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.

Michael, the pronounce is "oiler" (Leonhard Euler).

Try googling "Euler" and see the contribution to math made by this swiss mathematician, in the XVIII century !!!!

He exausted the analysis of various kind of graphs, and achieve to conclusions that are used to any network. And your problem is a "network".

However, in fact, it is not necessary to use their conclusions. In your problem, a little bit of logical reasoning (and Í like logic puzzles too, perhaps more than you !!!!!), one can "prove" the impossibility.