Chuckles

Long Walk to Forever ?*

While studying limits, a friend of yours, who fancies himself as a modern day Zeno**, proposes the following variation on Achilles and the Tortoise story.

Imagine a three-foot elastic band with a small tortoise sitting on one end (fixed) and Achilles, with a delicious flower, holding the other end of the band. Naturally, the hungry tortoise starts to walk toward the flower. However, when the tortoise reaches the one-foot mark, Achilles stretches the (whole) band an additional three feet in length. Undaunted, and perhaps a little slow, the tortoise walks another foot and once again Achilles stretches the band another three feet. If this situation continues stretching in this same manner, will the tenacious tortoise ever reach the end of the band and receive the flower from Achilles?

Find the distance the tortoise walks, and the ratio of the distance walked to the total length of the band.


*The title is borrowed from a short story by Kurt Vonnegut. The problem is derived from “Mind Benders” in Discover magazine.

** Search the Web for Zeno’s Paradox.



a hint is that the tortoise is always along for the ride. I figure this means that when the band moves the three feet, the tortoise also gets moved along with it because the whole band is moving universally at the same time.