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Originally Posted by politicophile
Confusing problem, especially at first glance. The tortoise will continuously be moving forward one foot for every three feet the band is lengthened. Watch the numbers:
Tortoise's distance from the flower (in feet):
3 - 2 - 5 - 4 - 7 - 6 - 9 - 8 - 11...
The tortoise will get further from the flower over time, not closer.
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I don't pretend to have the math intelligence to actually solve this thing, but I did notice something that might help someone smarter than me do it. The turtle walks 1 foot. Achilles then stretches the WHOLE band, including the part of the band the turtle has already walked over, 3 feet. Then the turtle walks another foot, so now even more of the band is behind him, so even more of that 3 foot stretch will be behind him. So each time the turtle moves, there's less stretching of the band in front of him since an ever-increasing part of the 3 foot stretch will be behind him.
I think he'll eventually reach Achilles, but like I said, I'm not good enough to figure out when.