Question 172
Let $n$ be a positive integer. Consider
$$
S = \{(x, y, z) : x, y, z \in \{0, 1, \dots, n\}, x + y + z \gt 0\}.
$$
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 $(0, 0, 0)$.
(This problem was suggested by the Netherlands.)
$$
S = \{(x, y, z) : x, y, z \in \{0, 1, \dots, n\}, x + y + z \gt 0\}.
$$
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 $(0, 0, 0)$.
(This problem was suggested by the Netherlands.)