Covering the \(k\)-skeleton of the 3-dimensional unit cube by six balls (Q740662)

Denote by \(b(n,d,k)\) the least number \(r\) such that the union of \(k\)-faces of the \(d\)-dimensional unit cube can be covered by \(n\) balls of radius \(r\). Let \(x_0=\frac{15+9a-a^2}{12a}\), where \(a=\sqrt[3]{215+3\sqrt{5559}}\). Then it is proved that \[ b(3,6,3)=b(3,6,2)=b(3,6,1)=\frac{1}{2}\sqrt{1+2x_0^2}=0.5379\ldots . \]