I am really struggle how to "permanently" eliminate this type of
weed on my brick patio.

Haven't really tackled my own brick patio (which has more weeds than
yours) yet. But a few techniques (in addition to those which were
posted) a

* boiling water

* cover with a tarp or other opaque object long enough to kill them
from lack of light

I'd love to figure out a way to outcompete the weeds with moss or
something attractive (most of my cracks are bigger than the ones at
http://tinypic.com/537zqsm.jpg although irregular due to poor
installation and/or years of settling). But I'm not sure how easy
that is going to be. My neighbor has a dwarf mondo grass which works
well for him, but his cracks are bigger (designed for this), and it
did take a while (as a slow-growing plant).