$23 \cdot 28$ On the surface of a cube, there exist skew diagonals of the square faces. If the distance between two such skew diagonals is 1, then the volume of the cube is
(A) 1.
(B) $3 \sqrt{3}$.
(C) 1 or $3 \sqrt{3}$.
(D) $3 \sqrt{3}$ or $3 \sqrt{2}$.
(2nd "Hope Cup" National Mathematics Invitational Competition, 1991)
［Solution］We discuss in two cases:
(1) If the two skew diagonals are on opposite faces of the cube (see Figure (1)), knowing that the edge length of the cube is 1, then
$$
V_{\text {cube }}=1^{3}=1 .
$$
(2) If the two skew diagonals are on adjacent faces of the cube (see Figure (2)), the edge length $a=\sqrt{3}$ can be found. Thus,
$$
V_{\text {cube }}=(\sqrt{3})^{3}=3 \sqrt{3} \text {. }
$$

Therefore, the correct choice is (C).