The vertices of a regular octahedron are the centers of the six faces of a cube with edge length $6$. A sphere is inscribed in the octahedron. What is the radius of the sphere?

1. Place the cube as $[-3,3]^3$; the octahedron has vertices $(\pm3,0,0),(0,\pm3,0),(0,0,\pm3)$, so it is the solid $|x|+|y|+|z|\le3$.
2. By symmetry the inscribed sphere is centered at the origin, and its radius is the distance from the origin to a face, for example the plane $x+y+z=3$.
3. That distance is $\dfrac{3}{\sqrt3}=\boxed{\sqrt3}$.