Project Euler Problem 599
The well-known Rubik's Cube puzzle has many fascinating mathematical properties.
Solution
Answer: 12395526079546335
Let $G$ be the full group of mechanically reachable transformations of the $2\times2\times2$ cube acting on the $24$ sticker positions.
The key idea is to use Burnside’s Lemma (equivalently, Pólya enumeration):
$$N(n)=\frac1{|G|}\sum_{g\in G} n^{c(g)},$$
where:
- $n$ is the number of available colours,
- $c(g)$ is the number of cycles in the permutation induced by $g$ on the $24$ stickers.
For the pocket cube, the action group on stickers has order
$$|G| = 88,179,840.$$
Using the standard six face-turn generators and computing the cycle index polynomial of the action on the 24 stickers, substituting $n=10$ gives the total number of essentially distinct colourings.
A direct computational implementation (e.g. with GAP / Mathematica / permutation-group code) reproduces the check value
$$N(2)=183,$$
matching the example in the problem statement.
Evaluating the cycle index at $n=10$ yields:
$$N(10)=12395526079546335.$$
Answer: 12395526079546335