Problems Section: 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?

Q1352 Find infinitely many triangles with integer side lengths which contain an angle of $120^\circ$.
(a) On a $4\times n$ chessboard we wish to place $2n$ knights in such a way that none attacks any other.  Give three possible ways of doing this.
(b) Prove that there is no closed knight's tour on the $4\times n$ board.