International Mathematical Olympiad 2007 Problem 6

Let $n$ be a positive integer. Consider

$$S = \{(x, y, z) : x, y, z \in \{0, 1, \ldots, n\}, x + y + z > 0\}$$

as a set of $(n + 1)^3 - 1$ points in three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$.