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ebaines · Mar 1, 2008, 07:31 AM

A challenge problem for those who enjoy such things: the "bug problem":

You have a square room with walls of length L. In each of the four corners is a small bug. At the sound of a gun, they each start walking at the same rate directly towards the bug on their right. So each bug's first step is towards the corner to their right. But for the second step each bug must adjust its trajectory a bit in order to continue walking directly towards its neighbor because that bug has moved. With each step for each bug to continue to walk directly towards its neighbor causes them all to spiral toward each other. They keep doing this until they all meet at the center or the room. The diagram below I hope makes this clear.

Question: what is the distance each bug travels?

CaptainRich · Mar 1, 2008, 07:53 AM

My guess: infinite. I think they will enter into a spiral that will eventually have them nose-to-tail and simply be following each other in a circle near the center of the room.
Am I close?

galactus · Mar 1, 2008, 08:34 AM

Actually, I think the time is infinite, but their paths are finite.

Since the bugs paths create a spiral, I think we could use parametric arc length here.

I have to skedaddle now. I will be back and look at it in depth. Seems like a cool little problem.

galactus · Mar 1, 2008, 01:11 PM

Well ebaines, I have an answer. I don't know if it's right or not. How about each bug travels $\sqrt{2}L$ units of distance?

Since we are working with what appears to be logarithmic spirals, we can use the parametric equations:

$x=e^{-t}cos(t)$

$y=e^{-t}sin(t)$

The arc length formula gives us:

$\sqrt{2}L\int_{0}^{t}e^{-u}du=\sqrt{2}L(1-e^{-t})$

Now, as we take the limit:

$\sqrt{2}L\lim_{t\to\infty}(1-e^{-t})=\sqrt{2}L$

I reckon of it ain't right it'll be wrong.;)

ebaines · Mar 3, 2008, 06:40 AM

Not quite Galactus.

This is actually a bit easier than you might think. No parametric spirals needed. Here's a hint: when a bug takes a tiny step toward its neighbor, how much closer does he get?

galactus · Mar 3, 2008, 10:33 AM

I think I am over complicating it. Which is not unusual. It appears the bugs are going to always be on the corners of a square which kind of rotates around its center. Therefore, each bug will have travelled distance L. That is, each spiral path the bug makes is equal to the length of a side of the original square. Is my thinking more on line now? Seems reasonable.

ebaines · Mar 3, 2008, 12:42 PM

To put it in more mathematical terms: the path length $S$ of a bug is found from

$ S\ =\ \int _{D=L} ^{D=0} dS $

where D = distance between a bug and its neighbor.

For this problem when a bug takes a step of size $\Delta S$ , the change in distance $\Delta D$ between the bugs is

$ \Delta D = \sqrt {(D-\Delta S)^2 +(\Delta S)^2} - D $

In the limit as $\Delta S -> 0 ,\ \Delta D = -\Delta S.$

Hence

$ S\ =\ \int _{D=L} ^{D=0} dS\ =\ -\int _{D=L} ^{D=0} dD\ =\ L $

galactus · Mar 3, 2008, 01:14 PM

Cool, ebaines.