# Some more interesting groups

Here are the results of an exploration of four groups, all of whom leave the edges in some M-symmetric class, for the face turn metric:

Fixes edges:

 d pos M M+inv 0 1 1 1 8 528 11 6 9 360 8 5 10 2521 63 43 11 5220 117 68 12 72008 1541 821 13 217374 4584 2378 14 1084624 22743 11690 15 4905894 102419 51896 16 16944222 353629 178786 17 20029566 418360 212304 18 827386 17502 9419 19 216 7 7 44089920 920985 467424
Pons Asinorums edges:
 d pos M M+inv 6 4 2 2 7 6 1 1 8 78 3 3 9 116 6 6 10 462 12 12 11 5708 130 78 12 32556 690 373 13 151618 3194 1668 14 896527 18790 9605 15 5065624 105810 53641 16 20437568 426585 215786 17 17156739 358427 182222 18 342844 7331 4023 19 70 4 4 44089920 920985 467424
Superflips edges:
 d pos M M+inv 14 4872 103 54 15 51216 1071 548 16 539792 11277 5727 17 6053450 126347 63806 18 29140880 608273 307613 19 8298629 173861 89626 20 1081 53 50 44089920 920985 467424
Pons Asinorums and Superflips edges:
 d pos M M+inv 14 1344 28 14 15 18216 381 200 16 345328 7221 3677 17 4490004 93668 47308 18 30168301 629581 318152 19 9066419 190085 98053 20 308 21 20 44089920 920985 467424
These are the result of a program I wrote that attempts to solve a bunch of cubes at once, that share something in common (in this case, a given position of the edges). It's remarkable to me how much the peaks of the distributions are shifted between the four. If nothing else, this just shows that if there is a distance-21 cube, it does not have the edges in an M-symmetric position.

## Comment viewing options

### Here's the results for the co

Here's the results for the coset that has the edges in Reid's position (quarter turn metric this time):
``` d      pos        M    M+inv
14      864       55       28
16    48104     3016     1524
18  2003896   125521    63317
20 32050973  2006185  1010090
22  9986060   626478   318134
24       22        4        4
26        1        1        1
44089920  2761260  1393098
```

what is "Reid's position" and what is the unique 26q position
that you found? Some of us need these things explained to us...

### Reid's position is U2 D2

Reid's position is

U2 D2 L F2 U' D R2 B U' D' R L F2 R U D' R' L U F' B' (26q*, 21f)

It can be found on Reid's home page.

### There is also a notational co

There is also a notational convention worth mentioning. The given maneuver is listed as 26q*. The 26q means that the maneuver is 26 quarter turns long (or we might also say that it's 26 moves long in the quarter turn metric), sort of like saying that an object is 3in. long for three inches or 5cm. long for five centimeters. The * says that the maneuver has been proven to be minimal in the given metric. The 21f means that this particular maneuver is 21 face turns long (or we might also say that it's 21 moves long in the face turn metric), and the absence of an * says that the maneuver has not been proven to be minimal in the given metric.

The q and f as units of measure originated in Cube-Lovers, and has been fairly widely adopted elsewhere -- including by Cube Explorer. I believe that the * to denote minimal maneuvers originated with Cube Explorer. Whether it originated there or not, it is certainly is used by Cube Explorer, and has been fairly widely adopted elsewhere.

As you watch Cube Explorer run, it displays progressively shorter maneuvers as it finds them. It is fairly common for the program to display (for example) an 18f maneuver and that a short time later the 18f becomes 18f*. The 18f says that an 18f maneuver has been found, but that it may or may not be minimal. When the 18f becomes 18f*, Cube Explorer has managed to prove that the maneuver is minimal. So the absence of the * does not mean that the maneuver is not minimal. It simply means that the maneuver may or may not be minimal, and if it's minimal it has not yet been so proven.

It turns out that the 21f maneuver given above for Reid's position is indeed not minimal in the face turn metric, but the same maneuver is minimal in the quarter turn metric at 26 quarter turns.

### Congratulations to this work!

Congratulations to this work! When are you doing the quarter turn metric? How long did it take to compute each table?

I also wanted to say that is better to call those tables for cosets rather then groups because if we call the group that fixes edges for H. Then you computed idH , g_1H , g_2H , g_3H. And only the first one is actually a group.

### Thanks! I'm doing the quarte

Thanks! I'm doing the quarter turn metric now; it should be done soon. This work took several days (not sure exactly; some of the above were done in less than 24 hours but some took a few days, and I'm using four machines, including two 1999-era Celeron boxes so even that timing is suspect.)

After the quarter-turn metric, I'm thinking of trying the cosets (is that right?) that have UD symmetry; I think there are 28 of them that don't have also M-symmetry. I think if there is a distance-21f or distance-27q position, it is likely it has some symmetry, so these are fertile grounds to mine, and solving 44M cubes at a go helps.

I'm not a mathematician by a long shot. Thanks for helping with the precision!