My son (Griffin, nearly 7) saw Hex Bugs in a store months ago and was instantly smitten. His dream came true recently when my wife bought him one.
The Hex Bug now does shows in his home/stadium. See video.
There is something quite lovely about this guy’s random walk. But I can’t quite figure out what scenario I can put him to generate a real probability problem. So I ask, What can you do with this?
Overthinking my teaching 
Nice video Chris. None of these question are great, but may spark others:
I’m wondering if he hits the long walls more frequently than short walls, or short walls more than the long ones? I can see arguments both ways… because longer wall is bigger (more places to hit) but short walls are farther apart (hard to get to without first hitting one of the long walls).
I’m also wondering if I can determine his mean free path. There’s also something about time spent in different locations… Like he spends a lot of time being trapped in corners.
I’m also intriqued by the short time where he bounces back and forth for a while only hitting top and bottom walls. What’s the longest we can expect him to go hitting only parallel walls?
I also just want to plot his trajectory for a while to get a feel for how it looks, which made me also wonder if he turns right just as often as he turns left?
The walk is not really a random walk. See http://express.howstuffworks.com/hex-bug-autopsy.htm for how the Hexbug works. Basically, the hexbug walks forward until it touches something with its antennae, then backs up on one side to turn until it is not touching, then walks forward again. Some randomness is introduced by the mechanical interaction with the environment, but the Hexbot itself is a fairly deterministic device.
Incidentally, I’ve seen a very similar approach used in an old RC toy cat: running forward turns both wheels, but reversing the motor disengages a clutch and only one wheel is driven backward.
gasstation, this guy’s different. See photo at this link. He has a little vibrating motor, rubberized springy legs and a rubberized nose. His locomotion comes from the springiness of his legs. The vibrating pushes him up, which allows the legs to expand and push him a bit forward. Then he falls back down, compressing the legs, etc. Many times each second. These interactions have quite a bit of meaningful and random variability in them.
When he hits a wall, he bounces (literally bounces) off of it and his exact next direction is not at all deterministic. Angle of reflection does not necessarily equal angle of incidence.
For example, at about 18 seconds into the video, as I am telling Griffy to keep his head out of the video for 30 seconds, the bug comes toward the wall at the bottom of the screen at about 15 degrees and bounces away at about 135 degrees-and continues to rotate until nearly perpendicular.
So this is leading me to think about directions to go with this. gasstation suggests that the bug’s position 5 seconds down the line can be predicted with pretty good accuracy. I disagree with that strongly. What experiment can we design to settle the matter? is a challenging statistics question.
I am interested in the kinds of questions Brian presents, but I’m not convinced students will be, and I don’t think this video leads very strongly to these kinds of questions. I can imagine modifying his environment, however. What if the bug is zooming around in a box with two holes-one on a short side and one on a long side? Which hole will he exit from and how long will it take? And what if the sizes of the holes are proportionate to the side lengths? Or inversely proportionate?
You have both helped me think about this. Thanks and keep it up!
Ah, sorry. I didn’t realize that it was a bristlebot design rather than the original hexbug. I only glanced at the video. You’re right that there is a pretty big stochastic element in bristlebot locomotion, and it might be reasonably modeled as a biased random walk. I think that sort of modeling is a bit tricky for high schoolers, though.
So gasstation’s vote is You can’t do anything with this because the best model is inaccessible without quite sophisticated mathematics. I’m curious whether this is a general principle.
If so, it may rule out most of the modeling that happens in the early study of algebra.
The model on gasstation’s mind seems to be one that addresses the question, what happens next? I am more inclined to look towards questions like what happens eventually? such as:
Does a naive probability model get close enough to answer these kinds of questions faithfully?
No, I wasn’t saying “you can’t do anything with this”. I was saying “you have to be careful—some problems that look easy aren’t”. That is what I meant by “a bit tricky for high schoolers.”
I don’t think that the students will be able to calculate the probability that of eventually exiting from the long-side hole or the short-side hole—that’s the sort of emergent property that Markov models are good for, but a lot of grad students would have trouble getting right. I’m not sure I could come up with a good model that both covered the behavior of the bristlebot and allowed a simple analytic solution. Simulating would be easier, if your students know Python.
You might be able to do some simpler problems, though, like how much time it spends in the right half of the box versus the left half, or closer to a long wall than to a short wall, both of which can be done by area comparison, if you assume that the random walk it takes covers all the bottom area uniformly (not guaranteed).