## Small subgroups and cosets

Submitted by brac37 on Sat, 10/16/2010 - 18:29.Square group table for FTM: Level | | SO | GO | inv | SO+inv | GO+inv | --------+--------+--------+--------+--------+--------+--------+ 0 | 1 | 1 | 1 | 1 | 1 | 1 |

## Cubic Symmetry Cycle Representations

Submitted by B MacKenzie on Fri, 10/08/2010 - 19:33.In responding to comments to a previous post it became of interest to represent cube states and cubic symmetry elements as facelet permutations in disjoint cycle form appropriate for GAP. I wrote a routine to dump facelet representations in disjoint cycle form and produced a table of the cubic symmetry group in cycle notation. It occured to me that this table might be of use to readers of this forum.

I number the cube facelets in the order they occur in the Singmaster-Reid identity configuration string:

12 34 56 78 90 12 34 56 78 90 12 34 567 890 123 456 789 012 345 678 UF UR UB UL DF DR DB DL FR FL BR BL UFR URB UBL ULF DRF DFL DLB DBR

The Up facelet of the Up-Front cubie is numbered 1 on through to the Right facelet of the Down-Back-Right cubie which is numbered 48. With this numbering the face turns are represented by the permutations:

## God's Algorithm out to 15f*

Submitted by tscheunemann on Sun, 08/15/2010 - 14:17.d (mod M + inv)* mod M + inv mod M positions --- --------------- ---------------- ---------------- ----------------- 0 1 1 1 1 1 2 2 2 18 2 8 8 9 243 3 48 48 75 3240 4 517 509 934 43239 5 6359 6198 12077 574908 6 81541 80178 159131 7618438 7 1067047 1053077 2101575 100803036 8 14034826 13890036 27762103 1332343288 9 184907170 183339529 366611212 17596479795 10 2437753739 2419418798 4838564147 232248063316 11 32135500721 31909900767 63818720716 3063288809012 12 423434369503 420569653153 841134600018 40374425656248 13 5575030719304 5538068321152 11076115427897 531653418284628 14 73286135656774 72805484795034 145610854487909 6989320578825358 15 957963000510751 951720657137855 1903440582318440 91365146187124313

## Slice turn metric, anyone?

Submitted by rokicki on Thu, 08/12/2010 - 22:51.dist mod M+inv mod M positions 0 1 1 1 1 4 4 27 2 13 19 501 3 150 236 9175 4 1920 3642 164900 5 31341 61457 2912447

## God's Number is 20

Submitted by rokicki on Sun, 08/08/2010 - 15:58.With about 35 CPU-years of idle computer time donated by Google, a team of researchers has essentially solved every position of the Rubik's Cube™, and shown that no position requires more than twenty moves.

This was a joint effort between Morley Davidson, John Dethridge,

Herbert Kociemba, and Tomas Rokicki.

More details are posted at http://cube20.org/.

## C3v Three Face Group

Submitted by B MacKenzie on Thu, 08/05/2010 - 23:42.
In a previous thread the C_{3v }
Three Face Group (RUF group, etc.) was discussed.
I have since been fooling around with the group and tried my hand at writing a coset solver for it.
I thought I might report some results from this.

Here are the states at depth enumerations for the three face edges only group and the three face corners only group:

C_{3v } Three Face Edges Group: States at Depth

## 15f* = 91365146187124313

Submitted by rokicki on Thu, 07/29/2010 - 00:34.This result is from a collaboration between Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki.

More details will be forthcoming in a future announcement.

## Relation between positions and positions mod M in FTM

Submitted by kociemba on Thu, 07/15/2010 - 05:10.## God's Algorithm out to 17q*

Submitted by tscheunemann on Wed, 07/14/2010 - 15:25.d (mod M + inv)* mod M + inv mod M positions -- --------------- --------------- --------------- ----------------- 0 1 1 1 1 1 1 1 1 12 2 5 5 5 114 3 17 17 25 1068 4 135 130 219 10011 5 1065 1031 1978 93840 6 9650 9393 18395 878880 7 88036 86183 171529 8221632 8 817224 802788 1601725 76843595 9 7576845 7482382 14956266 717789576 10 70551288 69833772 139629194 6701836858 11 657234617 651613601 1303138445 62549615248 12 6127729821 6079089087 12157779067 583570100997 13 57102780138 56691773613 113382522382 5442351625028 14 532228377080 528436196526 1056867697737 50729620202582 15 4955060840390 4921838392506 9843661720634 472495678811004 16 46080486036498 45766398977082 91532722388023 4393570406220123 17 426192982714390 423418744794278 846837132071729 40648181519827392

## God's Algorithm out to 16q*

Submitted by tscheunemann on Fri, 07/09/2010 - 05:06.d (mod M + inv)* positions -- -------------- ---------------- 0 1 1 1 1 12 2 5 114 3 17 1068 4 135 10011 5 1065 93840 6 9650 878880 7 88036 8221632 8 817224 76843595 9 7576845 717789576 10 70551288 6701836858 11 657234617 62549615248 12 6127729821 583570100997 13 57102780138 5442351625028 14 532228377080 50729620202582 15 4955060840390 472495678811004 16 46080486036498 4393570406220123

^{*}The (mod M + inv) column means symmetry reduced postions but only considering the edge cube positions. I just included it because I had those numbers. It is not comparable to earlier calculations because it may include duplicate positions of the full cube and is dependent on what cosets are used in the calculation.