Question

A typical football is made by stitching together 12 pentagons and 20 hexagons. How many vertices (junctions) are there in such a football?

Answer 1

Correct Answer: 60

To calculate the number of vertices in the football, we apply Euler's formula for polyhedra, which states: V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces.

In this problem, we have:

• F (Faces): The football consists of 12 pentagons and 20 hexagons, totaling F = 12 + 20 = 32 faces.
• E (Edges): Each pentagon has 5 edges and each hexagon has 6 edges. Therefore, the total count of face edges is 12 × 5 + 20 × 6 = 60 + 120 = 180. However, each edge is shared by 2 faces, so E = 180 / 2 = 90.

Substituting F and E into Euler's formula gives:

V - 90 + 32 = 2

V + 32 - 90 = 2

V = 90 - 30

V = 60

The number of vertices (junctions) in the football is 60, which fits within the given range of [60, 60].