Hints

1. Model the colors with small integers, storing them in a 3-dimensional array that models the n^3 number of smaller cubes inside of the large cube.

2. Here are the answers to the problem for selected values of n:

n     Colorable?

1        Yes
2        No
3        No
4        No
5        No
6        No
7        No
8        Yes
9        (You determine!)
10      (You determine!)
11      Yes
12      (You determine!)

3. How to solve an 8x8x8 cube.

4. 11-Cubed Case