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).

Wolfram's "New Kind of Science" book on cellular automata had a large accompanying list of open problems which must surely contain many suitable candidates (although note some of the questions are more in History of Mathematics territory).  I'd be surprised if they've been all ticked off in the last decade.
Wolfram's opus also serves as some inspiration as to how simple systems with complex emergent behaviour can create an enormous - and accessible - fresh new landscape of problems ripe for exploration.
