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.

This puzzle is unsolvable.

Let's call each section of the box a node, and each individual line segment as an edge.  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.

In this puzzle, the top center "node" and the two bottom "nodes" each have 5 "edges" comeing from them and therefor make this "network" not fully traversable without traversing the same edge twice.