  The mean king problem is a quantum mechanical retrodiction problem, in which
Alice has to name the outcome of an ideal measurement on a d-dimensional
quantum system, made in one of (d+1) orthonormal bases, unknown to Alice at the
time of the measurement. Alice has to make this retrodiction on the basis of
the classical outcomes of a suitable control measurement including an entangled
copy. We show that the existence of a strategy for Alice is equivalent to the
existence of an overall joint probability distribution for (d+1) random
variables, whose marginal pair distributions are fixed as the transition
probability matrices of the given bases. In particular, for d=2 the problem is
decided by John Bell's classic inequality for three dichotomic variables. For
mutually unbiased bases in any dimension Alice has a strategy, but for randomly
chosen bases the probability for that goes rapidly to zero with increasing d.
