17. (GDR 3) A ball $K$ of radius $r$ is touched from the outside by mutually equal balls of radius $R$. Two of these balls are tangent to each other. Moreover, for two balls $K_{1}$ and $K_{2}$ tangent to $K$ and tangent to each other there exist two other balls tangent to $K_{1}, K_{2}$ and also to $K$. How many balls are tangent to $K$ ? For a given $r$ determine $R$.
17. Centers of the balls that are tangent to $K$ are vertices of a regular polyhedron with triangular faces, with edge length $2 R$ and radius of circumscribed sphere $r+R$. Therefore the number $n$ of these balls is 4,6 , or 20 . It is straightforward to obtain that:
(i) If $n=4$, then $r+R=2 R(\sqrt{6} / 4)$, whence $R=r(2+\sqrt{6})$.
(ii) If $n=6$, then $r+R=2 R(\sqrt{2} / 2)$, whence $R=r(1+\sqrt{2})$.
(iii) If $n=20$, then $r+R=2 R \sqrt{5+\sqrt{5}} / 8$, whence $R=r[\sqrt{5-2 \sqrt{5}}+$ $(3-\sqrt{5}) / 2]$.