its inverse would map to the same dimension thus generating a dimension for each element/inverse element pair. Does that make sense? In this light, the dimensionality of the space generated by the group would be a thorny question.

### Dimension in group theory ?

I'm not sure I understand your question and what you mean by 'dimension'.

For me the word 'dimension' is a term of linear algebra. This means the common size of all minimal spanning sets (bases). In case of group theory, there is no theorem about common size of minimal spanning sets; (U, R, L, F, B) is a minimal spanning set of G (i.e. no strict sub-set of it does span G) but (UBLUL'U'B', R2FLD'R) also spans G (we cannot do better since G isn't cyclic). The usual way to represent a group is to draw its Cayley graph according to a given set of generators and you can do it 2D or 3D and it will be as symetrical and uncommutative as you said.

The problem here is that we find an euclidian nD-space easier to imagine than a simple finite group (by the way I think that finite sets are considered to be 0-dimensional aren't they ?)

Is the Cayley graph of G what you mean by your (20 or less)-dimensional space or do you want to know how many different cyclic subgroups there are in G (a cyclic group is a group of the form {g,,gg,...,g^(o(g)-1)=g^(-1)} where o(g) is the order of g) ?

### If you look at this way then

If you look at this way then probably this link will help you put it in mathematical words:

http://en.wikipedia.org/wiki/Order_(group_theory)

### h dimensional space

My conception of representing group elements as unit vectors in h dimensional space comes from my exposure to the application of group theory to quantum mechanics( which I will not hesitate to admit went right over my head ). See http://en.wikipedia.org/wiki/Great_orthogonality_theorem

### Identities and Duplicate Positions

Thanks to Herbert for posting the list of identities.  I haven't had time yet to double check his results against the results of my program.  I look forward to comparing the results from the two programs.

Just to be sure everybody's on the same page, I thought I would run through how Herbert's identities relate to duplicate positions and my EndsWith function.

Herbert's first identity is UURUR'F'UUL'U'LF=I.  We can associate the moves as (UURUR'F')(UUL'U'LF)=I.  If we define x=UURUR'F', then we have x'=UUL'U'LF and xx'=I.

That means that we have two processes for x:

x = UURUR'F'
x = (x')' = F'L'ULU'U' (simply inverting the last half of Herbert's process)

Therefore, (UURUR'F') and (F'L'ULU'U') are what my program calls duplicate positions.  As a first approximation, it appears that EndsWith(x)={F',U'}.  However, F'L'ULU'U' can also be written as F'L'ULUU.  So in reality, EndsWith(x)={F',U,U'}, and |EndsWith(x)|=3.

Looked at from the point of view of Herbert's identities, UURUR'F'UUL'U'LF may also be written as U'U'RUR'F'UUL'U'LF, as UURUR'F'U'U'L'U'LF, and as U'U'RUR'F'U'U'L'U'LF.  That's simply because the UU part of the process is not prime.  So I think a little work may be required in order to do a full comparison of Herbert's results with mine.

The calculation of EndsWith (and also StartsWith) for each position is an artifact of my optimal depth first God's algorithm enumerator that produces positions in lexicographic order.  Close to Start, the program simply creates positions such as RL' as both the process RL' and as the process L'R and makes note of the fact that RL' and L'R are duplicate positions.  Therefore, EndsWith(RL')={R,L'}. All positions are stored, as are all StartsWith and EndsWith values.  Positions are enumerated one level from Start at a time.  The current version of the program is able to follow this process out to 6q from Start before it runs out of memory.

Having stored all the data out to 6q from Start, the program can then calculate out to 12q from Start.  I've only run it out to 11q from Start so far because the 12q run would take so long.  To calculate out to 11q from Start, the program calculates all products of the form sitj in lexicographic order, where si is a position of length 6q and tj is a position of length 5q.  Because of producing the positions in lexicographic order, the positions don't have to be stored, duplicate positions sort together, and all the instances of a particular duplicate position are only counted once.

In addition to counting positions, the program is able to determine StartsWith and EndsWith values for each position.  Suppose for example that sitj=sktm and that there are no other duplicate instances.  Then, it is the case that EndsWith(sitj) = EndsWith(sktm) = EndsWith(tj) ∪ EndsWith(tm).

There are other complications, of course.  If |si|=6 and |tj|=5, then all we know for sure is that |sitj|<=11.  In order to assure that |sitj|=11, sitj must be compared against all products of shorter length, and the shorter length products are also being produced in lexicographic order at the same time.  In other words, to calculate out to 11q the program really has to produce all products of the form sitj in lexicographic order, where |si|=6 and 1<=|tj|<=5.

### Correction

My current program can store out to 7q from Start, and I have run it out to 13q from Start. It is 14q from Start that would currently take too long.

### I think I figured out how the

I think I figured out how the identities posted by Herbert relate to the missing 744 positions.

