Possible permutations of a 3x4 cube puzzle My kids have this 3x4 cube puzzle, you know, the one where a picture is formed if you assemble the cubes correctly.
In reality you can create 6 different pictures, using different sides of the cubes.
How many possible ways are there to combine these cubes in a 3x4 pattern?
Or in other words, what is the probability to randomly put the cubes together and get a correct picture?
 A: Consider first just the problem of orienting the correct faces upward.  There are $6^{12}$ choices to make, of which only six are valid; doing this at random will give only a probability of $\frac{6}{6^{12}}\approx \frac{1}{360,000,000}$.  Now we must arrange the twelve puzzle pieces into a $3\times 4$ grid correctly.  There are $12!$ ways to do this, and only one is correct (up to rotation, which I will consider at the end).   Then, after putting the twelve pieces into the correct locations, each piece must be oriented correctly, out of four possible ways.  There are $4^{12}$ ways to do this, and again only one of these is correct.  Putting this all together we get $$\frac{6}{6^{12}}\frac{1}{12!}\frac{1}{4^{12}}\approx 3.4\times 10^{-25}$$
However if they assemble the puzzle rotated $180^\circ$ we should count that as a solution too, so the answer really is twice that, or $6.9\times 10^{-25}$.  For comparison, there are half a million times more positions of this puzzle than there are for Rubik's cube, which has a paltry $4\times 10^{-19}$ probability of a randomly chosen position being solved.
