Discussions on the mathematics of the cube

Welcome to my Blog

Hello, I am Peter Jung, a physicist from Cologne University.

Only a couple of days ago I got very much interested in the Cube. At wikipedia I found a note that the diameter of the Cube group is not yet known, and a link to this site.

Great work!

Sniffing into the problem, it seems to be quite complex. But some ideas that came to me these days I could not find. That's the purpose of this visit: To ask whether attempts have been made along these lines of thought, and if so, what is the outcome. And if not, I would like to contribute some analysis.

Antisymmetry and the 2x2x2 Cube

Someone on the Yahoo forum asked about how to do a 2x2x2 God's algorithm calculation and mentioned the "1152-fold" symmetry for the 2x2x2. I got to looking at some of the messages in the Cube-Lovers archives that Jerry Bryan had made about B-conjugation and the 1152-fold symmetry of the 2x2x2. He found that there were 77802 equivalence classes for the 2x2x2.

I have used antisymmetry to further reduce the number of equivalence classes for the 2x2x2 to 40296. The following table shows the class sizes of these equivalence classes.

class size  class size/24  count
----------  -------------  -----
     24           1            1
     48           2            1
     72           3            3
     96           4            1
    144           6           14
    192           8           11
    288          12           49
    384          16           22
    576          24          337
    768          32            6
   1152          48         3353
   2304          96        36498
                           -----
                  total    40296

I then performed God's algorithm calculations (HTM and QTM) to find the number of equivalence classes at each distance from the solved 2x2x2 cube. The results are given below. Because the 2x2x2 has no centers that provide a reference for the positions of the other cubies, the number of positions of the corners for the 2x2x2 (the only cubies it has) can be considered to be 1/24 the number of positions of the corners of the 3x3x3 (3674160 instead of 88179840). So in the tables below, I use the factor-of-24 reduced numbers for simplicity. The tables further break down the positions with respect to different class sizes.

Odd Permutations of the Cube Shape of Square-1

Results are presented from an exhaustive search on the odd permutations of Square-1 in its solved shape. A maximum of 31 turns (U, D and /) are needed to solve these positions, which may be a new lower bound on the length of God's Algorithm for Square-1, in this metric.

The method I used was suggested by Tom and Silviu's coset searches for the Rubik's Cube: Starting from a cube-shaped, odd-parity position of Square-1, an iterated depth-first search was made for all even-parity cube-shaped positions, with the search being pruned on [shape]x[parity].

God's Algorithm to 13q, Summarized by Symmetry Class

God's Algorithm to 13q has been posted before, but it has not been summarized by symmetry class. The symmetry classes in the table below follow Dan Hoey's taxonomy.
           
|x| Symmetry  Size           Patterns        Positions
     Class     of
                xM


  0      M       1               1               1
      Total                      1               1
      
  1      CR     12               1              12
      Total                      1              12
      
  2      I      48               2              96
         Q       6               1               6

God's Algorithm to 11f, Summarized by Symmetry Class

God's Algorithm to 11f has been posted before, but it has not been summarized by symmetry class. The symmetry classes in the table below follow Dan Hoey's taxonomy.
           
|x| Symmetry  Size           Patterns        Positions
     Class     of
                xM

  0      M       1               1               1
      Total                      1               1
      
  1      CR     12               1              12
         Q       6               1               6
      Total                      2              18
      
  2      I      48               4             192

Dan Hoey's Taxonomy of Symmetry Groups and Shoenflies Symbols

I have received permission to post Dan Hoey's taxonomy of symmetry groups of Rubik's Cube.  Also, I will relate Dan's taxonomy to Shoenflies symbols as implemented in Herbert Kociemba's Cube Explorer.  (Go to http://kociemba.org/cube.htm and then click on Symmetric Patterns.)  To that end, some preparatory comments are in order.

In order to define any terminology for the symmetry groups of Rubik's Cube, it's necessary first to define some terminology for the symmetries of the cube.  To the best of my knowledge, no standard terminology has been adopted by the Rubik's cube community for the symmetries of the cube.  The terminology I'm going to use is very similar to some terminology I have seen before, but I can't remember the reference.  It may have been Christoph Bandelow's book, Inside Rubik's Cube and Beyond.  In any case, if I can find the reference I want to give proper credit.

Solving the 4x4x4 in 68 turns

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

Lower bound using the Edges antipode

One way of solving the cube is using two phases:

1) First solve all the edges [less than 18 moves as proved by Tom Rokicki ].
2) Take it to the cube identity [less than 22 moves as proved by Silviu Radu]. Which gives a maximum of 40 moves.

Is it possible to follow the same logic but use the Edges Antipode instead?. So the two phases become:

1) Solve all the edges by taking them to the Antipode edges instead of the identity edges. [this is guaranteed to be 18 moves].
2) From the edges antipode position take it to the cube identity. [Does anyone know the maximum moves needed for this phase?]. To find this maximum it would take the same computational effort to prove the 22 moves I guess. Let's call this maximum X.

A Little More on Odd and Even Permutations

One of the basic tenents of cube theory (a highly specialized branch of group theory - grin!) is that odd permutations of the corners can only occur with odd permutations of the edges, and that even permutations of the corners can only occur with even permutations of the edges. This fact is one of the three factors that causes the order of the cube group to be smaller than the order of what is sometimes the illegal cube group.

If you take a cube apart and put it back together, you can put it back together in twelve times as many ways as there are positions in the cube group. Twelve is because 3 * 2 * 2 = 12. Three of the ways are because the twist of the last corner cubie you put back together can be set in one of three different ways, only one of which is legal. Two of the ways are because the flip of the last edge cubie you put back together can be set in one of two different ways, only one of which is legal. The last two of the ways are because the corners and edges can by put back together either with the same odd/even parity (legal) or with the opposite odd/even parity (illegal).

X1+X3+X5+...

Hi,

Finding the diameter of God's Algorithm for the 3x3x3 cube is equivalent to finding the length of the sequence:

X0
X1
X2
X3
X4
X5
:
:
Xn

where X0=1; Xn is all positions reachable from the Xn-1 positions.

I have noticed and proved that for any rubik like puzzle , the following identity is true:

X1+X3+X5+....+X2n+1=X0+X2+X4+X6+....+X2n

This can be verified for the 2x2x2 cube sequence for example:

1
6
27
120
534
2,256
8,969
33,058
114,149
360,508
930,588
1,350,852
782,536
90,280
276

Though I have been reading many books and articles about rubik's like puzzles, I never came across the above identity. Did anyone see the above identity somewhere? Thanks.