First of all I hacked Herbert's code to generate the whole list of identities regardless of symmetries:

```U U R U R' F' U U L' U' L F  (12q*)
U U R B U' B' U U R' F' U F  (12q*)
U U R' U' R B U U L U L' B'  (12q*)
U U R' F' U F U U R B U' B'  (12q*)
U U F U F' L' U U B' U' B L  (12q*)
U U F R U' R' U U F' L' U L  (12q*)
U U F' U' F R U U B U B' R'  (12q*)
U U F' L' U L U U F R U' R'  (12q*)
U U L U L' B' U U R' U' R B  (12q*)
U U L F U' F' U U L' B' U B  (12q*)
U U L' U' L F U U R U R' F'  (12q*)
U U L' B' U B U U L F U' F'  (12q*)
U U B U B' R' U U F' U' F R  (12q*)
U U B L U' L' U U B' R' U R  (12q*)
U U B' U' B L U U F U F' L'  (12q*)
U U B' R' U R U U B L U' L'  (12q*)
U R U U B L U' L' U U B' R'  (12q*)
U R U R' F' U U L' U' L F U  (12q*)
U R U' B U' L' B L B' U B' R'  (12q*)
U R U' B' R R D' R' D B R R  (12q*)
U R U' B' R B R' U' B U B' R'  (12q*)
U R U' B' R B' D B D' R' B R'  (12q*)
U R R F R F' U' R R B' R' B  (12q*)
U R R F D R' D' R R F' U' R  (12q*)
U R B U' B' U U R' F' U F U  (12q*)
U R B B D B D' R' B B U' B'  (12q*)
U R B' R D' R' D B R' B U' B'  (12q*)
U R B' R' B U R' U' R B U' B'  (12q*)
U R B' R' B U' B L U' L' U B'  (12q*)
U R B' R' B B U' L' B L B B  (12q*)
U R' U F R' F' R U' R B U' B'  (12q*)
U R' U F' U' F R U' R B' R' B  (12q*)
U R' U' R R B' D' R D R R B  (12q*)
U R' U' R B U U L U L' B' U  (12q*)
U R' U' R B U' B L' B' L U B'  (12q*)
U R' U' R B U' B' U R B' R' B  (12q*)
U R' U' R B' R D B' D' B R' B  (12q*)
U R' F R F' U' R U' B U B' R'  (12q*)
U R' F R' D' F D F' R F' U' R  (12q*)
U R' F' U U L' U' L F U U R  (12q*)
U R' F' U F U U R B U' B' U  (12q*)
U R' F' U F U' R U' B' R B R'  (12q*)
U R' F' U F U' R' F R F' U' R  (12q*)
U R' F' U F' L F L' U' F U' R  (12q*)
U F U U R B U' B' U U R' F'  (12q*)
U F U F' L' U U B' U' B L U  (12q*)
U F U' R U' B' R B R' U R' F'  (12q*)
U F U' R' F R F' U' R U R' F'  (12q*)
U F U' R' F R' D R D' F' R F'  (12q*)
U F U' R' F F D' F' D R F F  (12q*)
U F R U' R' U U F' L' U L U  (12q*)
U F R R D R D' F' R R U' R'  (12q*)
U F R' F D' F' D R F' R U' R'  (12q*)
U F R' F' R U F' U' F R U' R'  (12q*)
U F R' F' R U' R B U' B' U R'  (12q*)
U F R' F' R R U' B' R B R R  (12q*)
U F F L F L' U' F F R' F' R  (12q*)
U F F L D F' D' F F L' U' F  (12q*)
U F' U L F' L' F U' F R U' R'  (12q*)
U F' U L' U' L F U' F R' F' R  (12q*)
U F' U' F R U U B U B' R' U  (12q*)
U F' U' F R U' R B' R' B U R'  (12q*)
U F' U' F R U' R' U F R' F' R  (12q*)
U F' U' F R' F D R' D' R F' R  (12q*)
U F' U' F F R' D' F D F F R  (12q*)
U F' L F L' U' F U' R U R' F'  (12q*)
U F' L F' D' L D L' F L' U' F  (12q*)
U F' L' U U B' U' B L U U F  (12q*)
U F' L' U L U U F R U' R' U  (12q*)
U F' L' U L U' F U' R' F R F'  (12q*)
U F' L' U L U' F' L F L' U' F  (12q*)
U F' L' U L' B L B' U' L U' F  (12q*)
U L U U F R U' R' U U F' L'  (12q*)
U L U L' B' U U R' U' R B U  (12q*)
U L U' F U' R' F R F' U F' L'  (12q*)
U L U' F' L F L' U' F U F' L'  (12q*)
U L U' F' L F' D F D' L' F L'  (12q*)
U L U' F' L L D' L' D F L L  (12q*)
U L F U' F' U U L' B' U B U  (12q*)
U L F F D F D' L' F F U' F'  (12q*)
U L F' L D' L' D F L' F U' F'  (12q*)
U L F' L' F U L' U' L F U' F'  (12q*)
U L F' L' F U' F R U' R' U F'  (12q*)
U L F' L' F F U' R' F R F F  (12q*)
U L L B D L' D' L L B' U' L  (12q*)
U L L B L B' U' L L F' L' F  (12q*)
U L' U B L' B' L U' L F U' F'  (12q*)
U L' U B' U' B L U' L F' L' F  (12q*)
U L' U' L F U U R U R' F' U  (12q*)
U L' U' L F U' F R' F' R U F'  (12q*)
U L' U' L F U' F' U L F' L' F  (12q*)
U L' U' L F' L D F' D' F L' F  (12q*)
U L' U' L L F' D' L D L L F  (12q*)
U L' B L B' U' L U' F U F' L'  (12q*)
U L' B L' D' B D B' L B' U' L  (12q*)
U L' B' U U R' U' R B U U L  (12q*)
U L' B' U B U U L F U' F' U  (12q*)
U L' B' U B U' L U' F' L F L'  (12q*)
U L' B' U B U' L' B L B' U' L  (12q*)
U L' B' U B' R B R' U' B U' L  (12q*)
U B U U L F U' F' U U L' B'  (12q*)
U B U B' R' U U F' U' F R U  (12q*)
U B U' L U' F' L F L' U L' B'  (12q*)
U B U' L' B L B' U' L U L' B'  (12q*)
U B U' L' B L' D L D' B' L B'  (12q*)
U B U' L' B B D' B' D L B B  (12q*)
U B L U' L' U U B' R' U R U  (12q*)
U B L L D L D' B' L L U' L'  (12q*)
U B L' B D' B' D L B' L U' L'  (12q*)
U B L' B' L U B' U' B L U' L'  (12q*)
U B L' B' L U' L F U' F' U L'  (12q*)
U B L' B' L L U' F' L F L L  (12q*)
U B B R D B' D' B B R' U' B  (12q*)
U B B R B R' U' B B L' B' L  (12q*)
U B' U R B' R' B U' B L U' L'  (12q*)
U B' U R' U' R B U' B L' B' L  (12q*)
U B' U' B L U U F U F' L' U  (12q*)
U B' U' B L U' L F' L' F U L'  (12q*)
U B' U' B L U' L' U B L' B' L  (12q*)
U B' U' B L' B D L' D' L B' L  (12q*)
U B' U' B B L' D' B D B B L  (12q*)
U B' R B R' U' B U' L U L' B'  (12q*)
U B' R B' D' R D R' B R' U' B  (12q*)
U B' R' U U F' U' F R U U B  (12q*)
U B' R' U R U U B L U' L' U  (12q*)
U B' R' U R U' B U' L' B L B'  (12q*)
U B' R' U R U' B' R B R' U' B  (12q*)
U B' R' U R' F R F' U' R U' B  (12q*)
U' R U R R F D R' D' R R F'  (12q*)
U' R U R' F R' D' F D F' R F'  (12q*)
U' R U R' F' U U L' U' L F U'  (12q*)
U' R U R' F' U F U' R' F R F'  (12q*)
U' R U R' F' U F' L F L' U' F  (12q*)
U' R U' B U B' R' U R' F R F'  (12q*)
U' R U' B' R B R' U R' F' U F  (12q*)
U' R R B' R' B U R R F R F'  (12q*)
U' R R B' D' R D R R B U R'  (12q*)
U' R B U U L U L' B' U U R'  (12q*)
U' R B U' B L' B' L U B' U R'  (12q*)
U' R B U' B' U U R' F' U F U'  (12q*)
U' R B U' B' U R B' R' B U R'  (12q*)
U' R B U' B' U R' U F R' F' R  (12q*)
U' R B' R D B' D' B R' B U R'  (12q*)
U' R B' R' B U R' U F' U' F R  (12q*)
U' R' U U F' L' U L U U F R  (12q*)
U' R' U F R R D R D' F' R R  (12q*)
U' R' U F R' F D' F' D R F' R  (12q*)
U' R' U F R' F' R U F' U' F R  (12q*)
U' R' U F' U L F' L' F U' F R  (12q*)
U' R' U' R B U U L U L' B' U'  (12q*)
U' R' F R F F U L F' L' F F  (12q*)
U' R' F R F' U F' L' U L U' F  (12q*)
U' R' F R F' U' R U R' F' U F  (12q*)
U' R' F R' D R D' F' R F' U F  (12q*)
U' R' F F D' F' D R F F U F  (12q*)
U' R' F' U F U U R B U' B' U'  (12q*)
U' F U F F L D F' D' F F L'  (12q*)
U' F U F' L F' D' L D L' F L'  (12q*)
U' F U F' L' U U B' U' B L U'  (12q*)
U' F U F' L' U L U' F' L F L'  (12q*)
U' F U F' L' U L' B L B' U' L  (12q*)
U' F U' R U R' F' U F' L F L'  (12q*)
U' F U' R' F R F' U F' L' U L  (12q*)
U' F R U U B U B' R' U U F'  (12q*)
U' F R U' R B' R' B U R' U F'  (12q*)
U' F R U' R' U U F' L' U L U'  (12q*)
U' F R U' R' U F R' F' R U F'  (12q*)
U' F R U' R' U F' U L F' L' F  (12q*)
U' F R' F D R' D' R F' R U F'  (12q*)
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U' F F R' F' R U F F L F L'  (12q*)
U' F F R' D' F D F F R U F'  (12q*)
U' F' U U L' B' U B U U L F  (12q*)
U' F' U L F F D F D' L' F F  (12q*)
U' F' U L F' L D' L' D F L' F  (12q*)
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U' F' L L D' L' D F L L U L  (12q*)
U' F' L' U L U U F R U' R' U'  (12q*)
U' L U L L B D L' D' L L B'  (12q*)
U' L U L' B L' D' B D B' L B'  (12q*)
U' L U L' B' U U R' U' R B U'  (12q*)
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U' L L F' L' F U L L B L B'  (12q*)
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U' L' U B L' B D' B' D L B' L  (12q*)
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U' L' B L B' U B' R' U R U' B  (12q*)
U' L' B L B' U' L U L' B' U B  (12q*)
U' L' B L' D L D' B' L B' U B  (12q*)
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U' B' U R B B D B D' R' B B  (12q*)
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R U U B U B' R' U U F' U' F  (12q*)
R U U B L U' L' U U B' R' U  (12q*)
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R U R' F' U F U' R' F R F' U'  (12q*)
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R U' B' R B R R U F R' F' R  (12q*)
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R R U R U' B' R R D' R' D B  (12q*)
R R U F R' F' R R U' B' R B  (12q*)
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R R F D R' D' R R F' U' R U  (12q*)
R R F' U' R U R R F D R' D'  (12q*)
R R F' R' F D R R B R B' D'  (12q*)
R R D R D' F' R R U' R' U F  (12q*)
R R D B R' B' R R D' F' R F  (12q*)
R R D' R' D B R R U R U' B'  (12q*)
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R R B' R' B U R R F R F' U'  (12q*)
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R F D D L D L' F' D D R' D'  (12q*)
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R F' U' R U R' F R' D' F D F'  (12q*)
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R F' R U F' U' F R' F D R' D'  (12q*)
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R F' R' F F D' L' F L F F D  (12q*)
R F' R' F D R R B R B' D' R  (12q*)
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R F' R' F D' F L D' L' D F' D  (12q*)
R D R R B U R' U' R R B' D'  (12q*)
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R D R' B R' U' B U B' R B' D'  (12q*)
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R D R' B' D B D' R' B R B' D'  (12q*)
R D R' B' D B' L B L' D' B D'  (12q*)
R D D F D F' R' D D B' D' B  (12q*)
R D D F L D' L' D D F' R' D  (12q*)
R D B R' B' R R D' F' R F R  (12q*)
R D B B L B L' D' B B R' B'  (12q*)
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R D B' D' B B R' U' B U B B  (12q*)
R D' R F D' F' D R' D B R' B'  (12q*)
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R D' R' D D B' L' D L D D B  (12q*)
R D' R' D B R R U R U' B' R  (12q*)
R D' R' D B R' B U' B' U R B'  (12q*)
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R D' F D F' R' D R' B R B' D'  (12q*)
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R D' F' R F R R D B R' B' R  (12q*)
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R D' F' R F' U F U' R' F R' D  (12q*)
R B U U L U L' B' U U R' U'  (12q*)
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R B B D L B' L' B B D' R' B  (12q*)
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R B' D' R D' F D F' R' D R' B  (12q*)
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R' U U F' L' U L U U F R U'  (12q*)
