  We show that all $n$-qubit entangled states, with the exception of tensor
products of single-qubit and bipartite maximally-entangled states, admit
Hardy-type proofs of non-locality without inequalities or probabilities. More
precisely, we show that for all such states, there are local, one-qubit
observables such that the resulting probability tables are logically contextual
in the sense of Abramsky and Brandenburger, this being the general form of the
Hardy-type property. Moreover, our proof is constructive: given a state, we
show how to produce the witnessing local observables. In fact, we give an
algorithm to do this. Although the algorithm is reasonably straightforward, its
proof of correctness is non-trivial. A further striking feature is that we show
that $n+2$ local observables suffice to witness the logical contextuality of
any $n$-qubit state: two each for two for the parties, and one each for the
remaining $n-2$ parties.
