Vivekanand Vellanki

If only it were this easy to realise infinity!!!

Lets estimate the number of combinations for the rubicks cube.

There are 3 types of cubes in a rubicks cube:

  • The corner cubes. There are a total of 8 of them
  • The middle cubes on each edge. There are a total of 12 of them
  • The middle cube on each face. There are 6 of them

In the worst case, the number of ways you can arrange the 8 corner cubes is 8!. Note that each corner cube can be rotated - so, there are 3 combinations in which it can be at each position. This gives a total of 3^8\times8! combinations involving the corner cubes.

Similarly, the number of ways in which the middle cubes of each edge can be arranged is 12!. Each middle cube can be in one of two positions for a total of 2^{12}\times12!

The middle cube on each face can be arranged in 6! ways.

Hence, the upper bound on the total combinations = 3^8\times8!\times2^{12}\times12!\times6!

I think this is a gross over-estimation of the number of combinations. The idea is to show that the number of combinations is not limitless, but has an upper bound.