3x3 Rubiks Cube
The standard Rubiks Cube size is a 3x3.
- A basic, layer-by-layer Rubiks Cube solution technique: Rubiks Cube/Layer Method
There are other, more advanced techniques:
The representation of the Rubik's Cube for use in mathematics (or computer programs) is an important topic. To turn a particular Rubik's Cube configuration into a mathematical representation agnostic to the particulars of the cube, we can label each face with an integer 1 to 36, and pick a particular order for each face. Then we can write a sorted cube as (1 2 3 4 5 ... 35 36). This also enables enumerating combinatoric properties of the cube.
See Rubiks Cube/Tuple for more on this tuple representation.
See Rubiks Cube/Permutations for more on representing permutations and uncovering their properties.
Counting the number of possible configurations of a Rubik's Cube requires thinking about how the puzzle works and the unique pieces that compose any Rubik's Cube puzzle. We can enumerate the state of each of the three types of pieces: corner pieces, edge pieces, and center pieces.
See Rubiks Cube/Numbers for the procedure and rundown of how we count the number of permutations of a 3x3 or 4x4 Rubik's Cube.
Patterns and Sequences
The symmetry properties of the Rubik's Cube lead to some interesting properties.
Notes on group theory and Rubiks Cube patterns: Rubiks Cube/Patterns
4x4 Rubiks Revenge
For the most part, solving the 4x4 is a lot like solving the 3x3, but complicated by two initial steps that need to happen first - arranging the center 2x2 cubies of each face, then orienting the two edge pairs of cubies of each of the twelve sides of the cube to match up.
Then you can think of these edge pairs as sticking together, and the whole thing becomes a 3x3 cube, but with 2 rows/columns instead of a single middle row/column.
However, the parity case complicates things. Whereas the 3x3 cube has a few ending scenarios with regard to the parity of the top layer and how the squares are oriented (memorizing these becomes crucial to fast solves), the 4x4 cube has some situations that are impossible on a 3x3 cube (such as two corner cubies being swapped) or a cube that's entirely solved except for two inside-out cubies on one of the top edges.
See Rubiks Revenge page for pictures.
To represent the Rubik's Revenge as a mathematical object, we can use a 96-tuple (that is, a permutation of the integers from 1 to 96, non-repeating). We have 8 corner pieces, 24 double-edge pieces (12 left-hand double edge pieces, 12 right-hand double edge pieces), and 24 center pieces, for a total of 56 pieces. However, some of these pieces have multiple faces, and we specify the state of the cube by enumerating the faces.
See Rubiks Cube/Tuple for information about how to turn a Rubik's Cube permutation, or arrangement, using a tuple of integers.
Rubiks Cubenotes on solving 3x3 and 4x4 Rubik's Cube
Cubes and Solutions:
2x2 Cube: Pocket Cube
4x4 Cube: Rubiks Revenge
Combinatorics and Permutations:
Counting cube permutations: Rubiks Cube/Numbers
Representation of cube permutations: Rubiks Cube/Tuple
Permutation Algebra: Rubiks Cube/Permutations
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