The maths of the Rubik's Cube: why it works.

How many positions are there?

The cube has 8 corners and 12 edges. The corners can be arranged in 8! ways and each can be twisted three ways, but the last twist is decided by the others, so there are 8! × 3⁷ corner states. The edges can be arranged in 12! ways and flipped two ways, again with the last flip decided, so 12! × 2¹¹ edge states. A parity rule says half of those combinations are impossible, so divide by 2.

43,252,003,274,489,856,000positions: about 43 quintillion

If you could try one position every second, it would take about 1.4 trillion years to see them all, which is far longer than the universe has existed.

God's number is 20

Out of all those positions, how many moves do you need in the worst case? In 2010 a team led by Tomas Rokicki, with Herbert Kociemba, Morley Davidson and John Dethridge, used a lot of computer time to show that the answer is 20, when a half turn counts as one move. No cube ever needs more than 20 moves, and some positions need all 20. Human solves use many more, because we trade moves for steps we can remember.

How many times until it repeats?

Every algorithm, repeated enough times, brings the cube back to solved. The number of repeats is called the order. Try an algorithm below.

Some orders to try: R U R' U' has order 6, R U has order 105 (surprisingly large for two moves), and R2 U2 has order 6. The order is the least common multiple of the cycle lengths of all the pieces it moves.

Inverses and commutators

Every sequence has an inverse that undoes it: the same moves in reverse order, each turned the other way. A commutator is a sequence of the form A B A' B'. When A and B touch only a few pieces in common, the commutator moves only those, which is how many algorithms are designed. The sexy move R U R' U' is a commutator of R and U, and it is the building block of the beginner method.

Parity

Parity is the idea that a permutation is either even or odd, depending on how many swaps it takes. A quarter turn of a face is a 4-cycle of corners and a 4-cycle of edges, so it is odd for both. Every move changes both parities together, which is why the cube can never be in a state with just two edges swapped. On bigger cubes the same idea produces parity cases that need special algorithms.