Solutions to Problems 1351 - 1360

Q1351 A city consists of a rectangular grid of roads, with $m$ roads running east--west and $n$ running north--south.  Every east--west road intersects every north--south road.  A construction vehicle travels around the city, visiting each intersection once (and only once) and finally returning to its starting point.  As it travels it builds a fence down the middle of each road it uses: thus it constructs, in effect, a single long fence which eventually loops back on itself.  How many city blocks are now inside the fence?
 
SOLUTION If we assume that each block is a square with side length $1$ unit, then the number of blocks inside the fence is equal to the area enclosed by the path.  Since the path is a polygon with lattice points for its vertices we can use Pick's Theorem:
$$A=I+{\textstyle\frac{1}{2}}P-1\ ,$$
where $A$ is the area enclosed, $I$ is the number of points inside the path and $P$ is the number of points on the path.  In this case there are $mn$ points altogether, and they are all on the path (because we are told that every intersection was visited).  Therefore $P=mn$ and $I=0$, and the number of blocks inside the path is $\frac{1}{2}mn-1$.