Almost two week ago, there was the Mystery Hunt. At school I wasn’t that into it. Not like these guys. Like just about everybody else, I was on a team, but I was the girl with the can of Oust air freshener that appeared every few hours to thoroughly spray down hunt headquarters.
A lack of hobbies might have something to do with it, but this year, I took more of an interest. I worked remotely, a few hours saturday a few hours sunday, pulled into it by John here in San Jose basically. But I didn’t have a bad time. I’ve been seeing a lot of Math is Fun propaganda lately, and I feel like sticking up for physics a bit. So here’s an example of a puzzle we solved and liked.
The puzzle is called Pesky Bugs. There’s that wav file, and there is a tag line which serves as a hint, “Bio Man even controls the insects flying around your head.”
The answer is “PLAGUE”.
I don’t know if you’ve ever seen a Mystery Hunt Puzzle before, but they don’t exactly come with instructions. Yet when you stumble upon the right way to solve one, you usually know it. There’s a certain elegance to these puzzles that I like.
The first thing to do here is split up the signal. Someone did this and found that the wav file is actually built up by overlapping six individual frequencies. The pure tones are at 220, 440, 660, 880, 1100, and 1320 Hz. They start up simultaneously at the beginning and end one at a time.
Next, you notice sound file is not in mono. The left and right channels are different; the amplitudes of the tones varying continuously but in a seemingly random way. Once you’ve gotten this far you’ll know that the solution must involve treating these two channels individually somehow, this feature is not accidental.
At this point I IMed Chris, my Sound Physics Go-To Guy. I asked, is the amplitude of a soundwave proportional to 1/r or 1/r^2, r being the straight line distance from the source? (1/r, he said) I also asked, is the electrical signal picked up by a mic proportional to the amplitude or the intensity of the wave? (amplitude, he said, what are you doing?) The idea was that there was something peculiar about the sound in the two channels, which shifted from one to the other, and if you leaned up very close to your speakers, it sounded like something was flying around your head. We were going to figure out the trajectory of the bugs.
The extraction was basically trigonometry, with some guess work involved. If the speakers were your ears how far apart were they? We initially constrained this by requiring that the distance D between the ears was no smaller than the maximum |R1-R2| and no larger than the minimum R1+R2 (bounding all possible triangles to be formed by the three sides). But this turned out not to be very stringent. We ended up with some funny looking curves that didn’t spell anything.
This is because there was another problem. Even knowing R1 and R2 and distance D between the ears the bug’s position is only constrained to a circle in the plane perpendicular to the axis of the ears. Assuming that it travels in a plane always we are still left with two ambiguous points for the current position of the bug. (If you look at the official solutions this ambiguity was actually eliminated with one extra piece of information: doppler shift, unfortunately we didn’t notice this about the tones) In any case, our solution was to look for momentum discontinuities at the x-axis that resemble reflections and stitch those manually.
We didn’t find them though, we found these smooth curves that, though definitely not the result of randomness, again, didn’t look like much:
Then there was a breakthrough! The appearance of the curves depended heavily on our choice of D. Though the upper bound varied widely, the D lower bound for each bug (frequency) was practically identical. This meant not only was D the same for each bug (as we suspected may be the case), but each bug also went all the way around the head. Pretty soon after realizing this we got our first letter: A
Due to the reflection ambiguity, upside-down, but unmistakable. A.
Some letters were confusing on account of that.
P or b?
Some were less so.
It’s exciting to me that so much information could be extracted from seemingly so little. Yet this is the kind of thing that physicists do every day. I remember from my time at LIGO using almost exactly the same technique to triangulate the location of gravitational wave sources using the time delay between signal detections at 4 detectors located around the world. In astrophysics tiny deflections in light from faraway galaxies can be used to determine the mass distributions of foreground dark matter. And in particle physics where just about everything is invisible, not just dark matter, all kinds of crazy techniques are used to infer the position and existence of a particle.
Anyways I just really like physics and also liked this puzzle. And in my mind there are many parallels. Seeing those letters emerge (especially that gorgeous G) after all that failed effort is astonishing but there’s the feeling that somehow it must. Much like finding order in the universe.