# Equiangular Spiral

Four bugs sit on the corners of a square table, wach facing the bug ahead on the next corner, clockwise as seen from the top. They begin walking, always at a constant speed, always directly toward the bug ahead. How far does each bug walk before they meet? (in the middle)
PS
I know the answer is that the distance walked is equal to the original distance between two consecutive bugs or one side of the original square. What I need is a good explanation mathematical or abstract. Thanx.

Here’s an intuitive way to think of it:

Say bug A is walking towards bug B. At each instant, the direction B is walking in is perpendicular to the direction A is walking. In other words, B is neither walking toward or away from A, so his actions aren’t doing anything to increase or decrease his distance from A. Therefore, A only has to walk the original distance between A and B–the length of the side of the square.

Anyway, like I said, that’s just an intuitive way to think of it, with a little hand waving to go along. If you want a more rigorous explanation, you can look here.

Well, if it’s a constant speed, you can just measure the time, then multiply. I suppose that only works in the real world though, not in math problems.

This exact problem is in a Martin Gardner book called Gotcha! (I think). He explains it quite well.