You may find
this problem similar.
In a cube of side 4, I pack eight spheres of unit radius.
What is the largest sphere I can place in the center (such that it doesn't overlap any of the other spheres)?
(In reply to
re(2): another dimension by Tristan)
That's pretty good, Tristan... I start having trouble imagining such a cube, when the dimensions exceed 3... :-)
-SK
P.S. yup... the sides of a 9-cube have 8 dimensions.