XP

We've always been told that a rubik cube has an absurd number of possibilities, but is that truly the case? That's why I decided to attempt to calculate the number of actual possibilities.

The middle pieces of a rubik cube are fixed in place. They can't be moved so we can just ignore them. There are 12 middle side pieces and 8 corners pieces. So we would need to multiply all the different middle side piece possibilities with the corner piece possibilities and we should get the answer. But as you can't just swap one piece of a rubik cube, you need an even number of pieces swapped so we then need divide the answer by 2 to get the true answer.

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The 8 corner pieces each have 3 sides so the first corner we look at would have 8 x 3 different possibilities. The second corner would then have 7 x 3 different possibilities and so on and so forth. But for the last one, you don't need to calculate 3 × 1 as it's rotation is already set by the others. You can't just have one rotated corner on a rubik cube.

So the extended formel would thus be 8 × 3 × 7 × 3 × 6 × 3 × 5 × 3 × 4 × 3 × 3 × 3 × 2 × 3, or simplified: 8! × 3^7 just to calculate the corner possibilities.

To calculate the middle side possibilities you just need to repeat what did for the corner pieces and you'd end up with 12! x 2^11.

Now after combining those 2 formals and dividing the answer by two, as already explained, you would end with: 12! x 2^11 × 8! x 3^7 × 1/2 = 4.3252003e+19, which seems to confirm that the cube truly has an absurd number of possibilities.