Don't worry about it, I thought exactly the same thing when I first saw the puzzle. It's very counter-intuitive.

]]>Sorry for all this posting.

Guests don't have an 'Edit' button.]]>

You're right, the distance does decrease.]]>

Logically, how can the caterpillar ever finish if every time it passes one cm, the distance increases by 100? ]]>

0 = ((((((100 - 1) * 2 - 1) * 3/2 - 1) * 4/3 - 1) * ... - 1) * (1 + 1/t))

But that doesn't change much.]]>

0 = (((((100 - 1) * 2) * 3/2) * ... * (1 + 1/t))

To make 'dt' zero, (1 + 1 / t) has to be zero, which is impossible.

Am I missing something?]]>

Mr. Big's reasoning is correct, but he forgets that each time the rope is stretched, the scale factor is less, which means the increase in distance is less. You can see it in his workings. First the caterpillar is 100cm away from the finish, then the finish gets 98cm further away, then 97.5cm and so on.

After a very long time, the increase in distance will become less than 1cm each time, meaning the caterpillar can start to gain ground again. Of course, it will still take ages to that because by that time the finish will be very far away indeed. But it will reach the finish.

]]>Let's assume there's a distance 'd' between the caterpillar and the end.

d = 100 cm.

The caterpillar travels 1 cm, d = 99 cm, the rope is stretched to 200 cm (twice as long), the distance scales by two, d = 198 cm.

The caterpillar travels another cm, d = 197, the rope is streched to 300 cm (3/2 as long), the distance scales by 3/2, d = 295.5 cm and so it goes on and on.

The distance only grows with time:

dt = (((((100 - 1) * 2) * 3/2) * (1 + 1/...)) * (1 + 1/t))]]>

Septimus wrote:478 689 242 000 000 000 000 000 000 000 000 000 years

Hmm...I doubt the caterpillar would actually live that long, though.

My answer: The caterpillar stands no chance of reaching the end.

I doubt the universe will live that long.

I could see that given enough time, the caterpillars progress each second would be greater than the rate at which the bungee lengthened, since the caterpillar's position with respect to time had a non-zero positive second derivative, while the second derivative of the bungee length was zero.

I wrote a C program to find the solution recursively, but after it ran for about twelve hours it was still not finished running.

]]>478 689 242 000 000 000 000 000 000 000 000 000 years

Hmm...I doubt the caterpillar would actually live that long, though.

My answer: The caterpillar stands no chance of reaching the end.

]]>Excellent work, humans.]]>

So, if we define the nth harmonic number as

then Hn grows about as fast as the natural logarithm of n.

And through some thinking, and some guidance from my math book, I got that there's little difference, so I went ahead and checked the difference.

After having compared various ones, I found out that there was a constant difference, which I later realized was Euler's constant (thanks wikipedia)

y or the greek letter gamma = ~0.577

SO!

ln (x) + y = 100 cm

ln (x) = 99.423

e^99.423 = 1,509594394 x 10^43 seconds

or

478 689 242 000 000 000 000 000 000 000 000 000 years

]]>So first it moves 1cm, then 1/2cm, then 1/3cm and so on.

You'd still get the same answer and it's easier to work out this way. There's probably an equation that will sum that series that you can use to work out how long it'll take.

]]>