Rubik's Cube Combinations

Date: 04/11/2001 at 10:12:57
From: Mr. Glapa
Subject: Combinations of possibilities

I read that a Rubik's cube has 4 quintillion (a 1 followed by 18 0's) 
different possible combinations. We are studying much simpler 
combinations, but I wanted to see if some of my students would be 
willing to try to get this answer. Is this number correct?  Here is 
how I would try to solve this problem. Am I doing it correctly?

     1 and 54 different cubes

     2 x 54   =  108
     3 x 108  =  324
     4 x 324  = 1296
     5 x 1296 = 6480
       :

and so on up to the number 54.

Date: 04/12/2001 at 09:12:04
From: Doctor Rick
Subject: Re: Combinations of possibilities

Hello, sir!

Your answer is 54! or "54 factorial". It represents the number of 
different cubes you could get by peeling off all the visible colored 
facelets of the cube (9 on each of the 6 faces), scrambling them, and 
sticking them all back on. 

This is very different from the question at hand, which is how many 
different patterns you can get by manipulating the cube in legitimate 
ways (turning one face, etc.) 

Even if we want to peel the colors off the cube, we are overcounting. 
We are counting cubes with the 9 red facelets switched among 
themselves, for instance, as different configurations. In fact they 
are indistinguishable. Therefore we should divide by 9!, the number 
of ways the 9 facelets can be interchanged, and repeat this for each 
of the six colors.

We are still overcounting. If we can get from one configuration to 
another just by turning the cube, they aren't really different, but we 
have counted them as different. How many ways can the cube be turned? 
Turn any of the 8 corners to the lower front left corner (just to pick 
one). Then, leaving that corner in place, rotate the cube to one of 3 
orientations. Thus there are 8*3 ways the cube can be oriented, and we 
must divide the total above by 8*3 = 24. The result (the number of 
distinct configurations we could get by peeling off the facelets and 
sticking them back on the cube) is

  54!/(24*(9!)^6) = 4.21239*10^36 (approximately)

Now let's get to the real question: How many different patterns can 
you get by manipulating the cube according to the rules? Here is what 
I would do to calculate the number of patterns.

There are 8 corner "cubies" on a Rubik's Cube, each different. We can 
choose a location for each in 8! ways. Each of the 8 cubies can be 
turned in 3 different directions, so there are 3^8 orientations 
altogether. The total number of ways the corner cubies can be placed 
is 8! * 3^8.

Turning our attention to the edge cubies, which have 2 faces showing, 
there are 12 of these, so we can choose locations for them in 12! 
ways. Each of the edge cubies can be turned in 2 different directions, 
so there are 2^12 orientations altogether. Thus the total number of 
ways the edge cubies can be placed is 12! * 2^12.

We're not multiplying counting rotations of the entire cube this time. 
That's because all the configurations we have counted leave the center 
cubies in the original positions. Rotating the entire cube results in 
a configuration that we have not counted (the center cubies are in 
different positions).

However, we are overcounting for a different reason. We have 
essentially pulled the cube apart and stuck cubies back in place 
wherever we please. In reality, we can only move cubies around by 
turning the faces of the cube. It turns out that you can't turn the 
faces in such a way as to switch the positions of two cubies while 
returning all the others to their original positions. Thus if you get 
all but two cubies in place, there is only one attainable choice for 
the positions of the remaining two cubies, and we must divide the 
number of attainable configurations by 2. Likewise, if you get all but 
one of the edge cubies into chosen positions and orientations, the 
orientation of the last edge cubie can only go into one of two 
orientations; thus we must divide by 2 again. Finally, if you get all 
but one of the corner cubies into chosen positions and orientations, 
only one of three orientations of the final corner cubie is 
attainable; thus we divide by 3. The number of legally attainable 
configurations is therefore

  8! * 3^8 * 12! * 2^12   
  --------------------- = 8! * 3^7 * 12! * 2^10
          2*2*3
                        = 43252003274489856000
                        = 4.32520*10^19 (approximately)

You can read more about the reason for the final constraints by 
reading an article by Douglas Hofstadter in Scientific American, March 
1981 (back when the Cube was all the rage). That article, by the way, 
is the source from which I got the term "cubies."

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