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, . . ., x_n in 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 x_i = 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+o_q(1))n/log_q n. Our result closes the gap between the lower bound attributed to Erdos and Rényi and the upper bounds developed subsequently by Lindström, Chvátal, 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.