VN-TST 2018 Problem 5

In a $m\times n$ square grid, with top-left corner is $A$ , there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$ .

a. Prove that this route exists if and only if at least one of $m,\ n$ is odd. b. If such a route exists, then what is the least possible of turning points?

*A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.