AustriaMO2013

Each arrangement of $B_1,\dots ,B_n$ begins with $B_1$ and ends with $B_k$ for some $k$ with $2\le k\le n$. From $B_1$ through $B_k$ the projections are arranged from "top to bottom" and from $B_k$ through $B_n$ and back to $B_1$ from "bottom to top". The $k-2$ projections $B_2,\dots ,B_{k-1}$ assume some $k-2$ of the $n-2$ intermediate positions between $B_1$ and $B_k$, and each of these choices of $k-2$ positions uniquely determines the entire sequence of the $B_i$. Since there are $\left(\genfrac{}{}{0ex}{}{n-2}{k-2}\right)$ such choices possible, we see that the total number of sequences of projections is equal to $$\sum _{k=2}^n\left(\genfrac{}{}{0ex}{}{n-2}{k-2}\right)=\sum _{i=0}^{n-2}\left(\genfrac{}{}{0ex}{}{n-2}{i}\right)=2^{n-2}.$$