Solving the Cube

Rubik said he spent weeks working out how to restore his own scrambled cube (Rubik's Cube and Hungarian Inventors 2). This chapter explains what a solution is, gives the move notation, outlines a beginner method in stages, and then steps back to the mathematics: why the cube has so many positions, what "God's number" is, and how speedcubing grew. The method is described in outline, so you will want a diagram or video to follow the moves. Figures and names are from general knowledge and flagged for checking.

What a Solution Is

A solved cube has each face showing a single colour. A scramble is a sequence of turns that mixes the colours, and a solution is a sequence of turns that undoes it. Because the six centre pieces never move relative to each other (see Chapter 2), a solution only has to put the twelve edges and eight corners in the right places and the right orientations.

Move Notation

Cubers use a standard notation, devised in the early days of the craze and now universal (check its origin; David Singmaster's notation is the one usually credited). Each letter is a face, turned a quarter-turn clockwise as seen when looking at that face:

A Beginner Method in Outline

The most common beginner approach is the layer-by-layer method. Each stage builds on the last without undoing it (check the details against a diagram, since small differences in how stages are taught are common):

1. The first-layer cross. Place the four edge pieces that include one colour (say white) around the white centre, each matching the centre colour on its other face.
2. The first-layer corners. Insert the four white corners, using a short repeated sequence of moves such as R U R' U' until each corner sits correctly (check).
3. The middle layer. Insert the four middle-layer edges, using a pair of mirrored sequences, one for the left and one for the right.
4. The last-layer cross. Make a cross on the opposite colour (say yellow), using a short sequence that turns edge pieces over without disturbing the first two layers.
5. The last-layer corners and edges. Position and orient the remaining corners and edges, each with a short sequence that changes only a few pieces and leaves the rest of the cube unchanged (check).

The key idea is the algorithm: a fixed sequence of moves that does a small, known job and preserves what you have already solved. The beginner method is slow but it works, and with practice it can be done in a minute or two (check).

The Mathematics

The cube is a standard example in group theory, the branch of mathematics that studies sets of operations that can be combined and undone. Each sequence of turns is an operation, doing one after another is combining, and every turn can be reversed (check the formulation).

• The number of positions. A 3x3x3 cube has about 43 quintillion reachable positions, 43,252,003,274,489,856,000 (check the figure). If you tried one a second, it would take longer than the age of the universe.
• Why not more. The pieces cannot be rearranged freely: only some combinations can be reached by turning the faces. A cube taken apart and put back wrongly may be unsolvable (check, and treat the exact constraints as a topic for further reading).
• God's number. The fewest moves that can solve any scramble, counted in a standard way, is 20. This was proved in 2010 by a team using large computing resources donated by Google (check the details and the counting method).

Speedcubing

Solving for speed became a sport with the internet. The World Cube Association organises competitions and keeps records (check the organisation, its founding and the current records). Top solvers use methods with many more algorithms than the beginner approach, and recognise patterns almost instantly (check the best-known methods, such as CFOP, and the other cube types).

• Times. The fastest official 3x3x3 solves are now a few seconds (check the current record, which changes).
• Variety. Events cover larger cubes, one-handed solves, blindfold solves and other puzzles.
• A community. Online forums, videos and shared algorithm lists are a large part of the culture.

Exercises