  The Mean King's problem asks to determine the outcome of a measurement that
is randomly selected from a set of complementary observables. We review this
problem and offer a combinatorial solution. More generally, we show that
whenever an affine resolvable design exists, then a state reconstruction
problem similar to the Mean King's problem can be defined and solved. As an
application of this general framework we consider a problem involving three
qubits in which the outcome of nine different measurements can be determined
without using ancillary qubits. The solution is based on a measurement derived
from Hadamard designs.
