III OBM

Let the cube have side $k$. $A$, $B$ are two adjacent vertices of the small octahedron, and $AX=BX=k/2$ and $\angle AXB=90^{\circ }$, so $AB=k/\sqrt{2}$.



The large octahedron has the vertices of the cube at the center of its faces. The line joining the centers of $OPS$ and $OQR$ is parallel to $PQ$ and $2/3$ the length. But it is also $k\sqrt{2}$, so the side of the large octahedron is $(3/2)k\sqrt{2}=3k/\sqrt{2}$ or $3$ times the side of the small octahedron. Hence the volume of the large octahedron is $3^3=27$ times the volume of the small.