Rubik's cube total combination tutorial

Consider the $8$ corner pieces. Dispite the orientation for the corners, we have $8$ unique pieces to place in $8$ different areas, that is $8$ factorial \(8!\). And then consider the orientation, each of the eight corner pieces has $3$ different directions, which leads to $3$ to the power of $8$ different situations \(3^8\) .

So this is $8! \times 3^8$

Similarly to how we calculate the corner pieces. This time consider the side pieces. In a standard $3 \times 3$ Rubik's cube, we have $12$ unique side pieces each contains two different colors. To put these $12$ pieces in 12 different positions, we have $12!$ situations. And consider orientation, that is $2^{12}$

In this case, we have $12! \times 2^{12}$

The above calculation leads us to $8! \times 3^8 \times 12! \times 2^{12}$.

Then let's step ahead to the part divided by $12$

Consider when you take the cube apart and recombine it, there is a chance that you make the correct one, i.e. you can solve this cube. And that is $\frac{1}{12}$ the chance to be exact.

And this situation is because in a standard $3 \times 3$ rubik's cube, there are some invalid positions for the pieces:

1. You can't rotate a single corner piece.

2. You can't rotate a single side piece.

3. You can't swap two side pieces.

None of these three situation can be achieved by just a sequence of valid rotation of a rubik's cube.(i.e. you have to achieve this by taking the cube apart)

And the first situation leads to three different combinations, the second leads to $2$ and the third leads to two.

The total situations is then $3 \times 2 \times 2 = 12$

In another word, There are totally $12$ different rubik's cube you can get if you take the cube apart and recombine it. And only one of them is actually SOVABLE

Hence from the formula we get(Contains every situations for the RIGHT and WRONG rubik's cube), divided by $12$ is the final answer to the total combinations of a Rubik's Cube.

FINAL ANSWER: $\frac{8! \times 3^8 \times 12! \times 2^{12}}{12} = 43,252,003,274,489,856,000$