Sunday, September 14, 2008

Circling Bugs

Here's another problem I liked from "How Would You Move Mount Fuji". I'll state the original problem, then the generalization, and hope that when I try to solve it tomorrow I can do a little better than I did yesterday with the boat and demon. I think this one is actually a famous problem.

There are four beetles located the the four corners of a square. At the same moment, they all start crawling directly towards the next beetle clockwise around from them. That beetle they're aiming for is also crawling, so each beetle continually adjusts its course so that it aims straight towards the next beetle.

Will they collide in finite time? How far will they walk before they collide?

Generalizations: Let there be "n" beetles at the corners of a regular n-gon. Same questions. Also, if you drew a path marking the route of a single beetle, how many degrees around the center of the square would it subtend (i.e. how many times does the beetle wrap around the center)? What is their exact path?


Nikita said...

I drew a diagram and this is what I figured from it: the beetles will spiral towards the center and they'll wrap themselves around the center an infinite number of times.
It can be said that they will collide at then center (within a finite time period) as the radius approaches zero though technically that would not happen.

Markkimarkkonnen said...

You're right - but here mathematically they really do run into each other. Even though they make an infinite number of circles, the circles also get infinitely small. It turns out the total length of the spiral is finite.