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R' U R U' B U' L' B L B' U B'  (12q*)
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R' U F R' F D' F' D R F' R U'  (12q*)
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R' U' R B' R D B' D' B R' B U  (12q*)
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R' U' B' R B R R U F R' F' R'  (12q*)
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R' F R F' U F' L' U L U' F U'  (12q*)
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R' F R F' U' R U' B U B' R' U  (12q*)
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R' F F D' L' F L F F D R F'  (12q*)
R' F D R R B R B' D' R R F'  (12q*)
R' F D R' D B' D' B R D' R F'  (12q*)
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R' F D' F L D' L' D F' D R F'  (12q*)
R' F D' F' D R F' R U' R' U F  (12q*)
R' F' U U L' U' L F U U R U  (12q*)
R' F' U F U U R B U' B' U U  (12q*)
R' F' U F U' R U' B' R B R' U  (12q*)
R' F' U F U' R' F R F' U' R U  (12q*)
R' F' U F' L F L' U' F U' R U  (12q*)
R' F' U' R U R R F D R' D' R'  (12q*)
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R' F' R U F' U L' U' L F U' F  (12q*)
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R' F' R' F D R R B R B' D' R'  (12q*)
R' D R D D F L D' L' D D F'  (12q*)
R' D R D' F D' L' F L F' D F'  (12q*)
R' D R D' F' R R U' R' U F R'  (12q*)
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R' D R D' F' R F' U F U' R' F  (12q*)
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R' D R' B' D B D' R D' F' R F  (12q*)
R' D D B' D' B R D D F D F'  (12q*)
R' D D B' L' D L D D B R D'  (12q*)
R' D B R R U R U' B' R R D'  (12q*)
R' D B R' B U' B' U R B' R D'  (12q*)
R' D B R' B' R R D' F' R F R'  (12q*)
R' D B R' B' R D B' D' B R D'  (12q*)
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R' D B' D L B' L' B D' B R D'  (12q*)
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R' D' R R F' U' R U R R F D  (12q*)
R' D' R F D D L D L' F' D D  (12q*)
R' D' R F D' F L' F' L D F' D  (12q*)
R' D' R F D' F' D R F' R' F D  (12q*)
R' D' R F' R U F' U' F R' F D  (12q*)
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R' D' F F L' F' L D F F R F  (12q*)
R' D' F D F F R U F' U' F F  (12q*)
R' D' F D F' R F' U' R U R' F  (12q*)
R' D' F D F' R' D R D' F' R F  (12q*)
R' D' F D' L D L' F' D F' R F  (12q*)
R' D' F' R F R R D B R' B' R'  (12q*)
R' B U R R F R F' U' R R B'  (12q*)
R' B U R' U F' U' F R U' R B'  (12q*)
R' B U R' U' R R B' D' R D R'  (12q*)
R' B U R' U' R B U' B' U R B'  (12q*)
R' B U R' U' R B' R D B' D' B  (12q*)
R' B U' B L U' L' U B' U R B'  (12q*)
R' B U' B' U R B' R D' R' D B  (12q*)
R' B R B B D L B' L' B B D'  (12q*)
R' B R B' D B' L' D L D' B D'  (12q*)
R' B R B' D' R R F' R' F D R'  (12q*)
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R' B R B' D' R D' F D F' R' D  (12q*)
R' B R' U R U' B' R B' D B D'  (12q*)
R' B R' U' B U B' R B' D' R D  (12q*)
R' B B U' L' B L B B U R B'  (12q*)
R' B B U' B' U R B B D B D'  (12q*)
R' B' R R D' F' R F R R D B  (12q*)
R' B' R D B B L B L' D' B B  (12q*)
R' B' R D B' D L' D' L B D' B  (12q*)
R' B' R D B' D' B R D' R' D B  (12q*)
R' B' R D' R F D' F' D R' D B  (12q*)
R' B' R' B U R R F R F' U' R'  (12q*)
R' B' D D L' D' L B D D R D  (12q*)
R' B' D B D D R F D' F' D D  (12q*)
R' B' D B D' R D' F' R F R' D  (12q*)
R' B' D B D' R' B R B' D' R D  (12q*)
R' B' D B' L B L' D' B D' R D  (12q*)
R' B' D' R D R R B U R' U' R'  (12q*)
F U U R U R' F' U U L' U' L  (12q*)
F U U R B U' B' U U R' F' U  (12q*)
F U F U' R' F F D' F' D R F  (12q*)
F U F F L D F' D' F F L' U'  (12q*)
F U F' L F' D' L D L' F L' U'  (12q*)
F U F' L' U U B' U' B L U U  (12q*)
F U F' L' U L U' F' L F L' U'  (12q*)
F U F' L' U L' B L B' U' L U'  (12q*)
F U L F' L' F F U' R' F R F  (12q*)
F U L L B L B' U' L L F' L'  (12q*)
F U L' U B' U' B L U' L F' L'  (12q*)
F U L' U' L F U' F' U L F' L'  (12q*)
F U L' U' L F' L D F' D' F L'  (12q*)
F U L' U' L L F' D' L D L L  (12q*)
F U' R U R' F' U F' L F L' U'  (12q*)
F U' R U' B' R B R' U R' F' U  (12q*)
F U' R' F R F F U L F' L' F  (12q*)
F U' R' F R F' U F' L' U L U'  (12q*)
F U' R' F R F' U' R U R' F' U  (12q*)
F U' R' F R' D R D' F' R F' U  (12q*)
F U' R' F F D' F' D R F F U  (12q*)
F U' F R U' R' U F' U L F' L'  (12q*)
F U' F R' F' R U F' U L' U' L  (12q*)
F U' F' U U L' B' U B U U L  (12q*)
F U' F' U L F F D F D' L' F  (12q*)
F U' F' U L F' L D' L' D F L'  (12q*)
F U' F' U L F' L' F U L' U' L  (12q*)
F U' F' U L' U B L' B' L U' L  (12q*)
F R U U B U B' R' U U F' U'  (12q*)
F R U F' U' F F R' D' F D F  (12q*)
F R U' R B' R' B U R' U F' U'  (12q*)
F R U' R' U U F' L' U L U U  (12q*)
F R U' R' U F R' F' R U F' U'  (12q*)
F R U' R' U F' U L F' L' F U'  (12q*)
F R R D R D' F' R R U' R' U  (12q*)
F R R D B R' B' R R D' F' R  (12q*)
F R F R' D' F F L' F' L D F  (12q*)
F R F F U L F' L' F F U' R'  (12q*)
F R F' U F' L' U L U' F U' R'  (12q*)
F R F' U' R U R' F' U F U' R'  (12q*)
F R F' U' R U' B U B' R' U R'  (12q*)
F R F' U' R R B' R' B U R R  (12q*)
F R' F D R' D' R F' R U F' U'  (12q*)
F R' F D' F' D R F' R U' R' U  (12q*)
F R' F' R U F F L F L' U' F  (12q*)
F R' F' R U F' U L' U' L F U'  (12q*)
F R' F' R U F' U' F R U' R' U  (12q*)
F R' F' R U' R B U' B' U R' U  (12q*)
F R' F' R R U' B' R B R R U  (12q*)
F R' D R D' F' R F' U F U' R'  (12q*)
F R' D R' B' D B D' R D' F' R  (12q*)
F R' D' F F L' F' L D F F R  (12q*)
F R' D' F D F F R U F' U' F  (12q*)
F R' D' F D F' R F' U' R U R'  (12q*)
F R' D' F D F' R' D R D' F' R  (12q*)
F R' D' F D' L D L' F' D F' R  (12q*)
F F U F U' R' F F D' F' D R  (12q*)
F F U L F' L' F F U' R' F R  (12q*)
F F U' R' F R F F U L F' L'  (12q*)
F F U' F' U L F F D F D' L'  (12q*)
F F R U F' U' F F R' D' F D  (12q*)
F F R F R' D' F F L' F' L D  (12q*)
F F R' F' R U F F L F L' U'  (12q*)
F F R' D' F D F F R U F' U'  (12q*)
F F D R F' R' F F D' L' F L  (12q*)
F F D F D' L' F F U' F' U L  (12q*)
F F D' F' D R F F U F U' R'  (12q*)
F F D' L' F L F F D R F' R'  (12q*)
F F L F L' U' F F R' F' R U  (12q*)
F F L D F' D' F F L' U' F U  (12q*)
F F L' U' F U F F L D F' D'  (12q*)
F F L' F' L D F F R F R' D'  (12q*)
F D R R B R B' D' R R F' R'  (12q*)
F D R F' R' F F D' L' F L F  (12q*)
F D R' D B' D' B R D' R F' R'  (12q*)
F D R' D' R R F' U' R U R R  (12q*)
F D R' D' R F D' F' D R F' R'  (12q*)
F D R' D' R F' R U F' U' F R'  (12q*)
F D F F R U F' U' F F R' D'  (12q*)
F D F D' L' F F U' F' U L F  (12q*)
F D F' R F' U' R U R' F R' D'  (12q*)
F D F' R' D R D' F' R F R' D'  (12q*)
F D F' R' D R' B R B' D' R D'  (12q*)
F D F' R' D D B' D' B R D D  (12q*)
F D D L D L' F' D D R' D' R  (12q*)
F D D L B D' B' D D L' F' D  (12q*)
F D' F L D' L' D F' D R F' R'  (12q*)
F D' F L' F' L D F' D R' D' R  (12q*)
F D' F' D R F F U F U' R' F  (12q*)
F D' F' D R F' R U' R' U F R'  (12q*)
F D' F' D R F' R' F D R' D' R  (12q*)
F D' F' D R' D B R' B' R D' R  (12q*)
F D' F' D D R' B' D B D D R  (12q*)
F D' L D L' F' D F' R F R' D'  (12q*)
F D' L D' B' L B L' D L' F' D  (12q*)
F D' L' F F U' F' U L F F D  (12q*)
F D' L' F L F F D R F' R' F  (12q*)
F D' L' F L F' D F' R' D R D'  (12q*)
F D' L' F L F' D' L D L' F' D  (12q*)
F D' L' F L' U L U' F' L F' D  (12q*)
F L F F D R F' R' F F D' L'  (12q*)
F L F L' U' F F R' F' R U F  (12q*)
F L F' D F' R' D R D' F D' L'  (12q*)
F L F' D' L D L' F' D F D' L'  (12q*)
F L F' D' L D' B D B' L' D L'  (12q*)
F L F' D' L L B' L' B D L L  (12q*)
F L D F' D' F F L' U' F U F  (12q*)
F L D D B D B' L' D D F' D'  (12q*)
F L D' L B' L' B D L' D F' D'  (12q*)
F L D' L' D F L' F' L D F' D'  (12q*)
F L D' L' D F' D R F' R' F D'  (12q*)
F L D' L' D D F' R' D R D D  (12q*)
F L L U L U' F' L L D' L' D  (12q*)
F L L U B L' B' L L U' F' L  (12q*)
F L' U L U' F' L F' D F D' L'  (12q*)
F L' U L' B' U B U' L U' F' L  (12q*)
F L' U' F U F F L D F' D' F  (12q*)
F L' U' F U F' L F' D' L D L'  (12q*)
F L' U' F U F' L' U L U' F' L  (12q*)
F L' U' F U' R U R' F' U F' L  (12q*)
F L' U' F F R' F' R U F F L  (12q*)
F L' F U L' U' L F' L D F' D'  (12q*)
F L' F U' F' U L F' L D' L' D  (12q*)
F L' F' L D F F R F R' D' F  (12q*)
F L' F' L D F' D R' D' R F D'  (12q*)
F L' F' L D F' D' F L D' L' D  (12q*)
F L' F' L D' L B D' B' D L' D  (12q*)
F L' F' L L D' B' L B L L D  (12q*)
F' U U L' U' L F U U R U R'  (12q*)
F' U U L' B' U B U U L F U'  (12q*)
F' U F U U R B U' B' U U R'  (12q*)
F' U F U' R U' B' R B R' U R'  (12q*)
F' U F U' R' F R F' U' R U R'  (12q*)
F' U F U' R' F R' D R D' F' R  (12q*)
F' U F U' R' F F D' F' D R F'  (12q*)
F' U F' L F L' U' F U' R U R'  (12q*)
F' U F' L' U L U' F U' R' F R  (12q*)
F' U L F F D F D' L' F F U'  (12q*)
F' U L F' L D' L' D F L' F U'  (12q*)
F' U L F' L' F U L' U' L F U'  (12q*)
F' U L F' L' F U' F R U' R' U  (12q*)
F' U L F' L' F F U' R' F R F'  (12q*)
F' U L' U B L' B' L U' L F U'  (12q*)
F' U L' U' L F U' F R' F' R U  (12q*)
F' U' R U R R F D R' D' R R  (12q*)
F' U' R U R' F R' D' F D F' R  (12q*)
F' U' R U R' F' U F U' R' F R  (12q*)
F' U' R U' B U B' R' U R' F R  (12q*)
F' U' R R B' R' B U R R F R  (12q*)
F' U' R' F R F F U L F' L' F'  (12q*)
F' U' F R U U B U B' R' U U  (12q*)
F' U' F R U' R B' R' B U R' U  (12q*)
F' U' F R U' R' U F R' F' R U  (12q*)
F' U' F R' F D R' D' R F' R U  (12q*)
F' U' F F R' D' F D F F R U  (12q*)
F' U' F' U L F F D F D' L' F'  (12q*)
F' R U F F L F L' U' F F R'  (12q*)
F' R U F' U L' U' L F U' F R'  (12q*)
F' R U F' U' F R U' R' U F R'  (12q*)
F' R U F' U' F R' F D R' D' R  (12q*)
F' R U F' U' F F R' D' F D F'  (12q*)
F' R U' R B U' B' U R' U F R'  (12q*)
F' R U' R' U F R' F D' F' D R  (12q*)
F' R R U' R' U F R R D R D'  (12q*)
F' R R U' B' R B R R U F R'  (12q*)
F' R F R R D B R' B' R R D'  (12q*)
F' R F R' D R' B' D B D' R D'  (12q*)
F' R F R' D' F F L' F' L D F'  (12q*)
F' R F R' D' F D F' R' D R D'  (12q*)
F' R F R' D' F D' L D L' F' D  (12q*)
F' R F' U F U' R' F R' D R D'  (12q*)
