Abstract. We consider the problem of searching on m current rays for a target of unknown location. If no upper bound on the distance to the target is known in advance, then the optimal competitive ratio is 1 + 2 m^m / (m - 1)^(m-1) We show that if an upper bound of D on the distance to the target is known in advance, then the competitive ratio of any search strategy is at least 1 + 2 m^m / (m - 1)^(m-1) - O(1 / log^2 D) which is again optimal---but in a stricter sense. To show the optimality of our lower bound we construct a search strategy that achieves this ratio. Surprisingly, our strategy does not need to know an upper bound on the distance to the target in advance; it achieves a competitive ratio of 1 + 2 m^m / (m - 1)^(m-1) - O(1/log^2 D) if the target is found at distance D Finally, we also present an algorithm to compute the strategy that allows the robot to search the farthest for a given competitive ratio C.