71. Cutting a Sphere. Into what maximum number of congruent pieces can a sphere be cut so that each side of each piece represents an arc of a great circle that is less than a quarter of such a circle?
71. Inscribed in this sphere is a regular dodecahedron or icosahedron, and perpendiculars are dropped from the center of the sphere to each face. Construct 60 isosceles triangles whose vertices are at the bases of these perpendiculars and whose bases are the sides of the corresponding faces. Now project these triangles from the center of the sphere onto the sphere. The resulting isosceles spherical triangles will be the congruent pieces into which the sphere should be cut.

[O. R. Ransom, A. M. M., 40, 114 (February 1933).]