
Ant On A Rubber Rope - yati
http://en.wikipedia.org/wiki/Ant_on_a_rubber_rope
======
DanielStraight
Since Wikipedia doesn't do a good job of simply stating _why_ the ant will
eventually reach the end:

The expansion of the rope is over its entire length. When the ant is halfway
across the rope, an expansion of 1 km only adds 0.5 km to the length the ant
must traverse. So the farther the ant walks, the less the expansion matters,
and eventually it won't even add the 1 cm the ant is moving per second.

At least that's as best I understand it.

Here's a Google spreadsheet that shows the process with a significantly faster
ant:

[https://docs.google.com/spreadsheet/ccc?key=0AsmKPGVX0X-0dFF...](https://docs.google.com/spreadsheet/ccc?key=0AsmKPGVX0X-0dFF2d1dVMUg2WTZWLUxzMFU2VFpfUFE&usp=sharing)

No guarantee of being free of off-by-one type of errors, but it should give
the general idea.

~~~
jre
That is a great intuitive explanation. Much better than the current wikipedia
informal solution.

You should add it to Wikipedia, maybe under a new "An intuitive explanation"
paragraph.

~~~
stilldavid
Or add it to the "simple" wikipedia language.

[http://simple.wikipedia.org/wiki/Ant_on_a_rubber_rope](http://simple.wikipedia.org/wiki/Ant_on_a_rubber_rope)

~~~
matznerd
relevant xkcd... [http://xkcd.com/547/](http://xkcd.com/547/)

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iandanforth
This is a great example of why I hate 'word problems' in math. The person
writing the problem is probably trying to demonstrate a nice mathematical
property and engage the reader, but what they are _doing_ is confusing the
hell out of people and providing a frustrating experience.

Almost every piece of information you get from this hypothetical scenario is
useless. Everything you know about ants, rubber, gravity, energy etc is
suddenly a distraction from the extremely narrow set of ideas which the author
allows to be relevant to the problem.

As someone who really really likes knowing material properties, insect
physiology, and basic physics, while caring little for abstract mathematical
properties, this kind of problem seems perfectly designed to tweak my nose.

~~~
Dirlewanger
This problem is all over the place in Wikipedia. Almost every page with any
kind of math on it assumes you're in grad school for a Master in Mathematics.
And I'm pretty sure any injection of a lay person explanation will anger those
who know the source material. Annoying but it is what it is with regards to
Wikipedia...

~~~
ceautery
Couldn't agree more. The WP:NOT page seems to address this with "Wikipedia is
not a manual, guidebook, textbook, or scientific journal", but with how
prevalent maths jargon appears without anything layman-accessible describing
what's being talked about, it seems like an uphill battle.

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mjn
One way of thinking of it is in terms of how far along the rope the ant is. If
the ant is 20% along the rope, and the rope stretches, the ant is still 20% of
the way along a now-bigger rope. Therefore the ant never loses ground, so to
speak.

The effect of the stretching is to reduce the ant's speed measured in units of
"proportion of the rope". E.g. if the rope is 100cm and the ant is going 1
cm/s, then the ant is covering 1% of the rope per second. If the rope doubles
in length and the ant keeps the same speed, the ant is covering 0.5% of the
rope per second. So the ant is decelerating in this set of units; the total
distance it travels becomes a matter of computing a limit.

~~~
SEMW
> But this will always be a positive number, therefore in some finite number
> of seconds, the ant will cover 100% of the distance.

Correct result, but that reasoning is dodgy.

It's easy to imagine a different problem where an ant always covers a positive
proportion of the rope every unit time, but the ant _doesn 't_ reach the end
of the rope. E.g. construct a situation where, in the 1st unit of time, the
ant covers a quarter of the rope, in the 2nd an eighth, in the 3rd a
sixteenth, ..., in the nth a 1/2^{n+1}th. In that scenario, as time tends to
infinity the ant asymptotically reaches the half-way point.

The reason it works in the problem in the article is that, there, the
additional proportions the ant covers in each unit of time form a _divergent
series_ (~the harmonic series), meaning the sum tends to infinity as n->
infinity. In my version above, they form a _convergent series_ , meaning the
sum tends to a finite number (in my example, 0.5) as n-> infinity. (Of course,
a convergent series with a sum >1 would also mean the ant reaches the end of
the rope in finite time).

~~~
mjn
Yeah, good point; I edited my comment a few minutes after I posted, but it
looks like you quoted the pre-edit version. :)

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jcampbell1
The harmonic series grows incredibly slowly but has no upper bound.

.00001/1 + .00001/2 + .00001/3 ... => infinity. The ant will eventually pass
100% of the band.

There are lots of counter intuitive results based on the fact the harmonic
series has no bound. I remember an article where someone showed you could
stack dominoes to produce infinite overhang, exploiting the same counter
intuitive bit of math.

