Maximum number of balls included in one ball

Question

I guess this question has received an answer since a long time, but I was not able to find it (bad queries on Internet, I suppose):

Take a ball $\mathcal{B}$ of radius $r$ in $\mathbb{R}^3$, for example centered in $0=(0,0,0,)$, $\mathcal{B}=\{u \in \mathbb{R}^3, d(u,0) \leqq r \}$, $"d"$ being the usual euclidean distance. What is the maximum number of balls of radius $x$ (of course, $0 \leqq x \leqq r$) that can be included in $\mathcal{B}$ ? (I mean: possibly tangent, but separate, not intersecting...)

Thanks for any guidance towards a solution.

Answer 1

This problem is known as the problem on dense (or optimal) packing of spheres in spheres, and its usual formulation (adapted to your notation) is given $r=1$ and number $n$ of spheres, find the largest $x$ such that $n$ spheres of radius $x$ can be included in $\mathcal B$. I remark that even two-dimensional counterpart of the problem is hard for not-so-small $n$, both finding the best $x$ (for which programs realizing special algorithms can be used) and proving its optimality. I guess this problem is well-studied and the respective results can be found in Internet. Offline I have only the following best computer found values of $x$ for some $n$ from Dave Boll’s page from 2002 (now dead).

14  0.323313
15  0.318305
16  0.310976
17  0.305694
18  0.301296
19  0.295332
20  0.287851
21  0.286833
22  0.279334
23  0.275081
24  0.271336
25  0.27112
26  0.2666792
27  0.26212
28  0.260096
29  0.257819
30  0.25478
31  0.253115
32  0.250712
33  0.248703
34  0.247006
35  0.244773
36  0.241787
37  0.240441
38  0.24036
39  0.236703
40  0.234871
41  0.232702
42  0.232119
43  0.229667
44  0.228083
45  0.226594
46  0.225164
47  0.223342
48  0.222363
49  0.220980

and pictures of packings for

$n=17$

$n=30$

$n=32$.