Unfortunately, (for the complete layman) since the last century, not only the answers but also the problems themselves have most often been impossible to understand.
I found the question interesting about simple open mathematical puzzles understandable to layman understandable by the complete layman: (Are there any (mathematical) puzzles that are still unresolved?)
I would like to add to the question: Are there any such, i.e. understandable, questions which have actually been answered by a layman (let us say after 1900)
In 1975 Martin Gardner published an article in Scientific American about tiling the plane with pentagons. It was believed that all such ways were found, but Marjorie Rice, an amateur with a high-school education, found four more through systematic study.
You can find a whole list of amateur mathematicians on Wikipedia.