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).
The sofa problem.
It asks for the rigid two-dimensional shape of largest area A that
can be maneuvered through an L-shaped planar region with legs of unit
width. The area A thus obtained is referred to as the sofa constant.
The exact value of the sofa constant is an open problem.
Author of the picture: Claudio Rocchini, see this link