The Race Between Achilles and the Tortoise
This is one quite puzzling thought experiment, dealing with the principle of asymptotic functions. Say that a tortoise (or any other suitably slow animal) agrees to a race with Achilles (or any other suitably faster person/animal/character). It is taken that Achilles is 10 times faster than the tortoise. To provide a handicap for the tortoise, it is given a head start of 10 meters. To keep all things simple, it is assumed (in terms of modern physics) that both the participants run at uniform speeds. The tortoise start the race, as Achilles watches on. When the tortoise reaches the 10 meter mark, Achilles begins his own running.
Now, Achilles will cover the distance between them in a given amount of time, being 10 times faster than the tortoise, but in that time, the tortoise would have moved another meter. Achilles would then cover that meter too, but the tortoise is now 10 cm ahead. Again, Achilles covers it, but the tortoise is a centimeter ahead still, and so on. The distance between Achilles and the tortoise is asymptotic, but by following the rational logic given, Achilles will never pass the tortoise, it being always a fraction of a distance ahead!
Using physics, the entire thing can be summed up into another situation. Two point objects are racing at uniform speeds. One has a head start, and the other is faster. Applying this logic again, the faster body will never cross the slower one, due to it's head start!
Now, using independent free body diagrams, we can say that Achilles would have crossed the tortoise easily. Assuming he ran at 10 m/s, in two seconds he would be 20 meters from the start, while the tortoise is 12 meters. Achilles also runs along with the tortoise at 1.11111111.... seconds, after which Achilles leads.
Therefore, applying logic and applying physics can give two different results. It also means that a seemingly asymptotic function (the distance between the racers) also grow closer faster. Every next 90% reduction of the distance between them took 90% less time. Hence, the function wasn't truly asymptotic.
Personally, I'm still baffled. Physics gives a rational answer, but applying normal, sound logic gives an irrational answer.
EDIT: Analysing a distance-time graph of the racers, and cross relating it with a time-distance between the runners graph gives an answer supporting physics (which is naturally inviolable). Hence, its the premises of the logic that makes no sense.
RE-EDIT: This logic will apply to a lot more than a race. The dichotomy paradox is a closely related theory, made by the same philosopher (Zeno). In this one, to get to a point a distance away, you need to cross the halfway mark between the points. And to get to the halfway mark, you need to get to the halfway mark of the halfway mark...and so on ad infinitum. This implies that motion cannot exist as a single process; you're doing an infinite number of processes.
An interesting fact that Diogenes the Cynic (someone whose simplicity was praised even by Alexander the Great), on hearing the theory and paradoxes. stood up and just walked across the podium, subtly showing what he felt about thought experiments and their accuracy.