## Optimal parity fix maneuvers on the 4x4x4 cube

Submitted by Bruce Norskog on Mon, 03/14/2011 - 21:09.I recently investigated optimal reduction parity fixes on the 4x4x4 cube.

First some explanation of terms. A common strategy used in solving the 4x4x4 cube is to solve the center pieces, and then pair up the 12 pairs of edge pieces. The puzzle can then be solved like a 3x3x3 cube by turning only the face layers, except for two possible types of parity conditions that can't normally occur on the 3x3x3. These parity conditions are often called *OLL parity* and *PLL parity* (since whether or not these parity conditions are present typically isn't recognized until attempting to solve the last layer). Since the 4x4x4 is "reduced" to a pseudo-3x3x3 cube, this strategy is generally referred to as *reduction*.

## 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: