The question that asks if a square of area $4n^2$ is the geometric figure with the most number of domino tilings, of all figures of the same area, [here][1],
is still unsolved. I find this question quite intriguing.


  [1]: https://mathoverflow.net/questions/162020/which-region-in-the-plane-with-a-given-area-has-the-most-domino-tilings