# Achilles & The Tortoise Paradox

“In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

—Aristotle, Physics VI:9, 239b15

Here is another interesting paradox which proves that mathematics can be beautifully strange sometimes.

This is a story of Achilles and a tortoise in a foot race. What this paradox says is that Achilles will never be able to overtake the tortoise if you give tortoise some headstart. Strange isn’t it? Let me prove it using mathematics

Imagine a straight line race between Achilles and the tortoise.

Assumptions:

1. Assume that the tortoise was given a 100m head start.

2. Assume both the Achilles and the tortoise are travelling at constant speed where Achilles is travelling faster.

After some fixed amount of time, say t0 – Achilles would have reached the initial position of tortoise (100 m). During this time t0 tortoise would have moved further by some distance x1, say 20 meters.

After some more time t1, Achilles crossed another 20 meters and reached tortoise’s previous position and that the tortoise moved further 4 meters

After some more time t2, Achilles moved further 4 meters and reached tortoise’s previous position and by that time tortoise would have moved further 0.8 meters

If you keep doing this infinitely you would see that Achilles will never be able to cross the tortoise because he will always have to reach tortoise’s last postion.

Take a look at this interesting video shown below and see for yourself.