Paths on a Rubik's cube 
An ant is initially positioned at one corner of the Rubik's cube and wishes to go to the farthest corner of the block from its initial position. Assuming the ant stays on the gridlines, how many different paths are possible for it to get to the far corner. Assume that there is no backtracking (it is always moving closer to its goal) and that the ant can travel on any of the six sides. (Be careful about not including paths more than once.)

I am uncertain I correctly calculated all the paths, please correct me if I made a mistake!
I started by realizing that the total number of paths on a rectangular gird is given by either the combination of: C(length+width, width) or C(length+width, length). I then concluded that any path going from one corner to it's opposite corner must to through exactly two faces of the cube. The number of path to go through two faces is then C(9, 3), and that there are six ways to make pairs of two faces starting from the ants corner.

C(9, 3) * 6

However, I believe I have included too many paths because some of the paths overlap. So I subtract every path that goes to the corners directly adjacent to the original corner, which is:

C(6, 3) * 3
So the grand total of paths should be: C(9, 3)*6 - C(6, 3)*3 = 444 paths


In general,
$f(n,n,n)=6(_{3n}C_n-_{2n}C_n)$
[1] http://www.maa.org/sites/default/files/pdf/pubs/Rubiks6.pdf
