Originally published at: The beautiful "Four Bugs Problem" - Boing Boing

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If each bug was crawling directly toward the bug in front of it *as the bug in front of it was moving laterally* relative to its position then wouldn’t each bug’s path be a curve rather than a straight line? What am I missing?

**Edit:** OK, I thought they were saying that the path of the insects formed the sides of the polygon. Never mind.

Intervals? It’s essentially selecting an anticipated vector at repeating intervals and marking that. If you just followed the path of the actual insect, it would be an arc.

but what’s the “problem”? i don’t see a “problem”, nor is one described.

The problem is that your lab is infested with bugs.

I think the “problem” (question) is “What is the shape of the path that is walked by each bug?”. I’m not sure that I could prove that it’s a logarithmic spiral… Hmmm…

Solved that one ages ago

me ol’ dad, he worked on navy planes and was always telling stories about the latest analog computers used to compute "*pursuit curves"*. So when you’re aiming things at moving things which can change course to avoid things you have to do a lot of trigonometry (with differentials), and in his days that meant odd shaped gears running over discs running over helices and such. Anyway he inspired a lot of my math interests and if the four bugs problem (which curiously i learned of as “the four dogs problem” hmh) strikes you as interesting, i’d humbly recommend this (new edition) book [link to amazon]: **Chases and Escapes: The Mathematics of Pursuit and Evasion (Princeton Puzzlers) by Paul Nahin (Author)**

with each bug moving directly toward its neighbor

Each bug has two neighbors.

Nope. The problem is, how long is the path walked by each bug? (Assuming the square has sides of length 1.) Answer:

Still exactly one, with more text here to make it nonobvious from the blur

ETA: Here’s why

As bug A is chasing bug B, and bug B moves a little bit *precisely sideways* from the direction B-A, the distance from bug A to bug B isn’t changing. So if every bug is moving at unit speed, the distance from each bug to the next is 1-t for time t from 0 to 1.

I have no idea what’s supposed to happen for n-gons other than n=4.

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