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).
The sofa problem.

From Wikipedia:


  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
Two envelopes problem

From wikpedia:
The two envelopes problem, also known as the exchange paradox, is a brain teaser, puzzle, or paradox in logic, philosophy, probability, and recreational mathematics. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. Historically, it arose as a variant of the necktie paradox.

The problem:


  You have two indistinguishable envelopes that each contain money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. You pick at random, but before you open the envelope, you are offered the chance to take the other envelope instead.


It can be argued that it is to your advantage to swap envelopes by showing that your expected return on swapping exceeds the sum in your envelope. This leads to the paradoxical conclusion that it is beneficial to continue to swap envelopes indefinitely.

Example -
Assume the amount in my selected envelope is \$20. If I happened to have selected the larger of the two envelopes, that would mean that the amount in my envelope is twice the amount in the other envelope. So in this case the amount in the other envelope would be \$10. However if I happened to have selected the smaller of the two envelopes, that would mean that the amount in the other envelope is twice the amount in my envelope. So in this second scenario the amount in the other envelope would be \$40. The probability of either of these scenarios is one half, since there is a 50% chance that I initially happened to select the larger envelope and a 50% chance that I initially happened to select the smaller envelope. The expected value calculation for how much money is in the other envelope would be the amount in the first scenario times the probability of the first scenario plus the amount in the second scenario times the probability of the second scenario, which is \$10 * 1/2 + \$40 * 1/2. The result of this calculation is that the expected value of money in the other envelope is \$25. Since this is greater than my selected envelope, it would appear to my advantage to always switch envelopes.

A large number of solutions have been proposed. The usual scenario is that one writer proposes a solution that solves the problem as stated, but then another writer discovers that altering the problem slightly revives the paradox. In this way, a family of closely related formulations of the problem have been created, which are discussed in the literature.
No proposed solution is widely accepted as correct. Despite this it is common for authors to claim that the solution to the problem is easy, even elementary. However, when investigating these elementary solutions they often differ from one author to the next. In the last two decades, several new papers have been published every year.