Maximizing Spheres: How Many Can Fit Around One?

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Suppose I have a bunch of spheres, all with the same radius r. Now, take one sphere and set it aside in R3. The question then, is, how many spheres can I place around this sphere such that at least one part of them is touching the central sphere?

I've arrived at a very strict upper bound of 27 spheres, because 27 spheres would fill the entire volume of space between the surface of the central sphere and the outer sufaces of the surrounding spheres. Other than that, I don't really know how to get to an actual number. Thoughts?
 
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