Vrangr wrote: Two unit cubes have a common center. Is it always possible to number the vertices of each cube from $1$ to $8$ so that the distance between each pair of identically numbered vertices would be at most $4/5$? What about at most $13/16$? BTW ,as they have given whether it is always possible , if we show that for atleast 1 case , distance is ${\geq}$ $\frac{13}{16}$ , aren't we done ??

We would be done but i don't think there are any counter examples to this problem...