Notes on Tuple Representation of Rubiks Cube

Let's first explain what we mean when we talk about a tuple representation of a cube, and why this is useful.

Tuple Representation

A tuple representation means, we are representing one possible permutation of the Rubik's Cube using a tuple, ideally a tuple of N items arranged in some particular way.

Now, if we think about how a 3x3 Rubik's Cube or 4x4 Rubik's Revenge is mechanically constructed, we see that the cube consists of:

Rubik's Cube: 26 total (mechanical) pieces

8 corner pieces
12 edge pieces
6 center pieces

Rubik's Revenge: 56 total (mechanical) pieces

8 corner pieces
24 double-edge pieces (12 left-hand, 12 right-hand)
24 center pieces

However, it is important to note that we are not trying to find the minimal representation of the Rubik's Cube, we are simply trying to find a unique representation of a Rubik's Cube. Representing faces requires more information than representing pieces, but it is a lot simpler and accomplishes what we need:

The 3x3 Rubik's Cube has 9 squares on each face, and 6 faces, for a total of 36 squares.
The 4x4 Rubik's Revenge has 16 squares on each face, and 6 faces, for 96 total squares.

Now, if we were looking for a minimal representation, we would utilize the fact that some of these squares are innately linked (for example, the three faces representing a corner piece are always positioned in the same way relative to one another, even though they may move relative to the rest of the pieces on the cube).

However, we simply want a unique representation, so we can represent the state of any 3x3 Rubik's Cube using a 36-tuple, or the state of any 4x4 Rubik's Revenge using a 96-tuple.

Why A Tuple Representation

Finding a tuple representation enables us to study the properties of various move sequences and understand how the cube works.

For example, if we repeatedly apply any sequence of moves to a Rubik's Cube, eventually it will return back to the solved state. To predict how many times a sequence must be applied to a cube to return to the solved state, we can use techniques demonstrated by Donald Knuth in Volume 3 of The Art of Computer Programming (see AOCP) to derive a permutation algebra, factor permutations into cycles, and find the sequence length via the lcm of each cycle length.

Tuple representation: https://github.com/charlesreid1/rubiks-cycles/blob/master/tup.py

4x4 Rubiks Cube Representation

Numbering System

Start with a numbering system for the cube. The nxnxn rubiks cube solver library I'm using (https://github.com/dwalton76/rubiks-cube-NxNxN-solver) implements the following numbering system:

             01 02 03 04
             05 06 07 08
             09 10 11 12
             13 14 15 16

17 18 19 20  33 34 35 36  49 50 51 52  65 66 67 68
21 22 23 24  37 38 39 40  53 54 55 56  69 70 71 72
25 26 27 28  41 42 43 44  57 58 59 60  73 74 75 76
29 30 31 32  45 46 47 48  61 62 63 64  77 78 79 80

             81 82 83 84
             85 86 87 88
             89 90 91 92
             93 94 95 96

corresponding to the following cube state:

         U U U U
         U U U U
         U U U U
         U U U U

L L L L  F F F F  R R R R  B B B B
L L L L  F F F F  R R R R  B B B B
L L L L  F F F F  R R R R  B B B B
L L L L  F F F F  R R R R  B B B B

         D D D D
         D D D D
         D D D D
         D D D D

Now we can write the solved cube as the following 96-tuple:

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96]

Moves as Permutations

Each move of a face - U, D, R, L, F, B, and other two-layer or second-layer moves, as well as sequences of moves - can now be thought of as a permutation of these 96 integers.

I modified the nxnxn rubiks cube library to print out the permutation corresponding to each type of move. For example, here is U:

In [7]: r.rotation_map('U')
Out[7]:
[(1, 13),
 (2, 9),
 (3, 5),
 (4, 1),
 (5, 14),
 (6, 10),
 (7, 6),
 (8, 2),
 (9, 15),
 (10, 11),
 (11, 7),
 (12, 3),
 (13, 16),
 (14, 12),
 (15, 8),
 (16, 4),
 (17, 33),
 (18, 34),
 (19, 35),
 (20, 36),
 (33, 49),
 (34, 50),
 (35, 51),
 (36, 52),
 (49, 65),
 (50, 66),
 (51, 67),
 (52, 68),
 (65, 17),
 (66, 18),
 (67, 19),
 (68, 20)]

Now the starting state of a cube can be written as the above tuple, and rotations of various faces can be written as permutations.

Once we can write a sequence of moves as a permutation of 96 integers, we can start to dig deeper into the effect that it has on the cube state.

Permutation Algebra

$a = \bigl(\begin{smallmatrix} 1 & 2 & 3 & \cdots & n-1 & n \\ 2 & 3 & 4 & \cdots & n & 1 \end{smallmatrix}\bigr)$

Flags

Rubiks Cube/Tuple

From charlesreid1

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