I am interested in maximally distant states of the cube. I have often
wondered just what a maximally distant state would look like. I also
wonder HOW MANY of them there are!

Interesting fact (offered without proof (it's not hard)):

Assuming we are counting quarter-twists. If I hand you a cube in a
maximally distant state, and ask you to solve it in as few twists as
possible, you don't have to think at all in order to know what to do
first! ANY first twist will bring it closer to home (after that it
gets harder).

Call a state with this property a "local maximum". Any maximally
distant state is also a local maximum. Also, any symmetric state is a
local maximum. This doesn't mean that a maximally distant state is
symmetric, but it does get you thinking along those lines!