How many colors do you need so no two points exactly one unit apart share a color?
Imagine coloring every point on an infinite flat plane so that any two points exactly one unit of distance apart always get different colors. How few colors could possibly make this work? Mathematicians have proven you need at least 5 and at most 7, but despite decades of effort since the problem was first posed, nobody has ever pinned the true minimum down to one exact number.
Reveal the answer
The answer is known only to be 5, 6, or 7, and remains genuinely unsolved: it's called the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson. A key 2018 improvement raised the known lower bound from 4 to 5 using a 1,581-vertex graph discovered with computer search, but closing the gap to a single exact value remains an open problem in mathematics today.
— Wikipedia contributors, Hadwiger–Nelson problem — Also called the chromatic number of the plane