CEMC Galois, 2007

A $(2k+1)$ by $(2k+1)$ by $(2k+1)$ cube, where $k$ is a positive integer, is painted in the same manner as the $5$ by $5$ by $5$ cube with white squares in the corners. Again, the cube is cut into $1$ by $1$ by $1$ cubes. (i) In terms of $k$, how many of these cubes have exactly two white faces? Explain how you obtained your answer. (ii) Prove that there is no value of $k$ for which the number of cubes having at least two white faces is $2006$.