Consider the problem of searching for a leprechaun that moves randomly between two sites. The movement is modelled with a two-state Markov chain. One of the sites is searched at each time t = 1,2,…, until the leprechaun is found. Associated with each search of site i is an overlook probability αi and a cost Ci Our aim is to determine the policy that will find the leprechaun with the minimal average cost. Let p denote the probability that the leprechaun is at site 1. Ross conjectured that an optimal policy can be defined in terms of a threshold probability P* such that site 1 is searched if and only if p ≥ P*. We show this conjecture to be correct (i) when α1 = α2 and C1 = C2, (ii) for general Ci when the overlook probabilities α, are small, and (iii) for general αi and Ci for a large range of transition laws for the movement. We also derive some properties of the optimal policy for the problem on n sites in the no-overlook case and for the case where each site has the same αi, and Ci.