Can a polyhedron have 14 faces, 24 edges and 33 vertices?

Answer

To determine if a polyhedron can have 14 faces, 24 edges, and 33 vertices, we can use Euler's formula for polyhedra, which states:

\[ V - E + F = 2 \]

Where:
- \( V \) = number of vertices
- \( E \) = number of edges
- \( F \) = number of faces

Given:
- \( F = 14 \) (faces)
- \( E = 24 \) (edges)
- \( V = 33 \) (vertices)

Now, we will substitute these values into Euler's formula.

### Step 1: Substitute the values into Euler's formula
Using the formula \( V - E + F = 2 \):

\[ 33 - 24 + 14 = 2 \]

### Step 2: Calculate the left-hand side
Now, let's calculate the left-hand side:

\[ 33 - 24 = 9 \]
\[ 9 + 14 = 23 \]

### Step 3: Compare with the right-hand side
Now, we compare the left-hand side with the right-hand side:

\[ 23 \neq 2 \]

### Conclusion
Since the left-hand side (23) is not equal to the right-hand side (2), the values do not satisfy Euler's formula. Therefore, a polyhedron cannot have 14 faces, 24 edges, and 33 vertices.

### Final Answer
No, a polyhedron cannot have 14 faces, 24 edges, and 33 vertices.

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