Lugo’s original post is so good it seems almost superflous to paraphrase it, but I can’t resist the temptation.

Drill a tunnel through the earth, from anywhere to anywhere — New York to Maine, or New York to Australia, or wherever else you like. Like so:

Now drop the object of your choice (Lugo suggests a burrito, but you might prefer a gravity-driven train) into the tunnel entrance and wait till it comes out the other side. It’s a standard calculus problem to calculate how long you’ll have to wait: The answer is 42 minutes, regardless of the length of the tunnel. I’m sure I once found it surprising that the tunnel length doesn’t matter, but I’ve known it long enough that I now take it in stride. So that’s not how Lugo surprised me.

The surprise is that if you change the size of the earth (while maintaining its density), the answer is still 42 minutes. Whether the earth is the size of a pea or the size of the solar system, it’s a 42 minute trip from one end of the tunnel to the other. (We’re — quite reasonably — ignoring the effects of relativity here. For an earth that was half the size of the universe, we’d have to make some corrections.)

Why so? You could, of course, discover this through a direct calculation. But Lugo provides a much slicker argument, namely:

Once you know that the travel time is independent of the tunnel length, it’s clear there are only three things it can depend on: The earth’s density, its radius, and the gravitational constant. Neither density nor radius have anything to do with time, so we’re really going to have to use that gravitational constant, which is measured in units of

(The first factor is the 32 feet per second squared that you learned about in elementary school; the second factor depends on distance from the center of the earth and the mass of the earth, neither of which varies in those elementary school problems, so it’s never explicitly mentioned.)

If you multiply that gravitational constant by density (which is pounds per cubic foot), you’ll get a bunch of cancellation, leaving you with

Now you’ve got something that depends only on time! To turn this into an actual number of seconds, you can just invert it and take the square root. At that point, you’re done! (Except maybe for multiplying by some dimensionless constant.) Any attempt to bring the radius of the earth into the formula is going to introduce units of length, which have no business popping up in a formula for time. Therefore the earth’s radius must be irrelevant.

Isn’t that cool? And amazing?

(And yes, there are hidden assumptions, e.g. that the solution must be not just a function of mass, length and the gravitational constant, but a rational function — so this is not quite a proof, but a highly convincing heuristic argument that does give the correct answer. That, I think, in no way detracts from its coolness.)

Lugo’s post is both more concise and more carefully worded, but I thought this slightly longer and less formal version might be helpful to at least a reader or two. Besides, I felt like writing it.

Do you have an equally amazing argument to share?

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Suppose you’re observing something that changes over time — say the Dow Jones average, or the temperature in Barrow, Alaska, or the number of people who have been shot by terrorists so far this year. Suppose you have absolutely no prior information about how this thing behaves — in particular, you might have no way of knowing whether it changes continuously (like the temperature) or whether it’s subject to sudden changes (like the number of terrorist victims). You have no formula for it; you don’t even know whether there is a formula. It could be absolutely anything.

(For those who want more precision: This thing you’re observing is a real-valued function defined on the positive real numbers — which we can think of as a function of time. It can be any function whatsoever.)

Now suddenly, at some randomly chosen moment, your ability to observe comes to a halt. If that randomly chosen moment is, say, 6:23 AM, then your observations go right up to , but do not include, 6:23 AM.

Your task is to accurately guess the value of the variable at that first moment that you can’t observe. In fact, let me make your job a little harder. If your observations stop at 6:23AM, then I want you to accurately guess the value at 6:23 AM and for a little while thereafter.

Now here’s the theorem: There is a strategy that will allow you to succeed at this task with probability 100%.

If that sounds even remotely plausible to you, then you haven’t understood it.

If you could implement this strategy, you could certainly beat the stock market, unless you preferred to spend all your time winning bar bets about the paths of raindrops along windowpanes. Unfortunately, the strategy is (provably) too complex to implement on any computer. But there’s a sense in which it’s easy to describe, though not to implement. Sometime soon I’ll blog the description and a sketch of why it works. If you just can’t wait, here is the paper I learned this from.

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Where can the panicked friend turn? To the Internet, of course. He posts a plea for help. Commenters jump into action, contacting police and social service agencies, pooling information to figure out what bus the girls are likely to be on, and arranging to have them escorted to a police station. A couple of hundred comments later, the girls are safe and sound. One commenter adds:

This is the best use of the Internet that I, personally, have ever seen. I’m so proud to be a member of this community.

Indeed.

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Like everyone else I know, I am of course a longtime fan of the webcomic XKCD. But somehow it took me until last week to become aware of the frequently brilliant competitor Luke Surl, of which the above is a delectable example. What else out there am I missing?

Hat tip to Harry Brighouse of Crooked Timber.

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“Between the Folds” has its national television debut tomorrow night (Tuesday, December 8 on PBS; check your local listings for the time). Or check here for additional showings.

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