# Paths of even and odd lengths between cube vertices

I have an ordinary cube with the standard 8 vertices and 12 edges. Say I define a path as a journey along the edges from one vertex to another such that no edge is used twice. Then I pick two vertices that are connected by a single edge, ie. a path of length 1. How would I go about proving that all the paths between these two points are always odd in length, ie. each such path consists of an odd number of edges.

Ideally I'm after an explanation that could be understood by a high school student. Which is another way of saying I'm trying to help my daughter with her maths homework :)