How to guess an n-digit number

In a deductive game for two players, SF and PGOM, SF conceals an n-digit number x = x1, ..., xn an base q, and PGOM, who knows n and q, tries to identify x by asking a number of questions, which are answered by SF. Each question is an n-digit number y = y1, ..., yn in base q; each answer is the number of subscripts i such that xi = yi. Moreover, we require PGOM send all the questions at once. We show that the minimum number of questions required to determine x is (2+oq(1))n/logq n. Our result closes the gap between the lower bound attributed to Erdos and Renyi and the upper bounds developed subsequently by Lindstrom, Chvatal, Kabatianski, Lebedev and Thorpe. A more general problem is to determine the asymptotic formula of the metric dimension of Cartesian powers of a graph. We state the class of graphs for which the formula can be determined, and the smallest graphs for which we did not manage to settle.