If you've ever redecorated a bathroom, you'll know that there are only so many ways in which you can tile a flat plane.
Proof: We show first that for any polyhedron we have 2E ≥ 3F and 2E ≥ 3V. The faces of the polyhedron are polygons, each bounded by a number of sides. Along each edge exactly two faces come together, so an edge corresponds to exactly two sides: the total number of sides is 2E. We also notice that any face has at least 3 sides, so the total number of sides is at least 3 times the number of faces.
The following diagrams show our algorithm as applied to the network arising from the dodecahedron.
Have a go at this Plus-themed Sudoku variation.