F' R F' U' R U R' F R' D' F D  (12q*)
F' R' F F D' L' F L F F D R  (12q*)
F' R' F D R R B R B' D' R R  (12q*)
F' R' F D R' D B' D' B R D' R  (12q*)
F' R' F D R' D' R F D' F' D R  (12q*)
F' R' F D' F L D' L' D F' D R  (12q*)
F' R' F' R U F F L F L' U' F'  (12q*)
F' R' D R D D F L D' L' D D  (12q*)
F' R' D R D' F D' L' F L F' D  (12q*)
F' R' D R D' F' R F R' D' F D  (12q*)
F' R' D R' B R B' D' R D' F D  (12q*)
F' R' D D B' D' B R D D F D  (12q*)
F' R' D' F D F F R U F' U' F'  (12q*)
F' D R F F U F U' R' F F D'  (12q*)
F' D R F' R U' R' U F R' F D'  (12q*)
F' D R F' R' F F D' L' F L F'  (12q*)
F' D R F' R' F D R' D' R F D'  (12q*)
F' D R F' R' F D' F L D' L' D  (12q*)
F' D R' D B R' B' R D' R F D'  (12q*)
F' D R' D' R F D' F L' F' L D  (12q*)
F' D F D D L B D' B' D D L'  (12q*)
F' D F D' L D' B' L B L' D L'  (12q*)
F' D F D' L' F F U' F' U L F'  (12q*)
F' D F D' L' F L F' D' L D L'  (12q*)
F' D F D' L' F L' U L U' F' L  (12q*)
F' D F' R F R' D' F D' L D L'  (12q*)
F' D F' R' D R D' F D' L' F L  (12q*)
F' D D R' D' R F D D L D L'  (12q*)
F' D D R' B' D B D D R F D'  (12q*)
F' D' F F L' U' F U F F L D  (12q*)
F' D' F L D D B D B' L' D D  (12q*)
F' D' F L D' L B' L' B D L' D  (12q*)
F' D' F L D' L' D F L' F' L D  (12q*)
F' D' F L' F U L' U' L F' L D  (12q*)
F' D' F' D R F F U F U' R' F'  (12q*)
F' D' L D L L F U L' U' L L  (12q*)
F' D' L D L' F L' U' F U F' L  (12q*)
F' D' L D L' F' D F D' L' F L  (12q*)
F' D' L D' B D B' L' D L' F L  (12q*)
F' D' L L B' L' B D L L F L  (12q*)
F' D' L' F L F F D R F' R' F'  (12q*)
F' L F L L U B L' B' L L U'  (12q*)
F' L F L' U L' B' U B U' L U'  (12q*)
F' L F L' U' F U F' L' U L U'  (12q*)
F' L F L' U' F U' R U R' F' U  (12q*)
F' L F L' U' F F R' F' R U F'  (12q*)
F' L F' D F D' L' F L' U L U'  (12q*)
F' L F' D' L D L' F L' U' F U  (12q*)
F' L D F F R F R' D' F F L'  (12q*)
F' L D F' D R' D' R F D' F L'  (12q*)
F' L D F' D' F F L' U' F U F'  (12q*)
F' L D F' D' F L D' L' D F L'  (12q*)
F' L D F' D' F L' F U L' U' L  (12q*)
F' L D' L B D' B' D L' D F L'  (12q*)
F' L D' L' D F L' F U' F' U L  (12q*)
F' L L D' L' D F L L U L U'  (12q*)
F' L L D' B' L B L L D F L'  (12q*)
F' L' U U B' U' B L U U F U  (12q*)
F' L' U L U U F R U' R' U U  (12q*)
F' L' U L U' F U' R' F R F' U  (12q*)
F' L' U L U' F' L F L' U' F U  (12q*)
F' L' U L' B L B' U' L U' F U  (12q*)
F' L' U' F U F F L D F' D' F'  (12q*)
F' L' F U L L B L B' U' L L  (12q*)
F' L' F U L' U B' U' B L U' L  (12q*)
F' L' F U L' U' L F U' F' U L  (12q*)
F' L' F U' F R U' R' U F' U L  (12q*)
F' L' F F U' R' F R F F U L  (12q*)
F' L' F' L D F F R F R' D' F'  (12q*)
D R R B U R' U' R R B' D' R  (12q*)
D R R B R B' D' R R F' R' F  (12q*)
D R F F U F U' R' F F D' F'  (12q*)
D R F D' F' D D R' B' D B D  (12q*)
D R F' R U' R' U F R' F D' F'  (12q*)
D R F' R' F F D' L' F L F F  (12q*)
D R F' R' F D R' D' R F D' F'  (12q*)
D R F' R' F D' F L D' L' D F'  (12q*)
D R D R' B' D D L' D' L B D  (12q*)
D R D D F L D' L' D D F' R'  (12q*)
D R D' F D' L' F L F' D F' R'  (12q*)
D R D' F' R R U' R' U F R R  (12q*)
D R D' F' R F R' D' F D F' R'  (12q*)
D R D' F' R F' U F U' R' F R'  (12q*)
D R' D B R' B' R D' R F D' F'  (12q*)
D R' D B' D' B R D' R F' R' F  (12q*)
D R' D' R R F' U' R U R R F  (12q*)
D R' D' R F D D L D L' F' D  (12q*)
D R' D' R F D' F L' F' L D F'  (12q*)
D R' D' R F D' F' D R F' R' F  (12q*)
D R' D' R F' R U F' U' F R' F  (12q*)
D R' B R B' D' R D' F D F' R'  (12q*)
D R' B R' U' B U B' R B' D' R  (12q*)
D R' B' D D L' D' L B D D R  (12q*)
D R' B' D B D D R F D' F' D  (12q*)
D R' B' D B D' R D' F' R F R'  (12q*)
D R' B' D B D' R' B R B' D' R  (12q*)
D R' B' D B' L B L' D' B D' R  (12q*)
D F F R U F' U' F F R' D' F  (12q*)
D F F R F R' D' F F L' F' L  (12q*)
D F D F' R' D D B' D' B R D  (12q*)
D F D D L B D' B' D D L' F'  (12q*)
D F D' L D' B' L B L' D L' F'  (12q*)
D F D' L' F F U' F' U L F F  (12q*)
D F D' L' F L F' D' L D L' F'  (12q*)
D F D' L' F L' U L U' F' L F'  (12q*)
D F L D' L' D D F' R' D R D  (12q*)
D F L L U L U' F' L L D' L'  (12q*)
D F L' F U' F' U L F' L D' L'  (12q*)
D F L' F' L D F' D' F L D' L'  (12q*)
D F L' F' L D' L B D' B' D L'  (12q*)
D F L' F' L L D' B' L B L L  (12q*)
D F' R F R' D' F D' L D L' F'  (12q*)
D F' R F' U' R U R' F R' D' F  (12q*)
D F' R' D R D D F L D' L' D  (12q*)
D F' R' D R D' F D' L' F L F'  (12q*)
D F' R' D R D' F' R F R' D' F  (12q*)
D F' R' D R' B R B' D' R D' F  (12q*)
D F' R' D D B' D' B R D D F  (12q*)
D F' D R F' R' F D' F L D' L'  (12q*)
D F' D R' D' R F D' F L' F' L  (12q*)
D F' D' F F L' U' F U F F L  (12q*)
D F' D' F L D D B D B' L' D  (12q*)
D F' D' F L D' L B' L' B D L'  (12q*)
D F' D' F L D' L' D F L' F' L  (12q*)
D F' D' F L' F U L' U' L F' L  (12q*)
D D R F D' F' D D R' B' D B  (12q*)
D D R D R' B' D D L' D' L B  (12q*)
D D R' D' R F D D L D L' F'  (12q*)
D D R' B' D B D D R F D' F'  (12q*)
D D F D F' R' D D B' D' B R  (12q*)
D D F L D' L' D D F' R' D R  (12q*)
D D F' R' D R D D F L D' L'  (12q*)
D D F' D' F L D D B D B' L'  (12q*)
D D L D L' F' D D R' D' R F  (12q*)
D D L B D' B' D D L' F' D F  (12q*)
D D L' F' D F D D L B D' B'  (12q*)
D D L' D' L B D D R D R' B'  (12q*)
D D B R D' R' D D B' L' D L  (12q*)
D D B D B' L' D D F' D' F L  (12q*)
D D B' D' B R D D F D F' R'  (12q*)
D D B' L' D L D D B R D' R'  (12q*)
D L D D B R D' R' D D B' L'  (12q*)
D L D L' F' D D R' D' R F D  (12q*)
D L D' B D' R' B R B' D B' L'  (12q*)
D L D' B' L L U' L' U B L L  (12q*)
D L D' B' L B L' D' B D B' L'  (12q*)
D L D' B' L B' U B U' L' B L'  (12q*)
D L L F U L' U' L L F' D' L  (12q*)
D L L F L F' D' L L B' L' B  (12q*)
D L B D' B' D D L' F' D F D  (12q*)
D L B B U B U' L' B B D' B'  (12q*)
D L B' L U' L' U B L' B D' B'  (12q*)
D L B' L' B D L' D' L B D' B'  (12q*)
D L B' L' B D' B R D' R' D B'  (12q*)
D L B' L' B B D' R' B R B B  (12q*)
D L' F L F' D' L D' B D B' L'  (12q*)
D L' F L' U' F U F' L F' D' L  (12q*)
D L' F' D F D D L B D' B' D  (12q*)
D L' F' D F D' L D' B' L B L'  (12q*)
D L' F' D F D' L' F L F' D' L  (12q*)
D L' F' D F' R F R' D' F D' L  (12q*)
D L' F' D D R' D' R F D D L  (12q*)
D L' D F L' F' L D' L B D' B'  (12q*)
D L' D F' D' F L D' L B' L' B  (12q*)
D L' D' L L B' U' L U L L B  (12q*)
D L' D' L B D D R D R' B' D  (12q*)
D L' D' L B D' B R' B' R D B'  (12q*)
D L' D' L B D' B' D L B' L' B  (12q*)
D L' D' L B' L U B' U' B L' B  (12q*)
D B R R U R U' B' R R D' R'  (12q*)
D B R D' R' D D B' L' D L D  (12q*)
D B R' B U' B' U R B' R D' R'  (12q*)
D B R' B' R R D' F' R F R R  (12q*)
D B R' B' R D B' D' B R D' R'  (12q*)
D B R' B' R D' R F D' F' D R'  (12q*)
D B D D R F D' F' D D R' B'  (12q*)
D B D B' L' D D F' D' F L D  (12q*)
D B D' R D' F' R F R' D R' B'  (12q*)
D B D' R' B R B' D' R D R' B'  (12q*)
D B D' R' B R' U R U' B' R B'  (12q*)
D B D' R' B B U' B' U R B B  (12q*)
D B B L U B' U' B B L' D' B  (12q*)
D B B L B L' D' B B R' B' R  (12q*)
D B' D L B' L' B D' B R D' R'  (12q*)
D B' D L' D' L B D' B R' B' R  (12q*)
D B' D' B R D D F D F' R' D  (12q*)
D B' D' B R D' R F' R' F D R'  (12q*)
D B' D' B R D' R' D B R' B' R  (12q*)
D B' D' B R' B U R' U' R B' R  (12q*)
D B' D' B B R' U' B U B B R  (12q*)
D B' L B L' D' B D' R D R' B'  (12q*)
D B' L B' U' L U L' B L' D' B  (12q*)
D B' L' D D F' D' F L D D B  (12q*)
D B' L' D L D D B R D' R' D  (12q*)
D B' L' D L D' B D' R' B R B'  (12q*)
D B' L' D L D' B' L B L' D' B  (12q*)
D B' L' D L' F L F' D' L D' B  (12q*)
D' R R F' U' R U R R F D R'  (12q*)
D' R R F' R' F D R R B R B'  (12q*)
D' R F D D L D L' F' D D R'  (12q*)
D' R F D' F L' F' L D F' D R'  (12q*)
D' R F D' F' D R F' R' F D R'  (12q*)
D' R F D' F' D R' D B R' B' R  (12q*)
D' R F D' F' D D R' B' D B D'  (12q*)
D' R F' R U F' U' F R' F D R'  (12q*)
D' R F' R' F D R' D B' D' B R  (12q*)
D' R D R R B U R' U' R R B'  (12q*)
D' R D R' B R' U' B U B' R B'  (12q*)
D' R D R' B' D D L' D' L B D'  (12q*)
D' R D R' B' D B D' R' B R B'  (12q*)
D' R D R' B' D B' L B L' D' B  (12q*)
D' R D' F D F' R' D R' B R B'  (12q*)
D' R D' F' R F R' D R' B' D B  (12q*)
D' R' D D B' L' D L D D B R  (12q*)
D' R' D B R R U R U' B' R R  (12q*)
D' R' D B R' B U' B' U R B' R  (12q*)
D' R' D B R' B' R D B' D' B R  (12q*)
D' R' D B' D L B' L' B D' B R  (12q*)
D' R' D' R F D D L D L' F' D'  (12q*)
D' R' B R B B D L B' L' B B  (12q*)
D' R' B R B' D B' L' D L D' B  (12q*)
D' R' B R B' D' R D R' B' D B  (12q*)
D' R' B R' U R U' B' R B' D B  (12q*)
D' R' B B U' B' U R B B D B  (12q*)
D' R' B' D B D D R F D' F' D'  (12q*)
D' F F L' U' F U F F L D F'  (12q*)
D' F F L' F' L D F F R F R'  (12q*)
D' F D F F R U F' U' F F R'  (12q*)
D' F D F' R F' U' R U R' F R'  (12q*)
D' F D F' R' D R D' F' R F R'  (12q*)
D' F D F' R' D R' B R B' D' R  (12q*)
D' F D F' R' D D B' D' B R D'  (12q*)
D' F D' L D L' F' D F' R F R'  (12q*)
D' F D' L' F L F' D F' R' D R  (12q*)
D' F L D D B D B' L' D D F'  (12q*)
D' F L D' L B' L' B D L' D F'  (12q*)
D' F L D' L' D F L' F' L D F'  (12q*)
D' F L D' L' D F' D R F' R' F  (12q*)
D' F L D' L' D D F' R' D R D'  (12q*)
D' F L' F U L' U' L F' L D F'  (12q*)
D' F L' F' L D F' D R' D' R F  (12q*)
D' F' R R U' R' U F R R D R  (12q*)
D' F' R F R R D B R' B' R R  (12q*)
D' F' R F R' D R' B' D B D' R  (12q*)
D' F' R F R' D' F D F' R' D R  (12q*)
D' F' R F' U F U' R' F R' D R  (12q*)
D' F' R' D R D D F L D' L' D'  (12q*)
D' F' D R F F U F U' R' F F  (12q*)
D' F' D R F' R U' R' U F R' F  (12q*)
D' F' D R F' R' F D R' D' R F  (12q*)
D' F' D R' D B R' B' R D' R F  (12q*)
D' F' D D R' B' D B D D R F  (12q*)
D' F' D' F L D D B D B' L' D'  (12q*)
D' L D L L F U L' U' L L F'  (12q*)
D' L D L' F L' U' F U F' L F'  (12q*)
D' L D L' F' D F D' L' F L F'  (12q*)
D' L D L' F' D F' R F R' D' F  (12q*)
D' L D L' F' D D R' D' R F D'  (12q*)
D' L D' B D B' L' D L' F L F'  (12q*)
D' L D' B' L B L' D L' F' D F  (12q*)
D' L L B' U' L U L L B D L'  (12q*)
D' L L B' L' B D L L F L F'  (12q*)
D' L B D D R D R' B' D D L'  (12q*)
