How many random questions are necessary to identify n distinct objects?

Abstract Suppose that X and A are two finite sets of the same cardinality n ⩾ 2. Assume that there is a bijective mapping φ: X → A which is unknown to us, and we must determine it. We are allowed to ask a sequence of questions each posed as follows. For a given B ⊂ A what is φ−1(B)? In this paper we study a case when the subsets B are chosen uniformly at random. The main result is: if each subset has to split all the atoms of a field generated by the previous subsets, then the total number of questions (needed to determine the mapping completely) is log 2 n + (1 + o p (1))(2 log 2 n) 1 2 . Here op(1) stands for a random term approaching 0 in probability as n → ∞.