4 colors problem is over a hundred years old and it is solved with more than 1200 hours of computer time. It concerns coloring of the countries on a map, using only 4 colors, such that no two neighboring countries would share the same color.
It is considered solved, though people kept finding errors in the steps used to find the solution, until in 1994 an independent method was used to prove it.
There's a very simple method that shows that in some situations, 4 color solution only works half the time and not the other half. No math is involved.