Three mutually tangent spheres of radius $3$ rest on a horizontal plane. A sphere of radius $1$ rests on top of them, tangent to all three. What is the distance from the plane to the top of the smaller sphere?
The centers of the three large spheres are at height $3$ and form an equilateral triangle of side $6$, with circumradius $\frac6{\sqrt3}=2\sqrt3$.
The small sphere's center lies above the center of this triangle at height $H$, at distance $3+1=4$ from each large center: $(2\sqrt3)^2+(H-3)^2=16$, so $H-3=2$ and $H=5$.
The top of the small sphere is at height $5+1=\boxed{6}$.
