Say that one polygon $P$ is inscribed in another one $Q$, if $P$ is contained entirely in (the interior and boundary of) $Q$ and every vertex of $P$ lies on an edge of $Q$. It's clear that a regular $m$-gon can be inscribed in a regular $n$-gon if $m\mid n$ or $m=2n$. Well-known results (e.g. the Square Peg problem) imply that it is always possible when $m$ is 3 or 4, and it's not hard to use the Intermediate Value Theorem to see that it can happen whenever $m\mid 2n$. It's also not hard to see that a regular pentagon cannot be inscribed in an equilateral triangle, square, or regular hexagon.

Is it ever possible to inscribe a regular $m$-gon in a regular $n$-gon when $m>4$ and $n$ are relatively prime? If not, are there even any cases where it can happen that are not on the above list? Note that this question is not original; it was asked for example as P220 of The Playground in Math Horizons, vol. 16, no. 1, 2008 Sep. But no answer was ever published in Math Horizons and I have not been able to find any other reference.