This reminded me of yet another problem, but a more discreet one. It goes a bit like this.
A pair of lovers married yesterday, but in the craziness of the after party, they wake up in different houses (A house would be a point in the lattice grid Z*Z). The groom knows well that the wife is an anxious person, and will walk A streets North, B streets East and sleep there (A and B can be negative, and -4 north is 4 south for example), each day with constant A and B. "Woe is me!" He exclaims because he does not know where is she, and he does not know A nor B.
The groom is very fast so he can move from any house to another in a matter of seconds, but he can only check a house per day because he is a slow searcher. (That means, you can ask each day one house "Is the bride here?"). Can the groom, with complete certainty, catch the bride?