How many ways can you select six points in the plane so that every subset of three points forms the vertices of an isosceles triangle? This is a question asked by Erdős and recently resolved.
One solution is to choose the five vertices of a regular pentagon and the center.
It’s easy to verify that this is a solution. The much harder part is to show that this is the only solution.
A uniqueness proof was published on arXiv last Friday: A note on Erdős’s mysterious remark by Zoltán Kovács.