Four bugs are located in the corners of a square, 10 inches on the side. They are arranged like this:

            A---B
            |   |
            D---C 

As the clock starts, A begins crawling directly toward B, which goes to C, C goes to D and D to A.

Each bug will home in exactly on its target, reguardless of the target's motion, so their paths will be curves spiraling toward the center of the square where they will meet.

What distance will each of the bugs have covered by then?

If every bug moves at the same time, then every bug follow the other at 90 degrees forming an exact square, but the trayectory resembles an spiral towards the center of the original configuration of the 10in square. My point of understanding the problem is different than that of levik's: If every bug is moving, than every "step" they take they are closer to each other and so on... They follow a bug that is getting closer and closer. I imagine this problem as a mere change of scale of the square respect to an angle of inclination. This angle of inclination opens from 0 to 45 degrees, so is just like a seen a growing square and at the same time rotating until it stops with its sides equal to 10in. In this particular problem, we assume that all the bugs walk with exactly the same speed, this gives the characteristic of a change of scale in the square. So measuring the position of a single bug respect to a reference angle (in this case from the center of the square), and at the same time adjusting the scale of the square respect to the same angle, we get to know the real trayectory of one bug. Makeing a simple program to sum every infinitesimal part of that trayectory would give a real aproximation of the perimeter or distance.
My result:
Distance covered of a single bug = 7.7422806275 in

Posted by Antonio on 2003-08-23 08:35:52