Given 4 spheres, assuming there is only a finite number of lines tangent
to all of them, how large can this number be? There is a well-known
upper-bound of 12, and a well-known lower bound of 12 with unit balls. If
we restrict the problem to *disjoint* unit balls, it is open. The best
known upper bound is still 12, and the best known lower bound is 8 with the
following example.

The four spheres have radius 80 and centers of coordinates
(0,0,0), (72,-171,-45), (-92,-130,38) and (-126,33,129). Notice that 4 of
the lines touch the spheres in one order and 4 in an other one.

With 4 spheres of radius 80 and centers (0,0,0), (-72,138,59),
(-82,21,172) and (-236,62,254), there are only 6 lines tangent to all the
balls, but all 6 touch the spheres in the same order.

These two exemples were obtained by generating 4 random balls and checking for lines tangent to them. The program never generated sets of balls with lines tangent to them in 3 different orders, or more than 8 tangents in total, and whenever there were 6 tangents in the same order, there were no other tangents.

The pictures were done in maple using code by Frank Sottile.