Village Road Network

In the nation of Chefland there are villages at co-ordinates $(x,y)$ whenever $x$ and $y$ are non-zero odd co-prime integers. $x$ can be either positive or negative, but $y$ is always positive. In each village $(x,y)$ there is a crossroads, with the roads marked North, South, East, West. Road East leads to $(x+2y, y)$. Road West leads to $(x-2y, y)$. Road North leads to either $(x, y+2