1985 AHSME Problem 20

A wooden cube with edge length n units (where n is an integer >2) is painted black all over. By slices parallel to its faces, the cube is cut into n^3 smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is n?

\mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 8 \qquad \mathrm{(E) \ }\text{none of these}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.