## Task 14/84

For which natural numbers $n$ does no polyhedron exist with exactly $n$ edges?
1. Statement: For all even numbers \( n \geq 6 \), there exists a polyhedron with exactly \( n \) edges.

Proof: Let \( n = 2k, k \in \mathbb{N}, k \geq 3 \). Every \( k \)-sided pyramid has exactly \( k \) edges from the base to the apex; thus, every \( k \)-sided pyramid with \( k \geq 3 \) has exactly \( n = 2k \geq 6 \) edges.

2. Statement: For all odd numbers \( n \geq 9 \), there exists a polyhedron with exactly \( n \) edges.

Proof: Let \( n = 2k + 1, k \in \mathbb{N}, k \geq 4 \). For every \((k-1)\)-sided pyramid, Statement 1 holds; it thus has exactly \( 2k - 2 \) edges. If one cuts off a corner at the base with a planar cut, 3 new edges appear. Therefore, the "mutilated" pyramid has exactly \( n = 2k - 2 + 3 = 2k + 1 \geq 9 \) edges.

3. Statement: For \( n < 6 \), there does not exist a polyhedron with exactly \( n \) edges.

Proof: Each face contains at least 3 edges, each polyhedron contains at least 4 faces, and each edge belongs to exactly 2 faces. The minimum number of edges is thus \( n_{\text{min}} = \frac{3 \cdot 4}{2} = 6 \).

4. Statement: It remains to check the existence of a polyhedron with exactly 7 edges. Suppose there were such a polyhedron. According to Euler's polyhedral formula, \( f + e = k + 2 \), for \( k = 7 \) thus \( f + e = 9 \). Since \( f \geq 4 \), it follows that \( e \leq 5 \).

There is only one polyhedron with \( f = 4 \), the tetrahedron, for which \( e = 4 \) and thus \( e + f = 8 \neq 9 \). If the number of faces is increased by 1, either a four-sided pyramid results with \( e = 5, k = 8 \), or a truncated pyramid (where the base and top faces are not necessarily parallel) with \( e = 6, k = 9 \). In no case is \( k = 7 \).

It is also immediately clear (a consequence of the polyhedral formula) that any further increase in the number of faces leads to an increase in the number of edges.

Result: For the natural numbers \( n < 6 \) and \( n = 7 \), there does not exist a polyhedron with exactly \( n \) edges.