Such a polyhedron cannot exist.
According to the Euler characteristics, V + F - E = 2, where V = vertices, F = faces, E = edges. This would require that the polyhedron had only two faces.
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12 vertices A prism with an n-sided base will have 2n vertices, n + 2 faces, and 3n edges.
Oh, dude, it's like a math riddle! So, if a polyhedron has 10 more edges than vertices, we can use Euler's formula: Faces + Vertices - Edges = 2. Since we know the relationship between edges and vertices, we can substitute that in and solve for faces. So, it would have 22 faces. Math can be fun... sometimes.
3 faces, 2 edges, and no vertices
4 faces, 6 edges, 4 verticesFour faces, six edges and four vertices.
Faces: 10 Vertices: 16 Edges: 24