It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement between Isaac Newton ("impossible") and David Gregory ("possible"). The cause of the dispute was, in part, that there seems to be a lot of room left over after 12 spheres.

How close of a call is it, so to speak? What is the largest possible $r$ so that it is possible to arrange 12 spheres of unit radius and a 13th sphere of radius $r$ all tangent to another unit sphere, without intersections? What does the optimal configuration look like?

rso that there are 13 disjoint unit tangent balls to a ball of radiusr-somehow more symmetric – Pietro Majer Apr 21 '12 at 15:08