Question 6
Let $n$ be a positive integer. Consider $$
S = \left\{ (x,y,z) \mid
x,y,z \in \{ 0, 1, \dots, n\}, \;
x+y+z \gt 0 \right\}
$$ as a set of $(n+1)^3-1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$.
S = \left\{ (x,y,z) \mid
x,y,z \in \{ 0, 1, \dots, n\}, \;
x+y+z \gt 0 \right\}
$$ as a set of $(n+1)^3-1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$.