A 3D shape has 12 vertices and 30 edges. Work out the number of faces on this shape.

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We have a 3D shape with 12 vertices and 30 edges. We need to find the number of faces. To solve this problem, we'll use Euler's formula for polyhedra, which relates vertices, edges, and faces of any convex polyhedron. Euler's formula states that for any convex polyhedron, the number of vertices minus the number of edges plus the number of faces equals 2. This is written as V minus E plus F equals 2. This remarkable formula applies to all convex three-dimensional shapes. Now let's substitute our known values into Euler's formula. We have V equals 12 and E equals 30. Substituting these values: 12 minus 30 plus F equals 2. This simplifies to negative 18 plus F equals 2. Adding 18 to both sides, we get F equals 20. Therefore, our 3D shape has 20 faces. This shape is actually an icosahedron, one of the five Platonic solids.