HMMT February

The centers of the smaller spheres lie on a tetrahedron. Let the points of the tetrahedron be $(1,1,1),(-1,-1,1),(-1,1,-1)$, and $(1,-1,-1)$. These points have distance $\sqrt{(} 3)$ from the center, and $\sqrt{(} 2)$ from each other, so the radius of the smallest sphere in which 4 spheres of radius $\sqrt{(2)}$ will fit is $\sqrt{(2)}+\sqrt{(3)}$. Scale this to the correct answer by dividing by $\sqrt{(} 2): \frac{2+\sqrt{6}}{2}$.