Authors’ abstract: “An object is hidden in one of two boxes and moves between the boxes in accordance with some specified continuous time Markov process. The objective is to find the object with a minimal expected cost, where the cost is a linear function of the following three variables: the expected time spent on searching and the expected total search effort in each box. The search policy is characterized by unbounded search intensities in each box as a function of time. The authors obtain an interesting characterization of the optimal policy based on several intervals of the posterior probability of the object being in the first box (there can be 2 to 5 such intervals). The optimal search intensity is either 0 or \(\infty\). Numerical comparisons with the bounded version of this problem, studied by Weber, show that the unbounded policy presented in this paper can sometimes be much more efficient”.