Please note my geometry background is very weak (high school geometry is all I have), so I would appreciate it if someone could explain it in very layman terms how to do this.
I am trying to solve the following problem
Prove that it is not possible to assign the integers $1,2,3,\ldots,20$ to the 20 vertices of a regular dodecahedron so that the five numbers at the vertices of each of the 12 pentagonal faces have the same sum.
I recognize how to solve this, but to do this I need to know how many times a single vertex is counted when adding all the faces up i.e. the number of edges that connect to a single vertex.
I also remember reading a side note somewhere in my Geometry course about something that looked like this: $v+e-f=2$ or something like that - can anybody help me figure out what this was and if I am remembering it right?
Source: Art of Problem Solving Volume 2, Ch.13 Equations and Expressions