Graph theory -- the number of verticies of graph G

Question

A connected planar graph G has 20 faces, and every vertex of G has degree exactly 4.

Find the number of vertices of G.

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How should I start? I have tried to draw some diagrams, but they lead to nothing.

Answer 1

Let $V,E,F$ be the numbers of vertices, edges, and faces of G, respectively. By Euler's theorem, we know that $$V-E+F=2.$$

We know $F=20$. Also, since every vertex has degree 4, we have $E=\frac{4V}2=2V$ and thus $$V-2V+20=2.$$ Solving this equation yields $V=\boxed{18}$.