The goal is to trace a line with your pencil across each edge on the box only once without crossing the vertices or picking your pencil up.

Note that this "box" contains 16 unique "edges".

Prove why this is an impossible task regardless of where you first place your pencil.

Erik, I know this kind of "challenge" quite well.

Michael asked to prove that it is unsolvable. Everyone that knows this problem knows that it is unsolvable.

However, the proof is too far from just saying "The only way to have a full traversable is if all nodes have an even number of edges or if all nodes except two are even". Everybody knows this too.

But, I think, the asked proof is in the "Theory of graphs", and in my oppinion, is too complex to be commented here.

I believe that Michael will accept your answer.