How do you calculate all the possible combinations on a Rubik's cube?

  1. I thought it would just be the number of faces multiplied by the nine cubes on each face? What am i doing wrong?
  2. jcsd
  3. berkeman

    Staff: Mentor

    Not all combinations are possible mechanically. I would probably try to solve this with a program. Are you comfortable writing a C program (or using some other programming language) to solve this?
  4. I know very little programming, a tiny but if Python but that's about it
  5. D H

    Staff: Mentor

    You're not going to be able to count the permutations on a computer. The number is too big.

    If you consider the problem of the number of permutations that can be made by pulling a Rubik's cube apart piece by piece and then reassembling it, this is a huge number. There are eight corner cubes which can be placed. That means 8! permutations just based on corner cube location. Each corner cube can be placed in one of three orientations. That's a factor of 38 permutations on top of the 8! location permutations. The twelve corner cubes lead to two more factors, 12! and 212. Altogether, there are ##8! \, 3^8 \, 12! \, 2^{12}## permutations of the ripped apart and resembled cube. That is a *big* number.

    Most of these permutations do not lead to the nice all colors on one face arrangement. There are constraints, but the final number is still huge.
  6. lurflurf

    lurflurf 2,326
    Homework Helper

    See here's_Cube
    There are
    $${8! \times 3^7 \times (12!/2) \times 2^{11}} = 43,252,003,274,489,856,000 \\

    {8! \times 3^8 \times 12! \times 2^{12}} = 519,024,039,293,878,272,000. $$
    the larger number is 12 times the smaller as there are 12 orbits
    that is any position can reach 1/12 positions though legal moves separating possible moves into 12 orbits
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