Cross your eyes and look at this
Here's a cool picture drawn by David Rowland. Cross your eyes! When the two images merge you'll get a nice 3d view of a doughnut with 7 hexagons drawn on it. It works better if you make the image as big as you can.
Each hexagon touches all the others. So, if the Earth were a doughnut divided into 7 countries this way, map-makers would need 7 colors of ink! That's the most they could need for a doughnut-shaped Earth, though.
If the Earth were a 2-holed doughnut, we might need as many as 8 colors. In general, for a doughnut with any number of holes, say g holes, the number is given by this wacky formula:
floor((7 + sqrt(1 + 48g)/2))
where "floor" means the largest integer less than or equal to the stuff in the parentheses.
This formula was conjectured by Percy John Heawood in 1890. The map in the picture here is called the Heawood graph
, and the conjecture is called the Heawood conjecture
The Heawood conjecture was proven by Gerhard Ringel and J. W. T. Youngs in 1968... except for the case g = 0, the case of a sphere, with no holes. That case, the 4-color conjecture
, turned out to be much harder! But that's another story for another day!
For more, try:https://en.wikipedia.org/wiki/Heawood_conjecture