Why is it impossible to cover a sphere that has radius $R$ with $3$ open semispheres of radius $R$? In my mind I have the pictorial image of the situation, but I can't find a formal proof.
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The union of two such semi-spheres always leaves an antipodal pair uncovered (where their boundaries intersect). This pair cannot be covered by a third semi-sphere since that does not contain any antipodal pair.