Quote:
Originally Posted by pdurrant
The gods have heard rumours that Achilles might not be as fast at running as they thought, so they decide to set him a task.
He must run a race on a special 100m racetrack. The special thing about this racetrack is that every time Achilles has run 10m along the race track, the whole track instantly stretches, as if made of rubber, by 100m.
Now Achilles can run indefinitely as a constant 10m/s. He really is a fast runner — good endurance too.
Have the gods set Achilles an impossible task? Can he ever finish the race?
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Here's the answer:
Spoiler:
The gods have not set Achilles an impossible task, he can finish the race.
It seems strange as first sight that he could ever finish, as the track grows by 100m every second, while Achilles only runs 10m every second.
The crucial point is that the 100m isn't added just to the end of the track, but is added equally all along the track, including the bits of the track that Achilles has already covered.
So, in the 1st second, Achilles runs 10m, which is 10% of the track. The track then increases to 200m, but because the extra length is added equally, Achilles is still 10% of the way along the track.
In the 2nd second, Achilles runs another 10m, which is 5% of the longer track, taking him to 15% overall. The track then increases to 300m, but Achilles is still 15% of the way along, now 45m along the track, despite having only run 20m.
In the 3rd second, Achilles runs another 10m, which is 3 1/3% of the longer track, taking him to 18 1/3% overall. The track then increases to 400m, but Achilles is still 18 1/3% of the way along, now 73 1/3m along the track, despite having only run 30m.
This continues. Every second Achilles runs a smaller and smaller percentage of the track, but the percentage he covers always increases.
The question is, does the percentage ever reach 100%? It might not. If he covered, in each second, 10%, 5%, 2.5%, 1.25%, 0.625%, 0.3125%, etc. he'd never get past 20% of the way along the track!
It's sometimes easier to look at fractions rather than percentages. With the conditions in this problem, every second from the start of the race, Achilles covers
1/10, 1/20, 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90, ...
So to find out if Achilles finishes the race, we need to know if the sum of all those fractions ever reaches 1.
Consider the following sequence of fractions:
1/10, 1/20, 1/40, 1/40, 1/80, 1/80, 1/80, 1/80, 1/160, ...
Each of these fractions is less than or equal to the corresponding fraction in the sequence we're interested in. But this sequence clearly (!) reaches and exceeds one — just look at it like this:
1/10 + 1/20 + (1/40 + 1/40) + (1/80 + 1/80+ 1/80 + 1/80) + (1/160 + ...
each of the bracked terms adds up to 1/20, and we obviously have more than twenty such bracketed terms, so this sequence, when added all together, is more than 1.
So the sequence of fractions we're interested in, since every fraction in it is greater than or equal to the corresponding term in the sequence that does exceed 1, also adds up to more than 1 — which corresponds to 100% of the track length.
So we know that Achilles will eventually reach the end of the track. It will take him some time though. Other answers in this thread give the results of numerical calculations to work out roughly how long it will take him.