2012 AMC 12A Problem 22

Distinct planes p_1,p_2,....,p_k intersect the interior of a cube Q. Let S be the union of the faces of Q and let P =\bigcup_{j=1}^{k}p_{j}. The intersection of P and S consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of Q. What is the difference between the maximum and minimum possible values of k?

\textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24

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