Are there any open mathematical puzzles? Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious efforts, by mathematicians (or laymen for that matter)?
My question does not ask for puzzles that have been shown to have either no solution or multiple solutions (or have been shown to be ambiguously formulated).

In his comment, user Vincent Pfenninger referred to a YouTube video that, amongst other fascinating, layman accessible puzzles, discusses packing squares problems proposed by Paul Erdős. I thought I'd include it among the answers (as a community wiki).

How big a square do you need to hold eleven little squares?

We don't even know if this is the best possible [solution.]

Which, to me, comes as a complete surprise. :)
Here is the link to Erich's Packing Center provided in grey below the picture. It contains lots of proposed solutions to packing problems like this one.
