This question is from the German mathematics competition Känguru der Mathematik. In this competition students have to solve 30 mathematical tasks like this in 90 minutes without calculator. Actually they are given 5 possible answers, but for this community a bit of additional challenge does not hurt.
A regular $15$-gon $A_1 \, A_2 \, ... \, A_{15}$ and a regular $n$-gon $B_1 \, B_2 \, ... \, B_n$ with side lengths of $1$ have a common side: $B_1 B_2 = A_2 A_1$. What value of $n$ makes the distance between $B_3$ and $A_{15}$ be $1$?