A basic, pure-Python Rubik's cube solver

# Overview

This is a Python 3 implementation of a (3x3) Rubik's Cube solver.

It contains:

• A simple implementation of the cube
• A solver that follows a fixed algorithm
• An unintelligent solution sequence optimizer
• A decent set of test cases

## Installation

The package is hosted on PyPI.

``````pip install rubik-cube
``````

Import from the `rubik` package,

```>>> from rubik.cube import Cube
>>> c = Cube("OOOOOOOOOYYYWWWGGGBBBYYYWWWGGGBBBYYYWWWGGGBBBRRRRRRRRR")
>>> print(c)
OOO
OOO
OOO
YYY WWW GGG BBB
YYY WWW GGG BBB
YYY WWW GGG BBB
RRR
RRR
RRR
```

## Implementation

### Piece

The cornerstone of this implementation is the Piece class. A Piece stores two pieces of information:

1. An integer `position` vector `(x, y, z)` where each component is in {-1, 0, 1}:

• `(0, 0, 0)` is the center of the cube
• the positive x-axis points to the right face
• the positive y-axis points to the up face
• the positive z-axis points to the front face
2. A `colors` vector `(cx, cy, cz)`, giving the color of the sticker along each axis. Null values are place whenever that Piece has less than three sides. For example, a Piece with `colors=('Orange', None, 'Red')` is an edge piece with an `'Orange'` sticker facing the x-direction and a `'Red'` sticker facing the z-direction. The Piece doesn't know or care which direction along the x-axis the `'Orange'` sticker is facing, just that it is facing in the x-direction and not the y- or z- directions.

Using the combination of `position` and `color` vectors makes it easy to identify any Piece by its absolute position or by its unique combination of colors.

A Piece provides a method `Piece.rotate(matrix)`, which accepts a (90 degree) rotation matrix. A matrix-vector multiplication is done to update the Piece's `position` vector. Then we update the `colors` vector, by swapping exactly two entries in the `colors` vector:

• For example, a corner Piece has three stickers of different colors. After a 90 degree rotation of the Piece, one sticker remains facing down the same axis, while the other two stickers swap axes. This corresponds to swapping the positions of two entries in the Piece’s `colors` vector.
• For an edge or face piece, the argument is the same as above, although we may swap around one or more null entries.

### Cube

The Cube class is built on top of the Piece class. The Cube stores a list of Pieces and provides nice methods for flipping slices of the cube, as well as methods for querying the current state. (I followed standard Rubik's Cube notation)

Because the Piece class encapsulates all of the rotation logic, implementing rotations in the Cube class is dead simple - just apply the appropriate rotation matrix to all Pieces involved in the rotation. An example: To implement `Cube.L()` - a clockwise rotation of the left face - do the following:

1. Construct the appropriate rotation matrix for a 90 degree rotation in the `x = -1` plane.
2. Select all Pieces satisfying `position.x == -1`.
3. Apply the rotation matrix to each of these Pieces.

To implement `Cube.X()` - a clockwise rotation of the entire cube around the positive x-axis - just apply a rotation matrix to all Pieces stored in the Cube.

### Solver

The solver implements the algorithm described here and here. It is a layer-by-layer solution. First the front-face (the `z = 1` plane) is solved, then the middle layer (`z = 0`), and finally the back layer (`z = -1`). When the solver is done, `Solver.moves` is a list representing the solution sequence.

My first correct-looking implementation of the solver average 252.5 moves per solution sequence on 135000 randomly-generated cubes (with no failures). Implementing a dumb optimizer reduced the average number of moves to 192.7 on 67000 randomly-generated cubes. The optimizer does the following:

1. Eliminate full-cube rotations by "unrotating" the moves (Z U L D Zi becomes L D R)
2. Eliminate moves followed by their inverse (R R Ri Ri is gone)
3. Replace moves repeated three times with a single turn in the opposite direction (R R R becomes Ri)

The solver is not particularly fast. On my machine (a 4.0 Ghz i7), it takes about 0.06 seconds per solve on CPython, which is roughly 16.7 solves/second. On PyPy, this is reduced to about 0.013 seconds per solve, or about 76 solves/second.