D' L B D' B R' B' R D B' D L'  (12q*)
D' L B D' B' D D L' F' D F D'  (12q*)
D' L B D' B' D L B' L' B D L'  (12q*)
D' L B D' B' D L' D F L' F' L  (12q*)
D' L B' L U B' U' B L' B D L'  (12q*)
D' L B' L' B D L' D F' D' F L  (12q*)
D' L' F F U' F' U L F F D F  (12q*)
D' L' F L F F D R F' R' F F  (12q*)
D' L' F L F' D F' R' D R D' F  (12q*)
D' L' F L F' D' L D L' F' D F  (12q*)
D' L' F L' U L U' F' L F' D F  (12q*)
D' L' F' D F D D L B D' B' D'  (12q*)
D' L' D F L L U L U' F' L L  (12q*)
D' L' D F L' F U' F' U L F' L  (12q*)
D' L' D F L' F' L D F' D' F L  (12q*)
D' L' D F' D R F' R' F D' F L  (12q*)
D' L' D D F' R' D R D D F L  (12q*)
D' L' D' L B D D R D R' B' D'  (12q*)
D' B R D D F D F' R' D D B'  (12q*)
D' B R D' R F' R' F D R' D B'  (12q*)
D' B R D' R' D D B' L' D L D'  (12q*)
D' B R D' R' D B R' B' R D B'  (12q*)
D' B R D' R' D B' D L B' L' B  (12q*)
D' B R' B U R' U' R B' R D B'  (12q*)
D' B R' B' R D B' D L' D' L B  (12q*)
D' B D B B L U B' U' B B L'  (12q*)
D' B D B' L B' U' L U L' B L'  (12q*)
D' B D B' L' D D F' D' F L D'  (12q*)
D' B D B' L' D L D' B' L B L'  (12q*)
D' B D B' L' D L' F L F' D' L  (12q*)
D' B D' R D R' B' D B' L B L'  (12q*)
D' B D' R' B R B' D B' L' D L  (12q*)
D' B B R' U' B U B B R D B'  (12q*)
D' B B R' B' R D B B L B L'  (12q*)
D' B' D D L' F' D F D D L B  (12q*)
D' B' D L B B U B U' L' B B  (12q*)
D' B' D L B' L U' L' U B L' B  (12q*)
D' B' D L B' L' B D L' D' L B  (12q*)
D' B' D L' D F L' F' L D' L B  (12q*)
D' B' D' B R D D F D F' R' D'  (12q*)
D' B' L L U' L' U B L L D L  (12q*)
D' B' L B L L D F L' F' L L  (12q*)
D' B' L B L' D L' F' D F D' L  (12q*)
D' B' L B L' D' B D B' L' D L  (12q*)
D' B' L B' U B U' L' B L' D L  (12q*)
D' B' L' D L D D B R D' R' D'  (12q*)
L U U F U F' L' U U B' U' B  (12q*)
L U U F R U' R' U U F' L' U  (12q*)
L U L U' F' L L D' L' D F L  (12q*)
L U L L B D L' D' L L B' U'  (12q*)
L U L' B L' D' B D B' L B' U'  (12q*)
L U L' B' U U R' U' R B U U  (12q*)
L U L' B' U B U' L' B L B' U'  (12q*)
L U L' B' U B' R B R' U' B U'  (12q*)
L U B L' B' L L U' F' L F L  (12q*)
L U B B R B R' U' B B L' B'  (12q*)
L U B' U R' U' R B U' B L' B'  (12q*)
L U B' U' B L U' L' U B L' B'  (12q*)
L U B' U' B L' B D L' D' L B'  (12q*)
L U B' U' B B L' D' B D B B  (12q*)
L U' F U F' L' U L' B L B' U'  (12q*)
L U' F U' R' F R F' U F' L' U  (12q*)
L U' F' L F L L U B L' B' L  (12q*)
L U' F' L F L' U L' B' U B U'  (12q*)
L U' F' L F L' U' F U F' L' U  (12q*)
L U' F' L F' D F D' L' F L' U  (12q*)
L U' F' L L D' L' D F L L U  (12q*)
L U' L F U' F' U L' U B L' B'  (12q*)
L U' L F' L' F U L' U B' U' B  (12q*)
L U' L' U U B' R' U R U U B  (12q*)
L U' L' U B L L D L D' B' L  (12q*)
L U' L' U B L' B D' B' D L B'  (12q*)
L U' L' U B L' B' L U B' U' B  (12q*)
L U' L' U B' U R B' R' B U' B  (12q*)
L F U U R U R' F' U U L' U'  (12q*)
L F U L' U' L L F' D' L D L  (12q*)
L F U' F R' F' R U F' U L' U'  (12q*)
L F U' F' U U L' B' U B U U  (12q*)
L F U' F' U L F' L' F U L' U'  (12q*)
L F U' F' U L' U B L' B' L U'  (12q*)
L F F D R F' R' F F D' L' F  (12q*)
L F F D F D' L' F F U' F' U  (12q*)
L F L F' D' L L B' L' B D L  (12q*)
L F L L U B L' B' L L U' F'  (12q*)
L F L' U L' B' U B U' L U' F'  (12q*)
L F L' U' F U F' L' U L U' F'  (12q*)
L F L' U' F U' R U R' F' U F'  (12q*)
L F L' U' F F R' F' R U F F  (12q*)
L F' D F D' L' F L' U L U' F'  (12q*)
L F' D F' R' D R D' F D' L' F  (12q*)
L F' D' L D L L F U L' U' L  (12q*)
L F' D' L D L' F L' U' F U F'  (12q*)
L F' D' L D L' F' D F D' L' F  (12q*)
L F' D' L D' B D B' L' D L' F  (12q*)
L F' D' L L B' L' B D L L F  (12q*)
L F' L D F' D' F L' F U L' U'  (12q*)
L F' L D' L' D F L' F U' F' U  (12q*)
L F' L' F U L L B L B' U' L  (12q*)
L F' L' F U L' U B' U' B L U'  (12q*)
L F' L' F U L' U' L F U' F' U  (12q*)
L F' L' F U' F R U' R' U F' U  (12q*)
L F' L' F F U' R' F R F F U  (12q*)
L D F F R F R' D' F F L' F'  (12q*)
L D F L' F' L L D' B' L B L  (12q*)
L D F' D R' D' R F D' F L' F'  (12q*)
L D F' D' F F L' U' F U F F  (12q*)
L D F' D' F L D' L' D F L' F'  (12q*)
L D F' D' F L' F U L' U' L F'  (12q*)
L D D B R D' R' D D B' L' D  (12q*)
L D D B D B' L' D D F' D' F  (12q*)
L D L D' B' L L U' L' U B L  (12q*)
L D L L F U L' U' L L F' D'  (12q*)
L D L' F L' U' F U F' L F' D'  (12q*)
L D L' F' D F D' L' F L F' D'  (12q*)
L D L' F' D F' R F R' D' F D'  (12q*)
L D L' F' D D R' D' R F D D  (12q*)
L D' L B D' B' D L' D F L' F'  (12q*)
L D' L B' L' B D L' D F' D' F  (12q*)
L D' L' D F L L U L U' F' L  (12q*)
L D' L' D F L' F U' F' U L F'  (12q*)
L D' L' D F L' F' L D F' D' F  (12q*)
L D' L' D F' D R F' R' F D' F  (12q*)
L D' L' D D F' R' D R D D F  (12q*)
L D' B D B' L' D L' F L F' D'  (12q*)
L D' B D' R' B R B' D B' L' D  (12q*)
L D' B' L L U' L' U B L L D  (12q*)
L D' B' L B L L D F L' F' L  (12q*)
L D' B' L B L' D L' F' D F D'  (12q*)
L D' B' L B L' D' B D B' L' D  (12q*)
L D' B' L B' U B U' L' B L' D  (12q*)
L L U L U' F' L L D' L' D F  (12q*)
L L U B L' B' L L U' F' L F  (12q*)
L L U' F' L F L L U B L' B'  (12q*)
L L U' L' U B L L D L D' B'  (12q*)
L L F U L' U' L L F' D' L D  (12q*)
L L F L F' D' L L B' L' B D  (12q*)
L L F' D' L D L L F U L' U'  (12q*)
L L F' L' F U L L B L B' U'  (12q*)
L L D F L' F' L L D' B' L B  (12q*)
L L D L D' B' L L U' L' U B  (12q*)
L L D' L' D F L L U L U' F'  (12q*)
L L D' B' L B L L D F L' F'  (12q*)
L L B D L' D' L L B' U' L U  (12q*)
L L B L B' U' L L F' L' F U  (12q*)
L L B' U' L U L L B D L' D'  (12q*)
L L B' L' B D L L F L F' D'  (12q*)
L B D D R D R' B' D D L' D'  (12q*)
L B D L' D' L L B' U' L U L  (12q*)
L B D' B R' B' R D B' D L' D'  (12q*)
L B D' B' D D L' F' D F D D  (12q*)
L B D' B' D L B' L' B D L' D'  (12q*)
L B D' B' D L' D F L' F' L D'  (12q*)
L B L L D F L' F' L L D' B'  (12q*)
L B L B' U' L L F' L' F U L  (12q*)
L B L' D L' F' D F D' L D' B'  (12q*)
L B L' D' B D B' L' D L D' B'  (12q*)
L B L' D' B D' R D R' B' D B'  (12q*)
L B L' D' B B R' B' R D B B  (12q*)
L B B U R B' R' B B U' L' B  (12q*)
L B B U B U' L' B B D' B' D  (12q*)
L B' U B U' L' B L' D L D' B'  (12q*)
L B' U B' R' U R U' B U' L' B  (12q*)
L B' U' L U L L B D L' D' L  (12q*)
L B' U' L U L' B L' D' B D B'  (12q*)
L B' U' L U L' B' U B U' L' B  (12q*)
L B' U' L U' F U F' L' U L' B  (12q*)
L B' U' L L F' L' F U L L B  (12q*)
L B' L U B' U' B L' B D L' D'  (12q*)
L B' L U' L' U B L' B D' B' D  (12q*)
L B' L' B D L L F L F' D' L  (12q*)
L B' L' B D L' D F' D' F L D'  (12q*)
L B' L' B D L' D' L B D' B' D  (12q*)
L B' L' B D' B R D' R' D B' D  (12q*)
L B' L' B B D' R' B R B B D  (12q*)
L' U U B' U' B L U U F U F'  (12q*)
L' U U B' R' U R U U B L U'  (12q*)
L' U L U U F R U' R' U U F'  (12q*)
L' U L U' F U' R' F R F' U F'  (12q*)
L' U L U' F' L F L' U' F U F'  (12q*)
L' U L U' F' L F' D F D' L' F  (12q*)
L' U L U' F' L L D' L' D F L'  (12q*)
L' U L' B L B' U' L U' F U F'  (12q*)
L' U L' B' U B U' L U' F' L F  (12q*)
L' U B L L D L D' B' L L U'  (12q*)
L' U B L' B D' B' D L B' L U'  (12q*)
L' U B L' B' L U B' U' B L U'  (12q*)
L' U B L' B' L U' L F U' F' U  (12q*)
L' U B L' B' L L U' F' L F L'  (12q*)
L' U B' U R B' R' B U' B L U'  (12q*)
L' U B' U' B L U' L F' L' F U  (12q*)
L' U' F U F F L D F' D' F F  (12q*)
L' U' F U F' L F' D' L D L' F  (12q*)
L' U' F U F' L' U L U' F' L F  (12q*)
L' U' F U' R U R' F' U F' L F  (12q*)
L' U' F F R' F' R U F F L F  (12q*)
L' U' F' L F L L U B L' B' L'  (12q*)
L' U' L F U U R U R' F' U U  (12q*)
L' U' L F U' F R' F' R U F' U  (12q*)
L' U' L F U' F' U L F' L' F U  (12q*)
L' U' L F' L D F' D' F L' F U  (12q*)
L' U' L L F' D' L D L L F U  (12q*)
L' U' L' U B L L D L D' B' L'  (12q*)
L' F U L L B L B' U' L L F'  (12q*)
L' F U L' U B' U' B L U' L F'  (12q*)
L' F U L' U' L F U' F' U L F'  (12q*)
L' F U L' U' L F' L D F' D' F  (12q*)
L' F U L' U' L L F' D' L D L'  (12q*)
L' F U' F R U' R' U F' U L F'  (12q*)
L' F U' F' U L F' L D' L' D F  (12q*)
L' F F U' R' F R F F U L F'  (12q*)
L' F F U' F' U L F F D F D'  (12q*)
L' F L F F D R F' R' F F D'  (12q*)
L' F L F' D F' R' D R D' F D'  (12q*)
L' F L F' D' L D L' F' D F D'  (12q*)
L' F L F' D' L D' B D B' L' D  (12q*)
L' F L F' D' L L B' L' B D L'  (12q*)
L' F L' U L U' F' L F' D F D'  (12q*)
L' F L' U' F U F' L F' D' L D  (12q*)
L' F' D F D D L B D' B' D D  (12q*)
L' F' D F D' L D' B' L B L' D  (12q*)
L' F' D F D' L' F L F' D' L D  (12q*)
L' F' D F' R F R' D' F D' L D  (12q*)
L' F' D D R' D' R F D D L D  (12q*)
L' F' D' L D L L F U L' U' L'  (12q*)
L' F' L D F F R F R' D' F F  (12q*)
L' F' L D F' D R' D' R F D' F  (12q*)
L' F' L D F' D' F L D' L' D F  (12q*)
L' F' L D' L B D' B' D L' D F  (12q*)
L' F' L L D' B' L B L L D F  (12q*)
L' F' L' F U L L B L B' U' L'  (12q*)
L' D F L L U L U' F' L L D'  (12q*)
L' D F L' F U' F' U L F' L D'  (12q*)
L' D F L' F' L D F' D' F L D'  (12q*)
L' D F L' F' L D' L B D' B' D  (12q*)
L' D F L' F' L L D' B' L B L'  (12q*)
L' D F' D R F' R' F D' F L D'  (12q*)
L' D F' D' F L D' L B' L' B D  (12q*)
L' D D F' R' D R D D F L D'  (12q*)
L' D D F' D' F L D D B D B'  (12q*)
L' D L D D B R D' R' D D B'  (12q*)
L' D L D' B D' R' B R B' D B'  (12q*)
L' D L D' B' L L U' L' U B L'  (12q*)
L' D L D' B' L B L' D' B D B'  (12q*)
L' D L D' B' L B' U B U' L' B  (12q*)
L' D L' F L F' D' L D' B D B'  (12q*)
L' D L' F' D F D' L D' B' L B  (12q*)
L' D' L L B' U' L U L L B D  (12q*)
L' D' L B D D R D R' B' D D  (12q*)
L' D' L B D' B R' B' R D B' D  (12q*)
L' D' L B D' B' D L B' L' B D  (12q*)
L' D' L B' L U B' U' B L' B D  (12q*)
L' D' L' D F L L U L U' F' L'  (12q*)
L' D' B D B B L U B' U' B B  (12q*)
L' D' B D B' L B' U' L U L' B  (12q*)
L' D' B D B' L' D L D' B' L B  (12q*)
L' D' B D' R D R' B' D B' L B  (12q*)
L' D' B B R' B' R D B B L B  (12q*)
L' D' B' L B L L D F L' F' L'  (12q*)
L' B D L L F L F' D' L L B'  (12q*)
L' B D L' D F' D' F L D' L B'  (12q*)
L' B D L' D' L L B' U' L U L'  (12q*)
L' B D L' D' L B D' B' D L B'  (12q*)
L' B D L' D' L B' L U B' U' B  (12q*)
