# Relation between positions and positions mod M in FTM

(Nm-Sm)*48 = N - S,

this is because (Nm-Sm)*48 is the number of positions with no symmetry. So we have

N = 48*Nm - (Sm*48 - S) = 48*Nm - C.

The constant C is tabulated below for all levels from 0 to 20:

depth C 0 47 1 78 2 189 3 360 4 1593 5 4788 6 19850 7 72564 8 237656 9 858381 10 3015740 11 9785356 12 35144616 13 122254428 14 436594274 15 1764160807 16 8037257961 17 37547823254 18 95403536079 19 21275288869 20 1140678

## Comment viewing options

### I am quite sure, that the num

kociemba.org/math/symmetrygroups.xls .

The computation of all symmetric positions in FTM was a big task, Silviu Radu did most of the computation for the large subgroups, completed by some Cube Explorer computations to sort out the 20f* from the >=19f* cubes, and I remember he used GAP to find for example the right subgroups for the pruning tables for C3. Together with his results for the large subgroups I was able to compute for each symmetry the number of cubes, which *exactly* have this symmetry, which was not trivial in some cases.

Doing the computation for QTM should be no problem, but I think we should not reinvent the wheel - maybe Silviu can help.

A problem for the antisymmetric positions is not only their size and the missing group property but also the fact, that there are so many different cases, this is combinations of the 33 essentially different symmetry subgroups and (sub)subgroups of index 2.

### Antisymmetric positions

At first, selfinverse positions are anti-symmetric positions whose anti-symmetry is trivial. So the problem is to solve anti-symmetric positions.

A void cube is selfinverse if it becomes a selfinverse cube after some coloring of the centers, where regular colorings of the face centers, which give a Hungarian cube, are allowed, but also colorings that give a mirrored Hungarian cube, which is a popular fake Hungarian cube. Corners that stay on their places need to be unrotated for selfinverse positions. Since a fake cube coloring only gives corners that are flipped and thus unrotated when they stay on their spots, the number of fake cube coloring solutions is about 5 times larger than the number of real cube coloring solutions.

Since only 20 of the 48 symmetries have order <= 2, 7/12 of the found solutions must be discarded. And the number of anti-symmetric positions including selfinverse is about 60 times larger than selfinverse only: 9 antisymmetries give about the same number of solutions as the trivial symmetry and 10 antisymmetries give about 5 times as many solutions.

Just make a big prune system for solving a void cube to a selfinverse one. Next, solve the clean void cube, reducing symmetry in the search path. I think it will take far less time than finding Gods number.

## Very nice

Would be very nice to get all symmetric positions analyzed in the QTM too, using Radu's and your ideas. Not sure who's going to do that, but that might also be a good way to find some "far" QTM positions. (Right now, only one 26q* and two 25q* positions are known, and only a very few

24q* positions [maybe a hundred; not sure.])

And then there's inverse positions, and antisymmetric positions. Of course, these aren't as "nice" as symmetric positions with respect to subgroups. But some day we'll figure it out, I'm sure.