2012 AMC 12A Problems/Problem 22

Problem

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 $k$ ?

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

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2012 AMC 12A Problems/Problem 22

Problem