## The Void Cube in GAP

Submitted by B MacKenzie on Thu, 03/10/2011 - 21:37.I have been amusing myself messing around with GAP and have modeled the void cube. The void cube is a standard cube with indistinguishable center cubie facelets. The void cube may be modeled by the group: < R , U , F , TR , RU , TF > , where the latter three generators are "Tier" or "Tandem" moves of a face and the adjacent middle slice. Note that the generators do not move the DBL cubie. As such, this is a fixed corner cubie model. The DBL cubie provides the necessary frame of reference which defines which face is Up, which face is Right and so forth. The tandem moves are the fixed corner cubie model counterparts of the L , D , B moves in the standard fixed center facelet model--they perform the same rearrangement of the cubies relative to one another.

## PSL(2,7) embeds in the 2x2x2 cube group

Submitted by secondmouse on Wed, 01/05/2011 - 16:47.that the simple group PSL(2,7) occurs naturally as a subgroup

of the 2x2x2 cube group of order 3674160 (well - with a slight

amount of wilful tinkering!). This is the model in which one

of the the 8 cubelets stays fixed.

One way of seeing how it is realised is to view to view the

corner cubelets as a single block, i.e. suppose all three

elements of each corner cubelet have the same colour.

Then taking the following labellings where all of 1 could be

coloured red, all of 2 yellow, etc. (UFR refers to the the

cubelet in the "Up" "Front" "Right" position, etc).

## Site URL changed

Submitted by cubex on Tue, 01/04/2011 - 13:23.The new URL is http://cubezzz.dyndns.org/drupal

Note that the numeric ip address will also work http://204.225.123.154 as I can't guarantee they won't make it necessary to switch to another service in the future as free services tend to disappear.

Let me know at cubexyz at gmail dot com if anything is broken.

Please update your links accordingly.

## The Fifteen Puzzle can be solved in 43 "moves"

Submitted by Bruce Norskog on Wed, 12/08/2010 - 16:43.Of course, it had been previously proved that some positions of the Fifteen Puzzle require 80 moves to solve, but in that work it was assumed that a move only affects one tile at a time. Since people commonly slide up to 3 tiles in the same row or column at once, it seems natural to count such an action as a single move. With this way of counting, which we call the "multi-tile metric," the maximum number of required moves is only 43, and of the 16!/2 = 10,461,394,944,000 valid configurations of the puzzle, there are only 16 antipodes, i.e., positions that actually require 43 moves.

The 16 antipodes include two positions that are mirror-symmetric to themselves. These two positions are those that are obtained by transposing the rows and columns with respect to either diagonal. The other antipodes consist of 7 pairs of positions that are mirror-symmetric with the other. These 14 positions also include 4 pairs of neighboring positions. So only 8 of the antipodes are "strict" antipodes having the property that any move gets you one move closer to the solved state.

## Cross-Check Patterns

Submitted by B MacKenzie on Wed, 12/08/2010 - 13:44.By applying the 24 rotation symmetries to the corner facelets of the cube one may generate the Cross Pretty Pattern Group. These patterns may be arranged into five conjugate classes: the identity cube, six order two 6-cross patterns, eight order 3 6-cross patterns, six order 4 4-cross patterns and three order 2 4-cross patterns.

By applying the 24 Th symmetries to the edge facelets of the cube one may generate the Check (or Checkerboard ) Pretty Pattern Group. These patterns may be arranged into six conjugate classes: the identity cube, pons asinorum, eight order three 6-check patterns, eight order six 6-check patterns, three order two 4-check patterns and three order two 2-check patterns.

## Banning gmail

Submitted by cubex on Thu, 11/18/2010 - 07:57.Send messages to the admin via cubexyz at gmail dot com

Thanks

## M_R,D Group

Submitted by mdlazreg on Wed, 11/03/2010 - 18:27.Analysis of theGroup ------------------------------ Level Number of Time Branching Positions Factor 0 1 0 s -- 1 4 0 s 4 2 10 0 s 2.5 3 24 0 s 2.4 4 58 0 s 2.416 5 140 2 s 2.414

## 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