I have completed a five-stage analysis of the 4x4x4 cube showing that it can always be solved using at most 68 turns. The analysis used the same five stages that were used in my prior posts where I claimed the 4x4x4 cube can be solved in 79 single-slice turns, or alternatively in 85 twists. The difference in this analysis is that it allows any single layer turn or double layer turn (where the two layers are any two adjacent layers and moved together) to be counted as a "turn." In some prior posts, I referred to these turns as "block turns." So the set of turns about the U-D axis that count as one turn are the following:
  U,U',U²,u,u',u²,(Uu),(Uu)',(Uu)²,
  D,D',D²,d,d',d²,(Dd),(Dd)',(Dd)²,
  (ud'),(u'd),(ud')²

On the old Cube-Lovers list, the terms Starts-with and Ends-with were defined as follows.  For a cube position x, StartsWith(x)=S(x) is the set of all moves with which a minimal maneuver can start and EndsWith(x)=E(x) is the set of all moves with which a minimal maneuver can end.

The concept is much older than Cube-Lovers, of course.  It's obvious that from any position except for Start itself, there must be at least one move which takes the Cube closer to Start.  The set of all such moves is simply the set of inverses of E(x).

Distance   Patterns Unique     Positions
 from      up to Symmetry
Start

  0                     1               1
  1                     2              18
  2                     9             243
  3                    75            3240      
  4                   934           43239      
  5                 12077          574908      
  6                159131         7618438      
  7               2101575       100803036      
  8              27762103      1332343288      
  9             366611212     17596479795      
 10            4838564147    232248063316      
 11           63818720716   3063288809012 

In my posting titled "The 4x4x4 can be solved in 79 moves (STM)," I reported about an analysis I did where the 4x4x4 cube is solved in five stages. In that analysis, a move was considered to be any quarter-turn or half-turn of a single slice.

I have now completed a similar analysis of the 4x4x4 cube where a move is considered to be any quarter- or half-turn twist of the cube, and where a twist is considered to be one portion of the cube (a face layer, or a block consisting of a face layer and the adjacent inner layer) being turned with respect to the rest of the cube. The analysis indicates that any valid position of the 4x4x4 cube can be solved via these five stages using no more than 85 twists.

I wanted to post a number of miscellaneous items about representing the cube, and I will also include a few other related items.

I'll start with the group S₃ as an example.  I will treat the group S₃ as acting on the set {0, 1, 2}.  As I have been doing recently, I'll use the notation (a b c) to represent the permutation 0→a, 1→b, 2→c. 

In this notation, the entirety of S₃ can be listed as follows:

(0 1 2)
(0 2 1)
(1 0 2)
(1 2 0)
(2 0 1)
(2 1 0)

This basic idea, or something very similar to it, is probably the way most people represent the cube in a computer program.  Variations on the theme could include an S₅₄ model, an S₄₈ model, an S₂₄ × S₂₄ model, and some sort of wreath product model.  The most common wreath product model would probably be something like (S₈ wr C₃) × (S₁₂ wr C₂).  In the wreath product model, S₈ and S₁₂ represent the corner cubies and the edge cubies, respectively, and C₃ and C₂ represent the twists of the corner cubies and the flips of the edge cubies, respectively.  Of course, none of these various models are isomorphic to the cube group.  Rather, the cube group is a subgroup of whatever group is chosen as the computer model.