My journey so far has taken me to the following cities in order: New York (where I started from), Baltimore, Cincinnati, Sioux City, Monroe (La.), and Columbus (Ga.). What's my next stop?
I can never go back to New York.
Another hint, added later:
I can never go back to New York. That's not just because the route is determined and it happens never to lead back to New York — though that's true — but because there's some quality of New York's that implies that a route of this sort can never lead to it.