How far are the corners of a $1 \mathrm{~m}$ edge length cube from the body diagonal?
When a cube is cut by a plane passing through a body diagonal and a vertex, the section is a right-angled triangle with side lengths of $1 ; \sqrt{2} ; \sqrt{3}$. The sought distance is the height of this triangle corresponding to the hypotenuse, which is $\frac{\sqrt{6}}{3}$.