We claim that this tiling of the Earth is impossible.

This is basically asking whether or not there exists a planar graph with all faces having size and each vertex having degree on a sphere. This is also equivalent to having the same conditions on a planar graph on a regular plane since it is possible to project any graph on a sphere onto a regular plane.

Assume for the sake of contradiction that this graph is planar. If it satisfies the conditions for a planar graph on a plane, then it should satisfy Euler's Formula, where:

Suppose we use tiles to tile Earth. Then for obvious reasons.

To count the number of edges with this configuration with faces, we realize that each face has edges, for a total of edges. However, we have over counted by a factor of two, since each edge borders two faces, for an actual value of edges. The number of vertices is also easily counted, with vertices over all the faces, but then we overcount by a factor of 4 since 4 faces share each vertex, for vertices.

We may plug our values of , , and into Euler's Formula:


This is clearly false, so we have a contradiction. Hence no planar graph of this sort exists, and thus we cannot tile the Earth with square tiles.