  We deal with a graph colouring problem that arises in quantum information
theory. Alice and Bob are each given a $\pm1$-vector of length $k$, and are to
respond with $k$ bits. Their responses must be equal if they are given equal
inputs, and distinct if they are given orthogonal inputs; however, they are not
allowed to communicate any information about their inputs. They can always
succeed using quantum entanglement, but their ability to succeed using only
classical physics is equivalent to a graph colouring problem. We resolve the
graph colouring problem, thus determining that they can succeed without
entanglement exactly when $k\leq3$.
