This is an Eulerian Path. Euler said:
Theorem: If a network has more than two odd vertices, it does not have an Euler path.
Euler also proved this:
Theorem: If a network has two or zero odd vertices, it has at least one Euler path. In particular, if a network has exactly two odd vertices, then its Euler paths can only start on one of the odd vertices, and end on the other.
Is solvable. As this has 2 odd vertices, you should start in one of them and end on the other. (top and bottom vertices)