Covering the sphere by equal zones

A zone of half-width w on the unit sphere S (2) in Euclidean 3-space is the parallel domain of radius w of a great circle. L. Fejes Tth raised the following question in [6]: what is the minimal w (n) such that one can cover S (2) with n zones of half-width w (n) ? This question can be considered as a spherical relative of the famous plank problem of Tarski. We prove lower bounds for the minimum half-width w (n) for all n a parts per thousand 5.