- 09 May '05 04:27Suppose an inchworm starts traveling along an elastomeric rubber rope (initially one meter in length) at 1 cm per second.

But, after each second, suppose the rope is**uniformly**stretched another meter in length. So the worm travels 1 cm, the rope gets uniformly stretched 1 meter, the worm travels another cm, the rope gets stretched another meter, etc., etc.

Will the worm ever make it to the end of the rubber rope? - 09 May '05 05:21

Yes. As the rope is stretched, the worm will get pulled with it.*Originally posted by davegage***Suppose an inchworm starts traveling along an elastomeric rubber rope (initially one meter in length) at 1 cm per second.**stretched another meter in length. So the worm travels 1 cm, the rope gets uniformly stretched 1 meter, the worm travels another cm, the rope gets stretched another meter, etc., etc.

But, after each second, suppose the rope is [b]uniformly

Will the worm ever make it to the end of the rubber rope?[/b]

After one second, rope length is doubled, so the distance the worm moved is doubled. The worm actually has moved 2cm. After 2 s, the worm has moved (2x1 + 3/2x1) cm ... after 3 s, the worm has moved (2x1 + 3/2x1 + 4/3x1)cm ... the stretching of the rope will not in any way modify the amount of time it takes the worm to traverse the rope. - 09 May '05 05:55

I am not disclosing yet whether I agree or disagree with your overall conclusion that the inchworm will eventually reach the end.*Originally posted by AThousandYoung***the stretching of the rope will not in any way modify the amount of time it takes the worm to traverse the rope.**

But I don't agree with your statement above. If we don't stretch the rope at all, then it takes the inchworm of course 100 seconds to traverse. Now consider that we only stretch once, after the first second. If your statement above is true, then it should still take him only 100 seconds; but it clearly will take him longer because immediately after the first pull, he is 2 cm along (as you pointed out), so he has another 198 cm to go. If we do not perform any more stretches, then it will take him an additional 198 seconds, for a total of 199 seconds. Clearly, the stretching process changes the time it takes him to traverse (if he traverses at all???) - 09 May '05 10:08

No the poor little worm won't ever make it.*Originally posted by davegage***Suppose an inchworm starts traveling along an elastomeric rubber rope (initially one meter in length) at 1 cm per second.**stretched another meter in length. So the worm travels 1 cm, the rope gets uniformly stretched 1 meter, the worm travels another cm, the rope gets stretched another meter, etc., etc.

But, after each second, suppose the rope is [b]uniformly

Will the worm ever make it to the end of the rubber rope?[/b]

He'll be eternally stuck at having only travelled 1% of the rope's total distance. - 09 May '05 15:19That worm will make it, provided he lives to the ripe old age of infinity.

At first glance, it seems like the worm will never make it: we just keep making his job harder and harder by stretching the rope underneath his inchy little feet. But the worm is going to keep on truckin', and we're going to keep stretching the rope, forever (or until we find out the answer). And when you deal with infinity, the first glance is rarely telling.

Consider this: no matter how long we make a static rope, the worm can traverse it. It may take him a heck of a long time, but he can traverse it. If we stretch it a bit, we've basically created another static rope that's longer - but the worm can still traverse it. By stretching it continually, we're basically creating many, many static ropes, each of which the worm can traverse. So the answer is "yes", the worm gets there in the end.

Now you may be saying "that's impossible - the rope is stretching faster than the worm can walk, and every stretch leaves the worm further and further behind!". This is true. But there is no time limit to this problem. It doesn't matter how far behind the worm falls, he will eventually make up the difference.

This problem deals with degrees of infinity. Every length the rope achieves can be traversed by the worm, given enough time. Therefore the sets of the lengths and the travel distances are 1-to-1 and onto (meaning they pair up element by element, with no overlaps and no double-counting), and therefore they are the same (infinite) "size". Also, the largest element in the length set corresponds with an element in the travel distance set, meaning the worm gets to the end!

Confusing? Convincing? Right? We'll see.

- 09 May '05 15:46I still don't understand how he'll ever make it to the end.

At any given point during his infintely long journey our little worm friend has only ever travelled 1% of the distance along the rope.

