The Four-Color Theorem Has Haunted Mathematicians for Decades. It Just Got a Rare New Proof

The famous problem asks whether a map can be filled in with only four colors such that the same color never borders itself. It has been solved with the help of computers, but some experts suggest an elusive, short answer is out there. The Four-Color Theorem Has Haunted Mathematicians for Decades. It Just Got a Rare New Proof, but Some Still Long for a Simpler Solution.

The four-color theorem is simple to state: Given a contiguous map, is it possible to color each region with one of four colors such that no neighboring regions share a color? Vico Santos for Quanta magazine The original version of this story appeared in Quanta magazine. Some math problems continue to haunt researchers long after they’ve been solved.

A proof emerges, is even celebrated, and yet dissatisfaction lingers. Perhaps the argument is too convoluted—the hunt persists for the elusive one-page paper—or perhaps it fails to give a deeper theoretical insight into why something is true. Whatever the reason, mathematicians return, again and again, to a case that is otherwise considered closed.

One of the most famous such cases is that of the four-color theorem, a problem that transformed how mathematicians think about their subject.