# Square subgroup in QTM

Analysis of the 3x3x3 squares group ----------------------------------- branching Moves Deep arrangements (h only) factor loc max (h only) 0 1 -- 0 1 6 6 0 2 27 4.5 0 3 120 4.444 0 4 519 4.325 0 5 1,932 3.722 0 6 6,484 3.356 1 (6 X pattern) 7 20,310 3.132 0 8 55,034 2.709 65 9 113,892 2.069 1,482 10 178,495 1.567 7,379 11 179,196 1.004 25,980 12 89,728 0.501 50,320 13 16,176 0.180 11,328 14 1,488 0.092 912 15 144 0.096 144 ------- ------ 663,552 97,611But I was not able to find a similar analysis in QTM. Does anyone have it? Is any code publicly available that will generate such QTM table as well as the distribution table of its cosets? Thanks.

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### HTM of the square group

I am copying it here for an easy access:

Square subgroup analysis using square moves only and all moves 0 1 1 1 1 1 2 2 2 3 5 5 4 18 18 5 56 62 6 162 214 7 482 693 8 1258 1871 9 2627 4093 10 4094 5394 11 4137 2774 12 2231 620 13 548 4 14 114 15 16 15752 15752

Schreier coset of the square subgroup analysis 0 1 1 1 2 3 3 23 4 241 5 3002 6 38336 7 490879 8 6298864 9 80741117 10 1028869318 11 12787176355 12 140352357299 13 781415318341 14 421980213679 15 330036864 16 17 1357981544340

And yes you are right I am interested in the QTM. The above is for HTM only.

Yes I started a brute force solve using Koceimba's optimal solver. However it is not as fast as I would like as it is solving them one by one. I was hoping I can use a coset solver that solves one coset at a time in a much faster way...

### Radu analyzed a larger group

I don't know if you were trying to solve all elements of the group, or just the 15752 symmetry class representatives, but even that of course is still a rather large number to brute force. Still, I figured the positions were not generally very deep, so I think it should not be too huge a task to do the symmetry class representatives. A coset solver, if you can find an appropriate one or write one, should of course be a lot more efficient.

Searching back in the forum, I found
this analysis of < U,D,L^{2},R^{2},F^{2},B^{2} > by Radu. The squares group is a subgroup of the group Radu analyzed, so all elements of the squares group are represented in Radu's analysis. The antipode of Radu's analysis is not an element of the squares group, and since all elements of the squares group are an even number of quarter turns from solved, that means 22q* is an upper bound for the elements of the squares group.

## Each half-turn is simply 2 quarter turns

That HTM analysis assumes only doing moves that keep the state of the cube within the subgroup (half-turns only). A quarter turn applied to a squares group position always puts the cube outside the squares group. For QTM, you can restrict moves to half-turns only, and count each half-turn as two moves, so you simply double the distance values in the HTM distribution (with the counts remaining the same).

Now, you might be really interested in what the distribution is if you don't restrict the analysis to staying within the subgroup. Removing the restriction can be considered for face turn metric as well as for quarter turn metric. Kunkle/Cooperman's paper provides that distribution for FTM (as well as the half-turns only version) in symmetry reduced terms. Maximum depth is 13 moves.

I'm not aware this has been done for QTM. You could generate all symmetry representatives of the squares group and brute force solve each one with a QTM optimal solver (and then take into account the class size for each representative).

If you also want the distribution for getting a scrambled cube into the squares group, I imagine that's only been done for FTM. (See the Kunkle/Cooperman paper.) That has over a trillion symmetry-reduced positions.