~~~
rafekett
Perhaps the most interesting and useful fact about the harmonic series is that
the nth harmonic number (the partial sum of the first n terms) is an extremely
accurately approximation of ln(n) (they are very close for all values of n, in
the limit they differ only by the Euler-Mascheroni constant, which is about
0.577). This is the only approximation I have seen that has constant error in
the limit.

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omegant
For the astronomical application of this problem, if the ants (photons) are
reaching the earth after crossing the expanding universe, there must be a deep
space observation (I don´t know if it´s technically achievable ) that will
appear as if new galaxies appear where previously none existed (was visible)?

Does this make sense?

Edit: added the question sign.

~~~
rwallace
It makes sense, and you would be right if the universe were expanding at
constant speed. But as the article mentions, the universe is actually
expanding at accelerating speed so the ant loses; thus we have the reverse
effect where as the eons go by, distant galaxies that used to be visible fade
from view.

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qwerty_asdf
A Rubber Band is 100,000 centimeters long, from _POINT A_ to _POINT B_.

An ant starts at one end of the rubber band ( _POINT A_ ), and walks toward
the other end ( _POINT B_ ), at 1 centimeter per second, for an infinite
period of time.

 _POINT B_ moves away from _POINT A_ at 100,000 centimeters per second, for an
infinite period of time.

    
    
      POINT B travels at: 100,000 centimeters / second
    
      The ant travels at: 1 centimeter / second
    
      In 100,000 seconds, the ant reaches the 100,000 
      centimeter mark, but POINT B is now 10,000,000,000 
      centimeters away. Both continue to move at the same 
      speed.
    

Additional Details:

If an average ant lives 90 days, then the ant only has 7,776,000 seconds to
reach the end of the rubber band. In this time, the ant will travel 7,776,000
centimeters, but the rubber band will have extended to 100,000^7,776,000
(that's one hundred thousand to the roughly seven-point-seven-millionth power)
centimeters long. Based on these facts, we may safely conclude that a mortal
ant will never reach the end of this hypothetical rubber band.

Did I miss a detail somewhere?

~~~
Roedou
Yes: because the band is expanding both in-front of AND behind the ant, it
will be much further on than X cm after X seconds.

(Also: this is not a problem concerned with the physics of rubber or biology
of ants. This ant is immortal, the rubber infinitely stretchy, etc.)

~~~
qwerty_asdf
...then _POINT A_ and _POINT B_ are both in motion, simultaneously, _while the
ant is walking_.

Which is great, except that when the original concept is framed to the reader,
the simple word problem doesn't specifically express that both ends are in
motion.

If both ends are in motion, then the ant's velocity changes. The ant might
move 1 centimeter per second, relative to a fixed point in space, but it isn't
moving 1 centimeter per second, relative to it's origin, _POINT A_.

The motion of the rubber band is influencing the speed of the ant, relative to
the band itself. Since the ant moves toward _POINT B_ , while _POINT A_
simultaneously accelerates away from the ant, the ant is moving faster than 1
centimeter per second, with each second spent traveling away from _POINT A_.

Once this detail is added, it completely changes the premise of the problem.
But the manner in which the problem is originally expressed is vague and
incomplete, misleading the reader (perhaps deliberately so).

Why would anyone claim the ant is moving 1 centimeter per second, RELATIVE TO
THE ONLY OTHER FRAME OF REFERENCE IN THE PROBLEM (the rubber band), when its
motion is clearly not 1 centimeter per second?

~~~
cookingrobot
It doesn't make a difference in the solution if Point A is fixed or moving.
The band is getting longer, and the and is standing on it - moving with it
either way.

~~~
qwerty_asdf
It does matter, because if you tack Point A down to a fixed position, and pull
Point B away at a fixed speed, and the ant's speed is explicitly measured as
relative to it's distance from A, then ant isn't catching a ride on the rubber
band.

In that case, the ant is only moving at a fixed speed, toward the other end
(100,000 times slower than the end's expansion), which will forever be too far
away.

~~~
thedufer
The ant's speed is not explicitly measured relative to its distance from A. It
is left unspecified what it is relative to, but the implication of "walking
speed" (well, crawling speed, I guess) is that it is relative to the point of
ground you are on.

