ÖMO 1983 Problem 6

Planes $\pi _1$ and $\pi _2$ in Euclidean space $\mathbb{R} ^3$ partition $S=\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2)$ into several components. Show that for any cube in $\mathbb{R} ^3$ , at least one of the components of $S$ meets at least three faces of the cube.