Discussions on the mathematics of the cube

Three Million Random Positions in the Quarter Turn and Half Turn Metric

Last year I solved one million random cubes in both the half-turn and the quarter-turn metric. Unformtately, the random number generator I used was good old |drand48()|, which is not of the highest quality. This time, I generated 3,000,000 positions using the Mersenne Twister random number generator and solved all of these with a new faster optimal solver. This is the result:
    12h 13h 14h  15h   16h    17h     18h    19h     sum
14q   -   1   2    -     -      -       -      -       3
15q   -   4  19   13     -      -       -      -      36
16q   1  11  47  124   126      -       -      -     309

NxNxN cubes in GAP

Since defining GAP definitions for large cube sizes can be very tedious, I have implemented some GAP code for defining NxNxN cubes. The main function is called GenCube and returns a group representing a cube of the size specified by the parameter n. This function has a 2nd parameter (center_ori) used for odd cubes that allows specifying whether or not you wish to have the orientation of the most central pieces on each face to be considered significant.

The code uses a face-based numbering system. The facelets on the U face are numbered 1 to n2, the L face uses numbers n2 + 1 to 2n2, and the remaining faces are similarly numbered in the order F, R, B, D. For handling orientation of the most central pieces on each face, 18 additional numbers are used, starting at 6n2 + 1.

Optimal parity fix maneuvers on the 4x4x4 cube

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

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

I don't see any explicit reference to this but I've noticed
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

I was forced to update the URL of the site this morning. http://dyndns.org saw fit to shut down access from the homelinux.org domain and I had to scramble to switch the site over to the allowed free domain name.

The new URL is http://cubezzz.dyndns.org/drupal

Note that the numeric ip address will also work 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"

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

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

We are getting tons of bogus accounts from bots using gmail as an email address so I decided to ban signups from using gmail. Old accounts already using gmail will still work. If anyone really wants to get an account using a gmail email address send me an email explaining why you want to join (so I know it's not bot).

Send messages to the admin via cubexyz at gmail dot com


M_R,D Group

Under the God's Algorithm Calculations link to the right of this page there is the following:
        Analysis of the  Group

     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