Under what conditions is the Dining Philosophers Problem unsolvable?
Well, if the philosophers are following the exact same strategy and there is nothing to distinguish them (like who is closest to the door etc.) then they will do exactly the same and deadlock must occur: Whenever one (Aristotle) picks up the left fork then the philosopher on the right (Beauvoir) takes Aristotle's right fork.

If you allow access to a source of randomness this should be easily fixable, however (everyone eats for a short time when possible and waits for a long, random, time thereafter).