Considering at every single one second interval he is only travelling far enough to keep up with the rate of stretching than isn't he technically never going to get to the end until he steps up a wormy gear and travels faster than the rate of stretching?! - 09 May '05 16:11

It's true that the rope is stretching faster than the worm can inch along, and at any point along the journey you can check the worm's position and see that it's falling behind. But "faster" only matters if there's a time limit.*Originally posted by Daemon Sin***I still don't understand how he'll ever make it to the end.**

At any given point during his infintely long journey our little worm friend has only ever travelled 1% of the distance along the rope.

Considering at every single one second interval he is only travelling far enough to keep up with the rate of stretching than isn't he technically never ...[text shortened]... o get to the end until he steps up a wormy gear and travels faster than the rate of stretching?!

The length of the rope will keep growing infinitely, and the worm will inch out an infinite distance. So the question is really, "which is bigger? Infinity or infinity?". Funny enough, the answer to this question is not always "infinity, stupid!", or even "infinity+1!". It depends on what type of infinity you're talking about.

In this case, both the length of the rope and the distance travelled by the worm are "countably infinite". Integers are "countably infinite". Real numbers are "uncountably infinite". I think there are a few more types of infinity too, but I don't know/understand what they are. But the important thing is that infinities of the same type can be correlated in a 1-to-1 and onto manner.

For example, after taking the stretching movement into consideration, the worm goes 100.21 cm (>100) after 26 seconds, 203.17 cm (>200) after 46 seconds, 303.61 cm (>300) after 64 seconds, 403.20 cm (>400) after 82 seconds, etc... (we can match up the times to the exact rope lengths if we're prepared to do more math, but this serves our purposes). Since every rope length can be matched up with a travel distance, every rope length will be traversed by the worm given enough time. And since the longest rope length is no exception, the worm will eventually make it! - 09 May '05 16:38

Basically, don't you have two diverging series subtracting (Length of the rope and length travelled)?*Originally posted by PBE6***It's true that the rope is stretching faster than the worm can inch along, and at any point along the journey you can check the worm's position and see that it's falling behind. But "faster" only matters if there's a time limit.**

The length of the rope will keep growing infinitely, and the worm will inch out an infinite distance. So the question is re ...[text shortened]... ough time. And since the longest rope length is no exception, the worm will eventually make it!

I think that the series that is left after the subtraction (length left to go) is diverging too, so the worm will never make it...

Right??? - 09 May '05 16:46

This statement is not true, which is why I think you are confused on the problem. If it were true, then it's clear that the worm would never make it to the end.*Originally posted by Daemon Sin***At any given point during his infintely long journey our little worm friend has only ever travelled 1% of the distance along the rope.**

But in this problem (one of my favorite problems because it is very counter-intuitive to me), we can show explicitly that the worm will make it to the end; furthermore, he will do so in a FINITE amount of time, ie., he doesn't need to live infinitely long -- only long enough.

The proof follows these lines:

Consider the fraction of rope that the worm has traversed after the Nth second. Since the stretching of the rope is uniform (ie, "affine" deformation), this fraction is preserved upon each stretch -- that is, the fraction is the same the instant before each stretch as it is the instant after each stretch.

After the first second the worm has traversed 1/100 of the rope. After the second second, the fraction is 1/100 + 1/200. After the 3rd second, the fraction is 1/100 + 1/200 + 1/300. And, generally, after the Nth second, the fraction is Summation(i=1 to i=N)[1/(100i)].

The question is then does this infinite sum ever get larger than 1? In fact, it must after finite N, because this sum is just 1/100 times the harmonic series, which is well-known to diverge (it blows up to infinity eventually). I think if you were to calculate at what N it actually exceeds 1, our worm would have to live longer than the universe has been in existence. Still, the important point is that he would only have to live a FINITE amount of time. Seems very counter-intuitive to me, but the math is what it is.

However, whether or not the worm gets to the end is not independent of how much we stretch the rope each time. If we stretch it sufficiently far each time, then we can make it so that the worm can never make it to the end, even if he lived infinitely long.

So a follow up question I would ask is:**what is the minimum amount by which we need to stretch the rope each time such that the worm will never make it to the end of the rope in finite time?**The limiting case would be when the fraction of rope traversed by the worm converges to 1 as N goes to infinity.