12. (2001 National High School Competition Question) The edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ is 1, then the distance between the line $A_{1} C_{1}$ and $B D_{1}$ is
12. Fill in $\frac{\sqrt{6}}{6}$. Reason: Consider the section $B B_{1} D_{1} D$ of the cube, then $A_{1} C_{1} \perp$ plane $B B_{1} D_{1} D$. Let $A_{1} C_{1}$ intersect $B_{1} D_{1}$ at point $O$, and in the plane $B B_{1} D_{1} D$, construct $O H \perp B D_{1}$ at $H$, then $O H$ is the common perpendicular of $A_{1} C_{1}$ and $B D_{1}$. Clearly, $O H$ is half the altitude of the hypotenuse of the right triangle $\triangle B B_{1} D_{1}$, i.e., $O H=\frac{\sqrt{6}}{6}$.