Prove that it is not possible to assign the integers $1,2,3,\cdots,20$ to the twenty vertices of a dodecahedron so that each face have constant sum

Question

Prove that it is not possible to assign the integers $1,2,3,\cdots,20$ to the twenty vertices of a regular dodecahedron so that the five numbers at the vertices of each of the twelve pentagonal faces have the same sum.

That's what I have done.

Let $k$ be the constant sum of each of the twelve faces of the dodecahedron,then: $$3(1+2+\cdots+20)=12k$$

$$3\left(\cfrac{20(1+20)}{2}\right)=12k$$

$$\cfrac{210}{4}=k$$

But $k$ is an integer,so we have a contradiction.

Is this proof correct and complete?Are there other ways to apprach the problem ?

(The problem has been taken from USAMTS 1)

Answer 1

Your solution is fine but I suggest you start by explicitly saying that

Since each vertex appears in exactly three faces, ...

Answer 2

Your solution looks perfect to me, I doubt there is a simpler way to approach this.