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.

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.

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.

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.

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.

        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

Square group table for FTM:
Level   |        | SO     | GO     | inv    | SO+inv | GO+inv |
--------+--------+--------+--------+--------+--------+--------+
    0   |      1 |      1 |      1 |      1 |      1 |      1 |