Let $ n$ be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of $ (n \plus{} 1)^{3} \minus{} 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)$.

[i]Author: Gerhard Wöginger, Netherlands [/i]
1. **Define the Set \( S \)**:
   The set \( S \) is defined as:
   \[
   S = \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x + y + z > 0 \right\}
   \]
   This set contains all points in the three-dimensional space where \( x, y, z \) are integers between 0 and \( n \), excluding the point \( (0,0,0) \).

2. **Calculate the Total Number of Points in \( S \)**:
   The total number of points in the cube \( \{0, 1, \ldots, n\}^3 \) is \( (n+1)^3 \). Since we exclude the point \( (0,0,0) \), the number of points in \( S \) is:
   \[
   |S| = (n+1)^3 - 1
   \]

3. **Constructing Planes to Cover \( S \)**:
   We need to find the smallest number of planes such that their union covers all points in \( S \) but does not include \( (0,0,0) \).

   - Consider the planes of the form \( x + y + z = i \) for \( i = 1, 2, \ldots, 3n \). These planes cover all points where the sum of the coordinates is between 1 and \( 3n \).
   - Additionally, consider the planes \( x = i \), \( y = i \), and \( z = i \) for \( i = 1, 2, \ldots, n \). These planes cover all points where one of the coordinates is fixed and the other two vary from 0 to \( n \).

4. **Counting the Planes**:
   - There are \( 3n \) planes of the form \( x + y + z = i \).
   - There are \( n \) planes for each of \( x = i \), \( y = i \), and \( z = i \), totaling \( 3n \) planes.

   Therefore, the total number of planes used in this construction is:
   \[
   3n + 3n = 6n
   \]

5. **Optimality of the Construction**:
   To prove that \( 3n \) planes are optimal, assume there exists a construction with fewer than \( 3n \) planes. Let these planes be of the form \( a_i x + b_i y + c_i z + 1 = 0 \).

   - Define the polynomial \( A(x,y,z) = \prod_{i=1}^n (x-i)(y-i)(z-i) \). This polynomial vanishes on all points where \( x, y, z \) are integers between 0 and \( n \).
   - Define the polynomial \( B(x,y,z) = \prod_{i=1}^N \left( a_i x + b_i y + c_i z + 1 \right) \). This polynomial vanishes on all points covered by the \( N \) planes.

   Since both polynomials vanish on \( S \), we have:
   \[
   A(0, 0, 0) = (-1)^{n} (n!)^3 \quad \text{and} \quad B(0, 0, 0) = 1
   \]

   However, \( A - (-1)^{n} (n!)^3 \cdot B(0, 0, 0) \) is nonzero and vanishes on all of \( \{0, 1, \ldots, n\}^3 \), which contradicts the combinatorial nullstellensatz since it has a nonzero coefficient \( x^n y^n z^n \).

   Therefore, \( 3n \) planes are indeed the minimum number required.

\(\blacksquare\)

The final answer is \( \boxed{ 3n } \).