Question 5
The vertices of an $m\times n$ rectangular grid $ABCD$ are road junctions, with $(m+1)(n+1)$ junctions in all. We want a route starting at $A$, moving along grid edges parallel to the sides of the rectangle, visiting every junction exactly once, and then returning to $A$.
(a) Prove that such a route exists if and only if $m$ or $n$ is odd.
(b) For $m,n$ satisfying (a), what is the smallest number of junctions at which the route turns?
(a) Prove that such a route exists if and only if $m$ or $n$ is odd.
(b) For $m,n$ satisfying (a), what is the smallest number of junctions at which the route turns?