## 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

## 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