L' B D' B R D' R' D B' D L B'  (12q*)
L' B D' B' D L B' L U' L' U B  (12q*)
L' B L B B U R B' R' B B U'  (12q*)
L' B L B' U B' R' U R U' B U'  (12q*)
L' B L B' U' L U L' B' U B U'  (12q*)
L' B L B' U' L U' F U F' L' U  (12q*)
L' B L B' U' L L F' L' F U L'  (12q*)
L' B L' D L D' B' L B' U B U'  (12q*)
L' B L' D' B D B' L B' U' L U  (12q*)
L' B B D' R' B R B B D L B'  (12q*)
L' B B D' B' D L B B U B U'  (12q*)
L' B' U U R' U' R B U U L U  (12q*)
L' B' U B U U L F U' F' U U  (12q*)
L' B' U B U' L U' F' L F L' U  (12q*)
L' B' U B U' L' B L B' U' L U  (12q*)
L' B' U B' R B R' U' B U' L U  (12q*)
L' B' U' L U L L B D L' D' L'  (12q*)
L' B' L U B B R B R' U' B B  (12q*)
L' B' L U B' U R' U' R B U' B  (12q*)
L' B' L U B' U' B L U' L' U B  (12q*)
L' B' L U' L F U' F' U L' U B  (12q*)
L' B' L L U' F' L F L L U B  (12q*)
L' B' L' B D L L F L F' D' L'  (12q*)
B U U L U L' B' U U R' U' R  (12q*)
B U U L F U' F' U U L' B' U  (12q*)
B U R R F R F' U' R R B' R'  (12q*)
B U R B' R' B B U' L' B L B  (12q*)
B U R' U F' U' F R U' R B' R'  (12q*)
B U R' U' R R B' D' R D R R  (12q*)
B U R' U' R B U' B' U R B' R'  (12q*)
B U R' U' R B' R D B' D' B R'  (12q*)
B U B U' L' B B D' B' D L B  (12q*)
B U B B R D B' D' B B R' U'  (12q*)
B U B' R B' D' R D R' B R' U'  (12q*)
B U B' R' U U F' U' F R U U  (12q*)
B U B' R' U R U' B' R B R' U'  (12q*)
B U B' R' U R' F R F' U' R U'  (12q*)
B U' L U L' B' U B' R B R' U'  (12q*)
B U' L U' F' L F L' U L' B' U  (12q*)
B U' L' B L B B U R B' R' B  (12q*)
B U' L' B L B' U B' R' U R U'  (12q*)
B U' L' B L B' U' L U L' B' U  (12q*)
B U' L' B L' D L D' B' L B' U  (12q*)
B U' L' B B D' B' D L B B U  (12q*)
B U' B L U' L' U B' U R B' R'  (12q*)
B U' B L' B' L U B' U R' U' R  (12q*)
B U' B' U U R' F' U F U U R  (12q*)
B U' B' U R B B D B D' R' B  (12q*)
B U' B' U R B' R D' R' D B R'  (12q*)
B U' B' U R B' R' B U R' U' R  (12q*)
B U' B' U R' U F R' F' R U' R  (12q*)
B R R U R U' B' R R D' R' D  (12q*)
B R R U F R' F' R R U' B' R  (12q*)
B R D D F D F' R' D D B' D'  (12q*)
B R D B' D' B B R' U' B U B  (12q*)
B R D' R F' R' F D R' D B' D'  (12q*)
B R D' R' D D B' L' D L D D  (12q*)
B R D' R' D B R' B' R D B' D'  (12q*)
B R D' R' D B' D L B' L' B D'  (12q*)
B R B R' U' B B L' B' L U B  (12q*)
B R B B D L B' L' B B D' R'  (12q*)
B R B' D B' L' D L D' B D' R'  (12q*)
B R B' D' R R F' R' F D R R  (12q*)
B R B' D' R D R' B' D B D' R'  (12q*)
B R B' D' R D' F D F' R' D R'  (12q*)
B R' U R U' B' R B' D B D' R'  (12q*)
B R' U R' F' U F U' R U' B' R  (12q*)
B R' U' B U B B R D B' D' B  (12q*)
B R' U' B U B' R B' D' R D R'  (12q*)
B R' U' B U B' R' U R U' B' R  (12q*)
B R' U' B U' L U L' B' U B' R  (12q*)
B R' U' B B L' B' L U B B R  (12q*)
B R' B U R' U' R B' R D B' D'  (12q*)
B R' B U' B' U R B' R D' R' D  (12q*)
B R' B' R R D' F' R F R R D  (12q*)
B R' B' R D B B L B L' D' B  (12q*)
B R' B' R D B' D L' D' L B D'  (12q*)
B R' B' R D B' D' B R D' R' D  (12q*)
B R' B' R D' R F D' F' D R' D  (12q*)
B D D R F D' F' D D R' B' D  (12q*)
B D D R D R' B' D D L' D' L  (12q*)
B D L L F L F' D' L L B' L'  (12q*)
B D L B' L' B B D' R' B R B  (12q*)
B D L' D F' D' F L D' L B' L'  (12q*)
B D L' D' L L B' U' L U L L  (12q*)
B D L' D' L B D' B' D L B' L'  (12q*)
B D L' D' L B' L U B' U' B L'  (12q*)
B D B D' R' B B U' B' U R B  (12q*)
B D B B L U B' U' B B L' D'  (12q*)
B D B' L B' U' L U L' B L' D'  (12q*)
B D B' L' D D F' D' F L D D  (12q*)
B D B' L' D L D' B' L B L' D'  (12q*)
B D B' L' D L' F L F' D' L D'  (12q*)
B D' R D R' B' D B' L B L' D'  (12q*)
B D' R D' F' R F R' D R' B' D  (12q*)
B D' R' B R B B D L B' L' B  (12q*)
B D' R' B R B' D B' L' D L D'  (12q*)
B D' R' B R B' D' R D R' B' D  (12q*)
B D' R' B R' U R U' B' R B' D  (12q*)
B D' R' B B U' B' U R B B D  (12q*)
B D' B R D' R' D B' D L B' L'  (12q*)
B D' B R' B' R D B' D L' D' L  (12q*)
B D' B' D D L' F' D F D D L  (12q*)
B D' B' D L B B U B U' L' B  (12q*)
B D' B' D L B' L U' L' U B L'  (12q*)
B D' B' D L B' L' B D L' D' L  (12q*)
B D' B' D L' D F L' F' L D' L  (12q*)
B L U U F U F' L' U U B' U'  (12q*)
B L U B' U' B B L' D' B D B  (12q*)
B L U' L F' L' F U L' U B' U'  (12q*)
B L U' L' U U B' R' U R U U  (12q*)
B L U' L' U B L' B' L U B' U'  (12q*)
B L U' L' U B' U R B' R' B U'  (12q*)
B L L D F L' F' L L D' B' L  (12q*)
B L L D L D' B' L L U' L' U  (12q*)
B L B L' D' B B R' B' R D B  (12q*)
B L B B U R B' R' B B U' L'  (12q*)
B L B' U B' R' U R U' B U' L'  (12q*)
B L B' U' L U L' B' U B U' L'  (12q*)
B L B' U' L U' F U F' L' U L'  (12q*)
B L B' U' L L F' L' F U L L  (12q*)
B L' D L D' B' L B' U B U' L'  (12q*)
B L' D L' F' D F D' L D' B' L  (12q*)
B L' D' B D B B L U B' U' B  (12q*)
B L' D' B D B' L B' U' L U L'  (12q*)
B L' D' B D B' L' D L D' B' L  (12q*)
B L' D' B D' R D R' B' D B' L  (12q*)
B L' D' B B R' B' R D B B L  (12q*)
B L' B D L' D' L B' L U B' U'  (12q*)
B L' B D' B' D L B' L U' L' U  (12q*)
B L' B' L U B B R B R' U' B  (12q*)
B L' B' L U B' U R' U' R B U'  (12q*)
B L' B' L U B' U' B L U' L' U  (12q*)
B L' B' L U' L F U' F' U L' U  (12q*)
B L' B' L L U' F' L F L L U  (12q*)
B B U R B' R' B B U' L' B L  (12q*)
B B U B U' L' B B D' B' D L  (12q*)
B B U' L' B L B B U R B' R'  (12q*)
B B U' B' U R B B D B D' R'  (12q*)
B B R D B' D' B B R' U' B U  (12q*)
B B R B R' U' B B L' B' L U  (12q*)
B B R' U' B U B B R D B' D'  (12q*)
B B R' B' R D B B L B L' D'  (12q*)
B B D L B' L' B B D' R' B R  (12q*)
B B D B D' R' B B U' B' U R  (12q*)
B B D' R' B R B B D L B' L'  (12q*)
B B D' B' D L B B U B U' L'  (12q*)
B B L U B' U' B B L' D' B D  (12q*)
B B L B L' D' B B R' B' R D  (12q*)
B B L' D' B D B B L U B' U'  (12q*)
B B L' B' L U B B R B R' U'  (12q*)
B' U U R' U' R B U U L U L'  (12q*)
B' U U R' F' U F U U R B U'  (12q*)
B' U R B B D B D' R' B B U'  (12q*)
B' U R B' R D' R' D B R' B U'  (12q*)
B' U R B' R' B U R' U' R B U'  (12q*)
B' U R B' R' B U' B L U' L' U  (12q*)
B' U R B' R' B B U' L' B L B'  (12q*)
B' U R' U F R' F' R U' R B U'  (12q*)
B' U R' U' R B U' B L' B' L U  (12q*)
B' U B U U L F U' F' U U L'  (12q*)
B' U B U' L U' F' L F L' U L'  (12q*)
B' U B U' L' B L B' U' L U L'  (12q*)
B' U B U' L' B L' D L D' B' L  (12q*)
B' U B U' L' B B D' B' D L B'  (12q*)
B' U B' R B R' U' B U' L U L'  (12q*)
B' U B' R' U R U' B U' L' B L  (12q*)
B' U' L U L L B D L' D' L L  (12q*)
B' U' L U L' B L' D' B D B' L  (12q*)
B' U' L U L' B' U B U' L' B L  (12q*)
B' U' L U' F U F' L' U L' B L  (12q*)
B' U' L L F' L' F U L L B L  (12q*)
B' U' L' B L B B U R B' R' B'  (12q*)
B' U' B L U U F U F' L' U U  (12q*)
B' U' B L U' L F' L' F U L' U  (12q*)
B' U' B L U' L' U B L' B' L U  (12q*)
B' U' B L' B D L' D' L B' L U  (12q*)
B' U' B B L' D' B D B B L U  (12q*)
B' U' B' U R B B D B D' R' B'  (12q*)
B' R R D' R' D B R R U R U'  (12q*)
B' R R D' F' R F R R D B R'  (12q*)
B' R D B B L B L' D' B B R'  (12q*)
B' R D B' D L' D' L B D' B R'  (12q*)
B' R D B' D' B R D' R' D B R'  (12q*)
B' R D B' D' B R' B U R' U' R  (12q*)
B' R D B' D' B B R' U' B U B'  (12q*)
B' R D' R F D' F' D R' D B R'  (12q*)
B' R D' R' D B R' B U' B' U R  (12q*)
B' R B R R U F R' F' R R U'  (12q*)
B' R B R' U R' F' U F U' R U'  (12q*)
B' R B R' U' B U B' R' U R U'  (12q*)
B' R B R' U' B U' L U L' B' U  (12q*)
B' R B R' U' B B L' B' L U B'  (12q*)
B' R B' D B D' R' B R' U R U'  (12q*)
B' R B' D' R D R' B R' U' B U  (12q*)
B' R' U U F' U' F R U U B U  (12q*)
B' R' U R U U B L U' L' U U  (12q*)
B' R' U R U' B U' L' B L B' U  (12q*)
B' R' U R U' B' R B R' U' B U  (12q*)
B' R' U R' F R F' U' R U' B U  (12q*)
B' R' U' B U B B R D B' D' B'  (12q*)
B' R' B U R R F R F' U' R R  (12q*)
B' R' B U R' U F' U' F R U' R  (12q*)
B' R' B U R' U' R B U' B' U R  (12q*)
B' R' B U' B L U' L' U B' U R  (12q*)
B' R' B B U' L' B L B B U R  (12q*)
B' R' B' R D B B L B L' D' B'  (12q*)
B' D D L' F' D F D D L B D'  (12q*)
B' D D L' D' L B D D R D R'  (12q*)
B' D L B B U B U' L' B B D'  (12q*)
B' D L B' L U' L' U B L' B D'  (12q*)
B' D L B' L' B D L' D' L B D'  (12q*)
B' D L B' L' B D' B R D' R' D  (12q*)
B' D L B' L' B B D' R' B R B'  (12q*)
B' D L' D F L' F' L D' L B D'  (12q*)
B' D L' D' L B D' B R' B' R D  (12q*)
B' D B D D R F D' F' D D R'  (12q*)
B' D B D' R D' F' R F R' D R'  (12q*)
B' D B D' R' B R B' D' R D R'  (12q*)
B' D B D' R' B R' U R U' B' R  (12q*)
B' D B D' R' B B U' B' U R B'  (12q*)
B' D B' L B L' D' B D' R D R'  (12q*)
B' D B' L' D L D' B D' R' B R  (12q*)
B' D' R R F' R' F D R R B R  (12q*)
B' D' R D R R B U R' U' R R  (12q*)
B' D' R D R' B R' U' B U B' R  (12q*)
B' D' R D R' B' D B D' R' B R  (12q*)
B' D' R D' F D F' R' D R' B R  (12q*)
B' D' R' B R B B D L B' L' B'  (12q*)
B' D' B R D D F D F' R' D D  (12q*)
B' D' B R D' R F' R' F D R' D  (12q*)
B' D' B R D' R' D B R' B' R D  (12q*)
B' D' B R' B U R' U' R B' R D  (12q*)
B' D' B B R' U' B U B B R D  (12q*)
B' D' B' D L B B U B U' L' B'  (12q*)
B' L U B B R B R' U' B B L'  (12q*)
B' L U B' U R' U' R B U' B L'  (12q*)
B' L U B' U' B L U' L' U B L'  (12q*)
B' L U B' U' B L' B D L' D' L  (12q*)
B' L U B' U' B B L' D' B D B'  (12q*)
B' L U' L F U' F' U L' U B L'  (12q*)
B' L U' L' U B L' B D' B' D L  (12q*)
B' L L U' F' L F L L U B L'  (12q*)
B' L L U' L' U B L L D L D'  (12q*)
B' L B L L D F L' F' L L D'  (12q*)
B' L B L' D L' F' D F D' L D'  (12q*)
B' L B L' D' B D B' L' D L D'  (12q*)
B' L B L' D' B D' R D R' B' D  (12q*)
B' L B L' D' B B R' B' R D B'  (12q*)
B' L B' U B U' L' B L' D L D'  (12q*)
B' L B' U' L U L' B L' D' B D  (12q*)
B' L' D D F' D' F L D D B D  (12q*)
B' L' D L D D B R D' R' D D  (12q*)
B' L' D L D' B D' R' B R B' D  (12q*)
B' L' D L D' B' L B L' D' B D  (12q*)
B' L' D L' F L F' D' L D' B D  (12q*)
B' L' D' B D B B L U B' U' B'  (12q*)
B' L' B D L L F L F' D' L L  (12q*)
B' L' B D L' D F' D' F L D' L  (12q*)
B' L' B D L' D' L B D' B' D L  (12q*)
B' L' B D' B R D' R' D B' D L  (12q*)
B' L' B B D' R' B R B B D L  (12q*)
B' L' B' L U B B R B R' U' B'  (12q*)
```

