Given a 3x3x3 cube (like a rubik's cube) where each of the 27 cubes has a distinct number, how many unique permutations are possible? Simple rotations of the entire cube should not be counted.
The solution might be related to
Rubik's cube and counting
Edit: to be clear about the the question I'll suggest an application.
I want to create a substitution cipher for english where I put the letters a-z and space, which is 27 characters, on each cube in a rubik's-type cube. I have a text I would like to encrypt. I have a starting orientation for the cube and I take the first letter and find it's location in the cube. I then rotate the cube 90 degrees in one direction and write down the resulting letter in that position. I repeat for all the letters in the message. I now have an encrypted message. To decode the message, I would have the same process, just in reverse. The process could be more complicated involving plane rotations and cumulative permutations if desired.
My question is how many different permutations of this cube could there be, that are not simple rotations of each other.
Best Answer
First, we figure out how many arrangements there are, including "simple rotations". There are $3^3 = 27$ blocks to be freely permuted. This means that there are $27!$ possible arrangements of the blocks (including simple rotations).
Now, note that any given arrangement can be rotated to yield
6$24$ (see comment below) superfically different arrangements. So, our count of $27!$ redundantly counts each rotated arrangement $24$ times. It follows that our desired total is $$ \frac{27!}{24} \approx 4.54 \times 10^{26} $$