What is the minimum number of colours required to draw a map so that no two adjacent regions have the same colour?

Even though we now know that 4 colors is enough to color any map, some maps can be colored with fewer colors--three colors or even two. Finding a method to determine exactly how many colors is needed for any map continues to daunt mathematicians today. Some mathematicians believe that a fast method to find out the minimum number of colors needed for a map is impossible (that is, it simply does not exist and hence can never be found).
This is widely regarded as the most important unsolved problem in computer science, mathematical logic, and the foundations of mathematics.

https://en.wikipedia.org/wiki/Four_color_theorem

Interestingly, that theorem only holds for the plane. If you got bored during breakfast and tried drawing a map on a doughnut (assuming it isn't jelly-filled), you could actually need as many as 7 colors.

[May 22 04 3:27 PM] kevinatilusa writes:

The place I got that from is a combinatorics textbook of mine by Van Lint and Wilson. I think it's called Heawood's conjecture/theorem or something similar (unfortunately, I don't have the book here with me).

Some people think the Four Color Theorem in the plane is proved but this is quite controversial.
To find out more try the following:
https://mathshistory.st-andrews.ac.uk/HistTopics/The_four_colour_theorem/