A surprising result
Surprisingly, the answer to all these questions is negative. We suggest that the reader does not proceed without first reflecting on what conclusions we can take form this information...
As we did in the planar case, we try to see if, using inversion in the three dimensional space, it is possible to reduce this general problem into one where conclusions are easy to draw. To so proceed in this case is more straightforward than in the previous problem. It is enough to check that, taking the tangency point of two of the three spheres as the center of inversion, those two spheres are transformed into two parallel planes and the third is transformed into a sphere \(S\) that is tangent to both planes (see figure 21 and use the following applet). We leave to the reader the task of reflecting on what will be the ring of spheres that are tangent to \(S\) and those two planes (see figure 21).
After all this, the name hexlet for this geometry problem seems rather natural. In figure 22 the same ring of spheres that are represented in figure 20 (corresponding to the triplet on figure 19) is represented but seen from a different viewpoint and without the fixed spheres.