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Disjoint Congruent Circles

A set of n disjoint, congruent circles Oi(r ) (i = 1,2,…n) packs the surface of a sphere S so that each region of the surface exterior to the circles is bounded by arcs of three of the circles.  Find the possible values of the number n of circles, and in each case express r in terms of the radius of S.

Japan, Edo Period, 1603-1867

"Disjoint Congruent Circles," Convergence (March 2006)