What is the nine square grid problem/exercise as it relates to
architecture?  What are the fundamental rules of this grid exercise
and how does a student go about solving the problem?

I'm not an architect, so I don't know firsthand, but a Google search
for "nine square grid architecture problem" turned up this article
from Harvard Design Magazine:
http://www.architecture.neu.edu/dbuploads/Kit-of-Parts_Conceptualism_Love.pdf

A discussion of the nine square grid exercise begins on the second page.