Imagine we have two size $m$ sets of points $X=\{x^i\}_{i=1..m}, Y=\{y^i\}_{i=1..m} \subset \mathbb{R}^n$ and we need to answer the question if they differ only by rotation: if there exists othogonal $O$ such that $X=OY$. Can it be answered in polynomial time? While it sounds simple, it isn't - ...