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iconjack
I saw this puzzle when I was kid, in a Martin Gardner book. Back then it was
an inch worm instead of an ant. When I saw the answer, it blew my mind at
first. Later I understood how it was related to the fact that the harmonic
series diverges, and it became less mind blowing, but it's still a little mind
blowing—because it's still a little mind blowing that the harmonic series
diverges.

It became one of my favorite puzzles, and I tried it on friends and family. I
certainly never thought it could have "real-world" applications.

But then it occurred to me, people were solving this puzzle every day.

Think of an ordinary loan. Each month, you make a payment that's part interest
and part principal. At first it's mostly interest, then the ratio shifts over
time, the final payments being nearly all principal.

You normally start with a loan amount (principal), an interest rate, and a
loan duration (typically 180 or 360 months). From these parameters, you figure
the monthly payment, a process called amortization. Part of the payment goes
towards interest, part towards the principal.

Each month you're required to service the loan, which means to pay (at least)
the interest. That's the cost of renting the money. Some loan arrangements
allow you to pay down the principal at your own pace. If you only ever pay
interest, the principal will remain unchanged, and the loan will go on
forever. This is the Netflix model: they don't care how long you keep a
disc—you're paying rent on it every month. Many people pay more than the
monthly payment from time to time. The principal will be reduced by this extra
amount.

The worm has taken out a loan. The twist is, we don't yet know the principal,
nor the total amount to be paid, which corresponds to the final rope length.
Instead, we know the payment: 1 yard. The interest portion varies, but the
worm consistently pays down the principal 1 inch each pay period.

Some of the added yard (monthly payment) appears in front of the worm
(interest portion) some behind it (principal portion). Stretching the rope
_uniformly_ has the effect of servicing the interest. At first, most of the
newly added rope appears in front of the worm, but as with the ordinary loan,
the back/front ratio increases over time.

That the worm will eventually reach the end of the rope is now evident. If
your interest payment is taken care of, then even a small monthly pay-down of
the principal will eventually pay off any size loan.

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PeterWhittaker
This is an interesting problem, but not that interesting, really.

The really interesting problem is an ant walking on the surface of an
expanding sphere, where the rate of the expansion of the sphere is
proportional to the radius of the sphere.

If the ant starts at one pole, will it ever reach the other pole?

Yes, I've underspecified the problem - we want the complete set of solutions.

Assume initial radius r, ant of length l, ant walking at speed w, and initial
rate of expansion e (remembering that de/dt is proportional to r).

Under what conditions will an ant at pole P1 reach P2? Under what conditions
is impossible for the ant to reach P2?

Presumably there is a range of values such that the ant can traverse the
sphere if it starts early enough or is big enough or if w is high enough.

Now make things even more interesting: Assume that there is a maximum speed M,
and that the closer w or e gets to M, the harder it is to get incrementally
close to M.

This is the expanding universe. We are the ant.

If we had developed high speed interstellar travel "early enough", would we
ever have been able to cross the universe? Or is it always too late?

Now that's an interesting problem. Solution left as an exercise for the
reader.

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adamtj
The moment the ant takes even one step, the far end will no longer be move
away from the ant at 1 km/s, so it will be able to move fast enough to catch
up and cross the rope.

After one step, the ant will be somewhere in the middle, and both ends will be
moving away from it. (Because when the rope stretches, every point on the rope
moves away from every other point on the rope.) If the far end is moving one
direction, and the near end is moving the other direction, and the two ends
are moving away from each other at 1 km/s, then neither of them can be moving
away from the ant at 1 km/s. The ant will then be able to catch up with either
end.

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dizzib2
The end of the rope is travelling at constant velocity but the ant is always
accelerating, therefore will eventually catch up and overtake the end.

~~~
ColinWright
That's true, but not enough. You can set up situations where the ant in always
accelerating, but at an ever decreasing rate, and never reaches the end.

Details matter.

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IgorPartola
Is the starting point of the rope fixed and the end being pulled away from it
or is the center of the rope fixed and the end points are being pulled away
from each other?

Other fun questions: does the rope stretch uniformly or are there waves? Do
the ants fore and rear legs get separated as the rope under it stretches? Is
there any sag in the rope?

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fsckin
Sounds like a vari-ant of Zeno's Dichotomy Paradox.[0]

[0]
[http://en.wikipedia.org/wiki/Zeno's_paradoxes](http://en.wikipedia.org/wiki/Zeno's_paradoxes)