In total they are 1440 identities.

If we look at positions generated by the first half of each identity we can split them as follow:

1440 = 1248 + 96 + 96

Group A : 1248 are positions that StartsWith=1 and EndsWith=1

Group B : 96 are positions that StartsWith=1 and EndsWith=2

Group C : 96 are positions that StartsWith=2 and EndsWith=1

Each position from group B merges with a position from group C to form one identity. The duplicate position formed this way has StartsWith=3 and EndsWith=3 and they are 96 of them.

The recurrent formula based on trivial identities predicts 96 positions that has StartsWith=3 and EndsWith=3. Your table however shows that they are 192 of them. This is now explained by the fact that Group B and Group C merge their positions to add the new 96 positions with StartsWith=3 and EndsWith=3.

The recurrent formula based on trivial identities predicts 0 position that has StartsWith=4 and EndsWith=4. Your table however shows that they are 24 of them. The reason is that group A which is:

Group A : 1248 are positions that StartsWith=1 and EndsWith=1

can be splitted as follow:

1248 = 1152 + 96

Group D : 1152

Group E : 96

Each 4 positions in group E combine to create a position that has StartsWith=4 and EndsWith=4. 96/4 is 24.

The positions in Group D, combine in pairs to form 576 positions that have StartsWith=2 and EndsWith=2. Your table says that they are 16992 positions with StartsWith=2 and EndsWith=2. The recurrent formula predicts however 16416 which is exactly 576 shorter.

So as a summary we have:

1440 = 1248 + [96 + 96] = 1152 + 96 + [96 + 96] = 2*576 + 4*24 + 2*96

out of the 2*576 positions, only 576 will be counted and 576 are duplicates. They have StartsWith=2 and EndsWith=2

out of the 4*24 positions, only 24 will be counted and 3*24 are duplicates. They have StartsWith=4 and EndsWith=4

out of the 2*96 positions, only 96 will be counted and 96 are duplicates. They have StartsWith=3 and EndsWith=3

So the missing positions are:

576 + 3*24 + 96 = 744

which is :

879,624 [What the trivial recurrent formula predicts] - 878,880 [The real number]

Now that I understand how the non trivial identities relate to the missing positions at level 6, I am trying to figure out if there is a way I can come up with a new recurrent formula that takes into account those non trivial identities.

Based on my initial tries this seemed to be a hard task...

### Identities at higher level

Great! Thanks a lot.

I did download your QTM solver but I am not sure what is the input to it to generate the above list?

Also is there an argument option to your QTM solver to ignore symmetry? as I would like to generate the full list without having to apply the 48 symmetries afterwards...

And is it possible to use your QTM optimal solver to find identities at higher levels? The next level 7 for example? If that is possible, what would be the input to the solver?

Thanks a lot.

### I used U U' for the input. T

I used U U' for the input.
There yet is no option to ignore symmetries or to search for suboptimal solutions, but you are right, I should add something like this.

### While I am happy your code ge

While I am happy your code generates the list it generates for the U U' input I am surprised that it does!

I would never have thought to give such an input to it, since the optimal sequence for U U' is do nothing instead of a 12 length sequence...

In any case thank you for your code, I was able to tweak it to ignore symmetries.

I was not able to make it search for suboptimal solutions. If you ever do that please let me know as I am very interested in non trivial identities at higher levels.

I wonder if you have any idea how much time it would take a modified code to generate all of them?

Thanks a lot

### It is not difficult to add su

It is not difficult to add support for suboptimal solutions. Hope I can do this at the weekend. I think you can count the time in minutes to generate 13 move and 14 move nontrivial identities.

### In fact I am not able to gene

In fact I am not able to generate suboptimal solutions. My change of the code generated identities like:

U U R U R' F' U U L' U' L F (12q*)
R U U R U R' F' U U L' U' L F R' (14q*)
R' U U R U R' F' U U L' U' L F R (14q*)
F F U U R U R' F' U U L' U' L F' (14q*)
F' U U R U R' F' U U L' U' L F F (14q*)
L U U R U R' F' U U L' U' L F L' (14q*)
L' U U R U R' F' U U L' U' L F L (14q*)
B U U R U R' F' U U L' U' L F B' (14q*)
B' U U R U R' F' U U L' U' L F B (14q*)

But these 14q* identities are not trivial since they have the 12q* identity in the middle...

I look forward to your change which hopefully will generate only non trivial identities.

Thanks

### Sorry, but I would call the i

Sorry, but I would call the identities above all suboptimal. How do you define what is a trivial identitiy and what is not?
Are these 14 move identities above trivial because they are a conjugation of some 12 move identities? But then

F*U U R U R' F' U U L' U' L F*F' which is

F U U R U R' F' U U L' U' L

is trival too. But you count it as nontrivial.
My program change will indeed provide all the (14q*) identities above!

### F*U U R U R' F' U U L' U' L F*F'

I want to discuss Herbert's example of F*U U R U R' F' U U L' U' L F*F', where a 14q identity has been constructed from a 12q identity in a trivial way.  Several other messages in this thread have also mentioned this issue.

My current depth first God's Algorithm enumerator includes code to calculate StartsWith and EndsWith because of an issue that arose on the old Cube-Lovers list.  Namely, is the inverse of a local maximum necessarily also a local maximum?  My program was able to determine that there are positions which are a local maximum and where the inverse is not a local maximum.  Such positions manifest themselves in my program when |StartWith(x)|<12 and |EndsWith(x)|=12.

Having identified such positions, it occurred to me (and I think I remember posting this to Cube-Lovers) that given such a position, a longer local maximum could always be constructed.  Namely, left multiply such an x by a quarter turn q such that the length of qx is one more than the length of x.  The existence of such a q is guaranteed by the fact that |StartWith(x)|<12, and qx is guaranteed to be a local maximum.  In fact, the process can be repeated over and over again to yield longer and longer local maxima, and the process only stops when |StartWith(x)|=12 .

Well, the exact same concept applies whether x is a local maximum or not.  Let x be any position such that |StartWith(x)|<12.  Left multiply x by a quarter turn q such that the length of qx is one more than the length of x.  It is guaranteed that EndsWith(qx) ⊇ EndsWith(x) and therefore that |EndsWith(qx)| ≥ |EndsWith(x)|.  It seems to me that when EndsWith(qx) = EndsWith(x), then we are deriving longer identities in only in the aforementioned trivial way, namely I=qxx'q'.  However, it also seems to me that when EndsWith(qx) ⊃ EndsWith(x), we can create a new and longer identity which is not trivially related to the shorter one.

The longer non-trivial identity arises as follows.  We suppose that r is a quarter turn that is not contained in EndsWith(x) and that r is contained in EndsWith(qx), fulfilling the condition that EndsWith(qx) ⊃ EndsWith(x).  Denote qx as y, remembering that EndsWith(y) contains r and that EndsWith(x) does not contain r.  Now, we write I=(qx)(x'q') as I=(qx)y' and choose a maneuver for y' that begins with the quarter turn r' and choose a maneuver for xq that does not end with the quarter turn r.  Such maneuvers are guaranteed to exist.  For example, a maneuver for qx can end with any quarter turn contained in EndsWith(x), and r is not in EndsWith(x).  Or vice versa, choose a maneuver for y' that does not begin with the quarter turn r' and choose a maneuver for xq that does end with the quarter turn r.

So I think here is the bottom line.  Let I=xx' be a non-trivial identity.  Non-trivial means that there is a way to write x and x' so that they don't simply collapse on each other from the middle.  Then, I=(qx)(x'q') may or may not be considered to have been derived from I=xx' in a trivial way.  If |EndsWith(qx)|=|EndsWith(x)|, then the derivation is trivial.  If |EndsWith(qx)|>|EndsWith(x)|, then there is a non-trivial derivation of a longer identity.

Note that if this is a reasonable approach, then it has an interesting implication for antipodes and indeed for all local maxima.  That is, non-trivial identities can be derived from some local maxima, namely from "short" local maxima that can't be derived from shorter local maxima.  But if a local maximum x can be derived from a shorter local maximum, then x can only be used to produce a trivial identity.

### Yes, I have the same feeling,

Yes, I have the same feeling, that the longer a local maximum is the less the probability it can give rise to a non trivial identity...

When you say:

the process can be repeated over and over again to yield longer and longer local maxima, and the process only stops when |StartWith(x)|=12 .

You are assuming that left multiplying x by q will yield the following:

|qx|=|x|+1

I do not see however why this should be true. It is very possible that multiplying x by q will yield a position lower than x... Or am I missing something?

If however what you are saying is true, in such a way that from a local maximum x with |StartWith(x)|<12 we can all the time get a new local maximum then looking at your table:

``` 13   1   1     81865248532   3929530575552
13   1   2     13402102133    643299207408
13   1   3       997279934     47869325232
13   1   4        42665046      2047739232
13   1   5          818454        39284064
13   1   6          229038        10972032
13   1   7           14457          693936
13   1   8           21816         1044144
13   1   9             383           18288
13   1  10            1075           51312
13   1  11              88            4224
13   1  12              21            1008
```

if we take a local maximum out of the available 1008, we can construct a new local maximum using your method that will have a length of 13+11=24 and |StartsWith(x)| = |EndsWith(x)| = 12.

if we take the 4224 positions that have |StartsWith(x)| = 1 and |EndsWith(x)| = 11 and right multiply each by its missing move, we will get new positions, if out of those positions we look for positions that have |StartsWith(x)| = 1 and |EndsWith(x)| = 12, we can construct new local maximums out of them. The final local maximum will have a length 14+11=25. if out of those positions we look for positions that have |StartsWith(x)| = 1 and |EndsWith(x)| = 11, we can also multiply each by its missing move, and look for local maximums out of the resulting positions. Once we find them, we apply your method to get a local maximum that has a length 15+11=26.

Finding a new position that is 26q* will be quite an achievement.

There is nothing that stops us from continuing such operations and we might even find a longer position!

I know that your program gets slow at 14q but if you work only on a smaller set of positions like the following for example:

``` 13   1   6          229038        10972032
13   1   7           14457          693936
13   1   8           21816         1044144
13   1   9             383           18288
13   1  10            1075           51312
13   1  11              88            4224
13   1  12              21            1008
```

then probably it will be faster yielding longer local maximums in a reasonable time.

So the algorithm for finding a new 26q* can be as follow:

for p in above_subset; do
right multiply p by all possible 2q* processes to get a new_p
if (new_p is a local maximum)
{
generate new local maximums using your method until you reach the last one last_local_maximum
exit the loop
}
done

Once the above loop return and assuming that a new_p that is local maximum was found, then we have:

|last_local_maximum| = 13 + 2 + 11 = 26

Of course if we have a subset of 14q* we can multiply by 1q* only and probably ending with a position that is:

|last_local_maximum| = 14 + 1 + 11 = 26

If the above is proven to be a fast program that can generate such positions, we can get greedy and go for attempts like:

|last_local_maximum| = 13 + 3 + 11 = 27

where we take the 13q* subset, right multiply it by all possible 3q*, look for a local maximum out of the resulting positions that is 16q*, then apply your method to that local maximum to get a 27q* position.

I hope I am not too optimistic.

