# If $G$ cubic planar graph with faces only $C_5$ and $C_6$, then $G$ has $12$ pentagons and $0$ hexagons

I want to prove the questions in the title:

If $$G$$ is a connected cubic planar graph composed of only pentagons and hexagons, then it has $$12$$ pentagons and $$0$$ hexagons.

I was able to show it has $$12$$ pentagons, from Euler formula. However I am unable to proceed.

I believe it has to do with an inner structure it forms, because simply counting degrees and using Euler formula does not seem to work

• Is that a planar graph? I say that because this is a question from a book, so I expect it to be true? – MTLaurentys Jun 9 at 15:50
• What about a soccer ball? – saulspatz Jun 9 at 17:03