Pie chart example makes a good case for not counting a point. In a real world situation. How many two countries share only a point between the two? So many borderlines to count between diagonally situated countries. In the simplified world, diagonally situated countries only share a point with each other, never a border. Thus the necessity to change the convention of not counting a point.
Coloring a map of the square world using a unit function following its rule for expanding the coloring scheme for more countries around it guarantees that the theorem works.
If the map has a country bigger than the rest on the map, bigger by a multiple of even number and there is at least one more column next to that country to its rightside, the guarantee no longer stands.