Question

Distinct planes $p_1, p_2, \ldots, p_k$ 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 the minimum possible values of $k?$

$8$

$12$

$20$

$23$

$24$

Solution:

On every face, the required segments join midpoints of edges. A plane cutting the cube meets the faces in one of four symmetric shapes: a square through midpoints ( $3$ such planes), a rectangle per edge ( $12$ planes), a triangle per vertex ( $8$ planes), or a regular hexagon per pair of opposite vertices ( $4$ planes).

Using all of them gives the maximum $k = 3 + 12 + 8 + 4 = 27.$

The full figure consists of $24$ short segments and $12$ long segments. The $4$ hexagon planes together contain all $24$ short segments, and the $3$ square planes contain all $12$ long segments, so the minimum is $k = 4 + 3 = 7.$

The difference is $27 - 7 = 20.$

Thus, the correct answer is C.

2012 AMC 12A Problem 22

Solution:

Problem 22 in Other Years