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.

Slavakion

I'm not sure if I understand the problem. The tortoise walks a foot, and then Achilles stretches the band three feet? If the tortoise doesn't move when the band is stretched, it will never reach the flower. Every foot that it moves results in it then losing two feet. The tortoise is essentially moving backwards.

If the tortoise does move with the band, how much does it move?

I thought the Zeno problem was more like -- the tortoise moves halfway towards Achilles, then half of the remaining distance, then half of the... and so on. So that technically, the tortoise will never get the flower. Although he should be close enough for all practical purposes before long.

Chuckles

the zeno problem is just an example given by the prof i believe, and i dont think he'll get it either. But another way i think about it is that when the band is moved three feet, the tortoise is "along for the ride" and moves with it too. Not sure how much though

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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shakran

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.

politicophile

Damn, you are entirely right: I somehow managed to neglect to consider that the other end of the band was fixed. This would work out exactly like Zeno's paradox - I don't have the math skillz to solve the thing, though...

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Gatorade Frost

It seems to me like it's just a Y=X and a Y=3X linear equation? They'd never intersect except at (0,0)

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shakran

No, because it's just like the old problem about the guy that falls in a hole, climbs 2 feet and slips back 1 foot. He ends up taking less time to get out than you think because the last time he climbs 2 feet, he's out, so he won't slip back down.

Once the turtle gets to the point that, after the band is stretched, he's within 1 foot of Achilles, there won't be another stretching because the turtle will just take that last step. . .

I think the graph would be curved. And the turtle would rapidly start making more progress once he reaches the halfway mark since now most of the stretching happens behind him.

Chuckles

Distance walked by Tortoise Total length of the band
1. 3
2. 6
3. 9
4. 12


General _expression for step x:
x 3(x + 1)
Ratio of distance walked to the total length of the band:
x/3(x + 1)


If the band continues stretching in this (incredible) manner, the tenacious tortoise will never reach the end of the band and will never get any nice flower. Instead, the furthest the tortoise will ever get as the total distance of the band approaches infinity, as a ratio of distance walked to the total length of the band, is calculated as follows:
Limit as xinfinity of x/3(x + 1)
Multiply each term by 1/x in numerator and denominator
= Lim x infinity x/x / (3x/x + 3/x)
= Lim xinfinity 1 / (3 + 3/x)
= 1/3
So the poor old tortoise will never cover more than 1/3 of the distance to Achilles and the flower. This assumes that the band stays fixed to the ground at the point where the tortoise starts.

Chuckles

my bud came up w/ this chart tho. for the distance the tortoise traveled is this:

xn* = 1 + xn-1 (6+3t)/(3+3t)

time x L L-x
t 3 + 3t

0 1 3 2
1 3 6 3
2 5.5 9 3.5
3 8.333333333 12 3.666666667
4 11.41666667 15 3.583333333
5 14.7 18 3.3
6 18.15 21 2.85
7 21.74285714 24 2.257142857
8 25.46071429 27 1.539285714
9 29.28968254 30 0.71031746
10 33.21865079 33 -0.218650794

xn is a previous distance x the tortoise walked, and says he does make it using old x over old L multiplied by new L

Pip

First trip, he goes 1 ft, and is within 2 ft of Achilles, then the band stretches 3 ft. He is then proportionally moved so that he is now 4 ft away from Achilles. He then walks another feet, now he's 3 ft from Achilles. The band stretches 3 ft, he's now 4.5 ft from Achilles, moving again... I put it up in Excel because my analytical skillz have dried away. Distance to Achilles before stretch got lower than one after travelling 10 ft, the band had then stretched to 33 feet. He only needs to go 0.781349 ft to reach Achilles then.

EDIT: Defining xn as the distance to Achilles after a stretch, and setting x0=3 (starting point) one gets
xn=(x(n-1)-1)*n/(n-1)

n will be how far the turtle has actually walked, the length of the rubber band after a stretch will be 3*(n+1).

MooseMan3000

Yeah, that looks right. I didn't go through the entire thing, but your logic is correct as far as I can see.

To explain it in another way:

First, the turtle walks a foot. He is now 1/3 of the way across the 3 foot band. Then the band is stretched 3 feet. He is still 1/3 (aka 2/6) of the way across, but now the band is 6 feet long.

The turtle is now 2/6 of the way across, and he moves 1 foot more. He is now 3/6 (or 1/2) of the way across. The band gets stretched 3 feet more, to 9 feet, but once again the turtle it still 1/2 of the way across. He is 4.5/9 (Yeah, it's improper. Forget it.) of the way across.

He moves once more, and is now 5.5/9 of the way across the band. (FINE. It's 11/18. Happy?) In this manner, he does continue getting closer to the flower every step he takes. This logic is what Pip's formula describes (and what Shakran was explaning earlier).

Xn=(X(n-1)-1)*n/(n-1)
(I didn't make that formula, I just edited Pip's to be a little more readable)

Then you just solve that for Xn=1. You'll get some decimal. You take the next whole number after that, and that's how many steps it takes the turtle to arrive at the flower. I didn't go through that, but I'm assuming Pip is right.

Edit: It's a tortoise. Oops.

Jinn

Since the math is beyond me at the moment, I'll just add that a tortoise is a turtle, but not all turtles are tortoises, at least in current usage. You'd be OKAY saying turtle.

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Pip

This is supposed to be math, not zoology!

KnifeMissile

Ater discussing (for fun) this problem with some friends, the answer is obvious.

Because the tortoise gets moved along the elastic band as it stretches, this situation will be easier to analyse if we merely measure the proportion of distance the tortoise has travelled across the elastic band, rather than its literal distance.

So, in the first iteration, the tortoise moves one foot across the three foot distance of the elastic band, making it one third of the way across. Then, Zeno stretches the band an extra three feet. However, this doesn't change the proportion of distance the tortoise has travelled, one thid. It then travels another foot across the, now, 3 + 3 = 6 foot elastic. In other words, it has travelled another 1/6 of the distance. After that, it will travel another foot across a 3 + 3 + 3 = 9 foot elastic band, making it another 1/9 the distance closer. As you can see, the length of the elastic band is 3n at each iteration n of the sequence, so the proportion of distance travelled will, thus, be 1/(3n). Sum this sequence and it becomes clear that this is 1/3 of the Hormonic series.

Because the Harmonic series diverges, we know that the tortoise will get to the flower and that it will do so the same number of times it will take the series to sum to three. Unfortunately, there's no simple formula to represent the sum of the Harmonic series, although it can be approximated by ln(n). This approximation is not very good for a sum as low as three but if the problem were just a little different, like stretching the elastic band twenty feet, instead of just three, then the number of iterations might be pretty close to e²⁰...

blizzak

Hmm...the method is beyond me right now, but a program could easily solve this, which makes me believe that a formula I once knew could as well.
Once the variable x is greater than or equal to 1, you take the n value and that is your number of iterations. The reason the x is only equal to 1 in this case and not 3 is because we are measuring the proportion, as KnifeMissile said:

x = 0;
y = 3;
n = 0;

while(x < 1){
x += 1/y;
y += 3;
n++;
}

Gah, that code looks horrible, I have to take some more CIS courses soon. But I think this is a solution, no?

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howdydave

Howdy!

We mustn't leave out the other sciences...

This isn't going to go on forever. One of two things will happen:

The elastic band will break (strength of materials) The band doesn't have infinite elasticity!
or
The tortoise will lose interest because eventually the flowers will move beyond the tortoise's field of vision (biology/optics.)

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