Problem:
A wooden cube with edge length units (where is an integer ) is painted black all over. By slices parallel to its faces, the cube is cut into smaller cubes each of unit edge 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
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
A smaller cube will have no face painted if it comes from the interior of the original cube, that is, if it is part of the cube of side length obtained by stripping away a one-unit layer from each face of the original. So there are ( unpainted smaller cubes. A smaller cube has one painted face if it comes from one of the faces of the original cube, but was not on the edge of the face. Thus there are such smaller cubes. Therefore, . Since , we can divide out the , leaving , or .