23.30 Connecting the centers of adjacent faces of a cube, a regular octahedron is obtained. Then, the ratio of the volume of this octahedron to that of the cube is 
(A) $\frac{\sqrt{3}}{12}$.
(B) $\frac{\sqrt{6}}{16}$.
(C) $\frac{1}{6}$.
(D) $\frac{\sqrt{2}}{8}$.
(E) $\frac{1}{4}$.
(40th American High School Mathematics Examination, 1989)
[Solution] Let the edge length of the cube be 2. It is easy to find that the edge length of the inscribed regular octahedron is $\sqrt{2}$. The volume of a regular octahedron with edge length $\sqrt{2}$ is $V_{1}=\frac{4}{3}$.

The volume of the cube is $V_{0}=8$.
$$
\therefore V_{1}: V_{0}=\frac{4}{3}: 8=1: 6 \text {. }
$$

Therefore, the answer is $(C)$.