Can you color every two-dimensional map using just four colors so that no areas that share a common boundary are colored the same?
According to the Four-Color Map Theorem, and a bit of graph theory, the answer is yes, as explained by James Grime in the following video:
The cool thing about the proof is that it matters to more than just the visual presentation of maps in the real world. For instance, if you have a mobile phone network, you can keep the transmission signals from its cell towers from interfering with each other over the regions they overlap using just four sets of frequencies.
But since the theorem was originally conceived in connection to its use on maps, here's the world, in just four colors:
Image Credit: Fibonacci (on Wikimedia) via Creative Commons Attribution-Share Alike 3.0 Unported license.