The Achilles paradox is a famous philosophical puzzle created by the ancient Greek philosopher Zeno of Elea around 450 BCE. It argues that a swift runner like the hero Achilles can never overtake a slow tortoise in a footrace if the tortoise is given a head start.

Because Achilles is much faster, he generously grants the tortoise a 100-meter head start.

Zeno argues that Achilles can never overtake the tortoise due to a continuous cycle:

1. The First Gap: To catch the tortoise, Achilles must first run the 100 meters to reach the tortoise’s starting point. During the time it takes Achilles to get there, the tortoise slowly creeps forward by a smaller distance—say, 10 meters.
2. The Second Gap: Achilles must now run those additional 10 meters. By the time he reaches that second point, the tortoise has moved forward again, perhaps by 1 meter.
3. The Infinite Loop: Achilles must then run that 1 meter, during which the tortoise advances a few centimeters.

Zeno claimed that because space can be infinitely subdivided, this sequence repeats forever. Every time Achilles arrives where the tortoise just was, the tortoise has moved a tiny bit further ahead. Therefore, Achilles can narrow the distance, but mathematically, he can never catch up.