### Another Clarification

> if we take a local maximum out of the available 1008,
> we can construct a new local maximum using your method
> that will have a length of 13+11=24 and |StartsWith(x)| = |EndsWith(x)| = 12.

For a position x of length 13 such that |StartsWith(x)|=1 and |EndsWith(x)|=12, the only guarantee is that it is possible to create a local maximum of length 14.  There is no guarantee of being able to go any further than that.

One way to think of it is that there is no way in general to guarantee that |StartsWith(qx)| is not equal to 12, in which case you couldn't go again.

Another way to think about it is to try to go directly from position x of length 13 to position sx of length 15 all in one go without first making a position of length 14.  Namely, pick an s of length 2.  But the length of sx could in general be 15 or 13 or 11, and I can't think of a way to assure that there exists an s that will yield a length of 15 for sx.  If |StartsWith(x)|<12 for |x|=13, we can guarantee that there exists q such that |qx|=14, but I can't think of a way to guarantee anything more than that.

### Yes I know that there is no g

Yes I know that there is no guarantee to go further once you move one level up.

But what I am suggesting is to do an exhaustive search, by taking all local maximums of length 13 such that |StartsWith(x)|=1, and apply your method to go one level up. Out of the resulting positions look for positions that have |StartsWith(x)| < 12 and repeat the whole process.

Of course it is possible that all generated positions will have |StartsWith(x)| = 12 in which case the program will simply stop. But who knows it might find new positions each time with |StartsWith(x)| < 12 that would allow it to go one level up each time...

### A Clarification

> You are assuming that left multiplying x by q will
> yield the following:

> |qx|=|x|+1

> I do not see however why this should be true. It is very
> possible that multiplying x by q will yield a position
> lower than x... Or am I missing something?

In general, you are correct that |qx| can be either |x|+1 or |x|-1.  But the supposition is that |StartsWith(x)|<12.  Under those circumstances, there is by definition at least one q such that |qx|=|x|+1.  Simply choose such a q with which to left multiply x.

### I see. Thanks. Mathematica

I see. Thanks.

Mathematically this can be proven as follow:

|StartsWith(x)|<12 => |EndsWith(x')|<12

Since |EndsWith(x')|<12 we can find a move A such that |x'A| = |x'| + 1

We then have |A'x| = |x'A| = |x'| + 1 = |x| + 1. The chosen q is simply A'.

### Distinction between Physical Models and Mathematical Models

I think your proof is correct but really unnecessary.  The issue arises because on a physical cube it is possible to right multiply a position by a quarter turn but it is not possible to left multiply a position by a quarter turn, or at least not easily or obviously.  After right multiplying a position x by q you have the position xq, and xq corresponds simply to performing the move q on a physical cube that was already in the position x.  There is not such a straightforward correspondence to create the position qx on a physical cube.

However, with a mathematical or computational model of the cube, it is just as easy to calculate qx as it is to calculate xq.  That's why if |StartsWith(x)|<12, we know that there exists a quarter turn q such that |qx|=|x|+1.

But what if we really did want to model the multiplication qx on a real, physical cube.  I can think of two ways to do it.  One way would be to begin with a cube in the Start state rather than a cube in position x.  From the Start state, perform the chosen quarter turn q, and then perform a maneuver that would yield the position x.

The other way would be to begin with a cube in position x and to do a little disassembly of the cubies.  Having done so, move the cubies with respect to each other in the way indicated by the move q and then reassemble the cubies back into the cube.  For example, suppose q is the quarter turn F.  We normally think of F as moving the contents of the fl cubicle to the fu cubicle, the contents of the fu cubicle to the fr cubicle, the contents of the fr cubicle to the fd cubicle, and the contents of the fd cubicle to the fl cubicle (and similarly for the corners).  To get the effect of left multiplying by F, disassemble the fl, fu, fr, and fd cubies, irrespective of where they are currently located on the cube.  Move the fl cubie to where the fu cube was, move the fu cubie to where the fr cubie was, etc., and reassemble the cube.  And similarly disassemble and move the corner cubies involved in the quarter turn F.

### I consider the identities abo

I consider the identities above as trivial because they are produced using lower identities.

This means that if I know all identities at lower levels I will be able to deduce them.

Non trivial identities however can not be deduced from previous identities... They are really something new that says something new about the cube group. Trivial identities say nothing about the cube group other than what is already said by lower identities...

This means also that trivial identities are not responsible for any missing positions at half their length. The missing positions at half their length are all explained by lower identities.

F*U U R U R' F' U U L' U' L F*F' is a trivial identity of length 14 and is not responsible for any missing positions at length 7.

F U U R U R' F' U U L' U' L is a non trivial identity of length 12 and is responsible for missing positions at length 6.

Of course the two sequences are the same position but when we talk about identities we are talking about sequences not about positions since all identities describe the same position which is START.

If you take a look at Jerry's example:

FFBBRRLLUUDD FFBBRRLLUUDD is a non trivial identity of length 24

However

FFBBRRLLUUDD DDUULLRRBBFF is a trivial identity of length 24

I hope I explained my self well and I really hope there is a way to produce only non trivial identities...

### Identities and Commutivity

Some thoughts about non-trivial identities. If a product is communitive then an identity may constructed by:

A][B] = [B][A]
[A][B] x [A'][B'] = I

It is not unusual that turn sequences may be divided into two sections which commute. For example:

( F U' R' U' R U U F' R' F U' F' ) ( R ) = ( R ) ( F U' R' U' R U U F' R' F U' F' )

Thus:

( F U' R' U' R U U F' R' F U' F' ) ( R ) ( F U F' R F U' U' R' U R U F' ) ( R' ) = I

also

( U B' F' U U B F ) ( R L U' U' L' R' U ) = ( R L U' U' L' R' U ) ( U B' F' U U B F )

Thus:

( U B' F' U U B F ) ( R L U' U' L' R' U ) ( F' B' U' U' F B U' ) (U' R L U U L' R' ) = I:

Would these examples be considered trivial identities?

### I would consider them as non

I would consider them as non trivial because they can not be deduced from lower identities.

### Commutitive Identities

So, the above aren't trivial but higher identities construct as:

• q a b = q b a
• a b q = b a q
• q a b g = q b a g

would be considered trivial because they would be constructed from a known identity: a b = b a?

### Correct. At least that is my

Correct. At least that is my understanding of non trivial identities.

I think to be mathematically correct, a non trivial identity is a simple cycle.

### I was able to generate subopt

I was able to generate suboptimal solutions. Thanks. Here are some initial results:
```levels patterns identities
12       32        1440
14      766       28608
16    10821      385314
18   128211     4629960
20  1481522           :
:         :           :
:         :           :
```

The questions that come to my mind are:

What are the longest non trivial identities?

How do they relate to the antipodes?

Is an antipode position half the sequence of a non trivial identity necessarily?

Is there a way to estimate the time needed to generate all non trivial identities at all levels?

Have been there any attempts to come up with a new recurrent formula that takes into account identities at level 12?

### Antipodes and Non-trivial Identities

I would suggest that an antipode would necessarily yield only a trivial identity. I think we are still talking about the quarter turn metric here, and in the quarter turn metric an antipode is necessarily a local maximum. Therefore, the EndsWith value for an antipode in the quarter turn metric is necessarily {F,F',B,B',R,R',L,L',U,U',D,D'}. Any process you might combine an antipode with, such as its own inverse to create an identity, will include a sequence in the middle such as FF' or D'D that trivially collapses into a shorter sequence.

I know that Dan Hoey has worked on a recurrent formula that takes into account identities of length 12q, and I know that I have. Others may have as well. I don't know of anybody that has succeeded.

### Backwards

With apologies, I think I got this exactly backwards.

Consider xx'=I where x is a local maximum.  I was thinking that because x is a local maximum, we can therefore always choose an optimal process for x that has a last move that will make xx' collapse in a trivial fashion.  That's a true statement, of course, but it's of no consequence.  That's because if x is any position whatsoever, we can always choose an optimal process for x that has a last move that will make xx' collapse in a trivial fashion.

The proper question is the following.  Given a position x, can we choose an optimal process for x that has a last move that will not make xx' collapse in a trivial fashion?  For example, for the position whose only optimal process is UR', the only way to write xx'=I is UR'RU' which collapses in a trivial fashion.  For an example in the other direction (and has recently been discussed), for the position that has the optimal processes FF and F'F', we have two ways to write xx'=I that don't collapse in a trivial fashion, namely FFFF=I and F'F'F'F'=I.

And as a longer example, Pons Asinorum is a local maximum that gives us many ways to write a 24q identity that doesn't collapse in a trivial fashion, for example (FFBBRRLLUUDD)(FFBBRRLLUUDD)=I.  So I think that the correct thing to say is that any local maximum can be used to produce a non-trivial identity that is twice as long as the local maximum itself, and any antipode is a local maximum.

### I think your example of Pons

I think your example of Pons Asinorum shows that a local maximum can be used to produce a non trivial identity.

I do not see however any reason why the following statement should be true:

"ANY local maximum can be used to produce a non trivial identity"

### Hopefully, I haven't messed u

Hopefully, I haven't messed up again, but here's the idea. A local maximum can end with any quarter turn.  Therefore, its inverse can begin with any quarter turn.  So we write our identity as xx'=I using minimal processes for x and x' and where x is a local maximum, we end x with any arbitrary quarter turn (such as F), and then we start x' with any of the eight quarter turns that are orthogonal to F, namely R, R', L, L', U, U', D, or D'.  Therefore, the middle of the identity cannot collapse in a trivial way.  The identity cannot collapse in a trivial way anywhere else because the processes for x and x' are minimal.

I am not sure about your argument and I think the following is a counter example:

x is a local maximum
x=abF where abF is an optimal sequence for x
x'=Rcd where Rcd is an optimal sequence for x'

so we have:

xx'=abFRcd=I

So here we have xx'=I constructed in such a way that it cannot collapse in a trivial way in the middle or anywhere else exactly as you required. But let's say that we have chosen a,b,c and d in such a way that we have:

bFRc=I

this means that xx' is constructed using two lower identities and hence is a trivial identity even though it has all your conditions.

I think the question should be asked as follow:

If x is a local maximum , is there necessarily an identity AB'=I where A=x such that A and B are optimal sequences to x and have no position in common except x?

In other words, is there an identity that can be represented as:

```
/\
/  \
/    \
/      \
/        \
/          \
START/            \x
\            /
\          /
\        /
\      /
\    /
\  /
\/

```

or is it that all identities AB=I can be represented using two or more lower identities, something like:

```

/\
/\     /  \
/  \   /    \
START/    \C/      \x
\    / \      /
\  /   \    /
\/     \  /
\/
```

in such a way that A and B have a common position C before x.

The question can also be asked as follow:

if x is a local maximum, is there a simple cycle linking x to START. The definition of a simple cycle is here:

http://en.wikipedia.org/wiki/Path_(graph_theory)

### Simple Cycle or Unique Simple Cycle and Symmetry

Ok, let's assume that the existence of a simple cycle is the definition we are looking for.  Unless I'm mistaken, the identity maneuver I gave in my Pons Asinorum example is a simple cycle.  What my example is not is that it is not a unique simple cycle.  Are you looking for positions that have a unique simple cycle?

An identity maneuver for Pons Asinorum obviously can be performed in a gazillion different ways, many of which do not yield a simple cycle.  But I believe Pons Asinorum can be performed in lots of ways that do yield a simple cycle.

I also think symmetry intrudes into this discussion.  The identity maneuver I gave for Pons Asinorum does not double back on itself at any point, so it satisfies the definition of a simple cycle.  But the identity maneuver I gave does have the following characteristic.  Think of the maneuver as a simple path from Start to Pons Asinorum followed by another simple path from Pons Asinorum back to Start.  Then, every vertex on the path from Pons Asinorum back to Start in my example is an M-conjugate of a vertex on the path from Start to Pons Asinorum.  So ignoring symmetry, there is a simple cycle.  But taking symmetry into account, the return path from Pons Asinorum to Start in my example doubles back upon itself at every vertex and it's hard to think of that as a simple cycle.  In a certain sense, such a return path is just as trivial as something like I=FFF'F'.

### Yes I agree, the identity man

Yes I agree, the identity maneuver you gave for Pons Asinorum is a simple cycle and I do consider it as non trivial.

What I am looking for is an answer to the following question:

Do ALL local maximums give rise to a simple cycle?

I am specially interested in antipodes... Is there always a simple cycle between START and an antipode?

If this is true it means that the longest non trivial identity has a length twice the diameter. If this is false the question becomes what is the longest non trivial identity.

I am not considering symmetry here, and that is why I changed Koceimba's code to generate all identities regardless of symmetry.

Unfortunately I was not able to generate non trivial identities only and the list I generated contained many identities that can simply deduced from lower identities...

### Yes I was talking about QTM.

Yes I was talking about QTM.

Thanks for your explanation. I understand now why an antipode can not be the result of a non trivial identity.