Colors on the Rubix Cube

Date: 01/15/2005 at 12:52:16
From: Jay
Subject: How many color combinations are there in a Rubix cube?

How many color combinations are there in a Rubix cube?  Each of the
six faces is a different color, and there are nine cubes of that color
on each side.

I thought that six to the fifty-fourth power was it because there are
six sides on a cube and nine different colors.  But then I started to
think about the rows and columns and each large face combinations.  My
mind was beginning to boggle so I thought you would help me get 
started at least with the correct formula procedure--then I would do
the actual math.  Thank you very much.

Date: 01/18/2005 at 11:43:34
From: Doctor Vogler
Subject: Re: How many color combinations are there in a rubix cube?

Hi Jay,

Thanks for writing to Dr. Math.  Your suggestion of 6^54 is a good
guess, but you might notice that it also includes the possibility of
all 54 faces being red.  That can't happen.

Look closely at your Rubik's Cube and you'll find that those 54 colors
lie on 8 corner pieces which have 3 colors each, 12 edge pieces which
have 2 colors each, and 6 center pieces which have 1 color each.  Then
you might also notice that the center pieces don't move.  All of these
observations will also help you to learn how to solve the cube.

Now you might notice that you can't swap corner pieces with edge 
pieces, or other odd combinations, but you can rearrange the corner
pieces, and you can rearrange the edge pieces.  (The corner pieces
don't move, recall, except by rotating the whole cube.)  You can also
rotate the corner pieces into three different positions, and you can
flip over the edge pieces.  So all of this might lead you to believe
that there are

  12! ways to permute the corners,
  8! ways to permute the edges,
  3^8 ways to rotate the corners,
  2^12 ways to flip over the edges,

for a grand total of

  12! * 8! * 3^8 * 2^12 total positions of the Cube.

But you should first ask if all of these positions are possible.  In
fact, they aren't.  First you should check that the last corner's
rotation will always be determined by the other 11 rotations, and the
last edge's position (how it's flipped) will always be determined by
the other 7.  Furthermore, half of the 12! permutations of the corners
are what mathematicians call "odd" permutations, and half of them are
even.  Same with the 8!.  It turns out that if the corners are in an
odd permutation, then so are the edges, and vice-versa.  So that means
that we really have

  12! * 8! * 3^7 * 2^10 total positions of the Rubik's Cube.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/