2017 Pascal Problem 20

A solid cube is made of white plastic and has dimensions n \times n \times n, where n is a positive integer larger than 1. The six faces of the cube are completely covered with gold paint. This cube is then cut into n^3 cubes, each of which has dimensions 1 \times 1 \times 1. Each of these 1 \times 1 \times 1 cubes has 0, 1, 2, or 3 gold faces. The number of 1 \times 1 \times 1 cubes with 0 gold faces is strictly greater than the number of 1 \times 1 \times 1 cubes with exactly 1 gold face. What is the smallest possible value of n?

\textbf{(A)}\ 7\quad \textbf{(B)}\ 8\quad \textbf{(C)}\ 9\quad \textbf{(D)}\ 10\quad \textbf{(E)}\ 4

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.