The foraging strategies used by animals are key to their success in spatially and temporally heterogeneous environments. We hypothesise that when a food source at a known location ceases to be available, flying insects will exhibit search patterns that optimise the rediscovery of such resources. In order to study these searching patterns, foraging honeybees were trained to an artificial feeder that was then removed, and the subsequent flight patterns of the bees were recorded using harmonic radar. We show that the flight patterns have a scale-free (Lévy-flight) characteristic that constitutes an optimal searching strategy for the location of the feeder. It is shown that this searching strategy would remain optimal even if the implementation of the Lévy-flights was imprecise due, for example, to errors in the bees' path integration system or difficulties in responding to variable wind conditions. The implications of these findings for animal foraging in general are discussed.