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