Question 163
Consider a rectangle board $ABCD$ of size $m \times n$ with $(m + 1) \times (n + 1)$ intersections. Some engineers want to build a route from $A$ which goes along the segments parallel to the sides of the board, passes through each intersection exactly once and finally turns back to $A$.
a. Prove that they can build the route if and only if either $m$ is odd or $n$ is odd.
b. With $m, n$ satisfying the condition in (a), find the least number of intersections at each there is a turn.
a. Prove that they can build the route if and only if either $m$ is odd or $n$ is odd.
b. With $m, n$ satisfying the condition in (a), find the least number of intersections at each there is a turn.