Can a polyhedron have 8 faces, 26 edges and 16 vertices ?

Answer

To determine if a polyhedron can have 8 faces, 26 edges, and 16 vertices, we can use Euler's formula for polyhedra, which states:

\[ E = F + V - 2 \]

Where:
- \( E \) is the number of edges,
- \( F \) is the number of faces,
- \( V \) is the number of vertices.

### Step-by-Step Solution:

1. **Identify the values given in the problem:**
- Number of faces \( F = 8 \)
- Number of edges \( E = 26 \)
- Number of vertices \( V = 16 \)

2. **Write down Euler's formula:**
\[ E = F + V - 2 \]

3. **Substitute the values into the formula:**
\[ 26 = 8 + 16 - 2 \]

4. **Calculate the right-hand side:**
- First, add the number of faces and vertices:
\[ 8 + 16 = 24 \]
- Then subtract 2:
\[ 24 - 2 = 22 \]

5. **Compare both sides of the equation:**
\[ 26 \neq 22 \]

6. **Conclusion:**
Since the left-hand side (LHS) does not equal the right-hand side (RHS), the values do not satisfy Euler's formula. Therefore, a polyhedron cannot have 8 faces, 26 edges, and 16 vertices.

### Final Answer:
No, a polyhedron cannot have 8 faces, 26 edges, and 16 vertices.