Rubiks Cube/Tuple

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.

See also: https://github.com/charlesreid1/rubiks-cycles

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

4x4 Rubiks Cube Representation

             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