Fermi problems and b***sh*t filtration

You’re one of the world’s greatest physicists, a Nobel Prize winner while still in your 30s. You built the first nuclear fission reactor in the squash courts at the University of Chicago in 1942, a key event in the development of the nuclear weapon that’s about to be tested.

For better or worse, the world is about to profoundly change forever, and you are as responsible as any single person alive for it. The blast goes off; from your vantage point ten miles away, the remote New Mexico desert glows brighter than day through your welding glasses. Your boss, Robert Oppenheimer, is moved to think of a line in the Bhagavad Gita - “Now I am become Death, the destroyer of worlds”. What do you do?

If you’re Enrico Fermi, you start dropping little bits of paper and noting where they land.

Why?

As Fermi explains here, he was trying to estimate how powerful the explosion had been:

About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during and after the passage of the blast wave. Since at the time, there was no wind I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2½ meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T.

Fermi’s answer was not precise - in the end, weeks of calculations by a number of different people taking advantage of all the instrumentation set up to monitor the test resulted in a final estimate of a blast a little under twice the size. But in a couple of minutes, with the most primitive of instruments, Fermi calculated an estimate that a) indicated that the bomb had worked roughly as designed, and b) was sufficiently accurate to serve as a check against gross errors in other, later calculation.

Fermi was reknowned for these estimation abilities, and thus the practice of making preliminary calculations with rough, estimated data has become known as a Fermi problem, or, alternatively, a back-of-the-envelope calculation. It’s a fundamental skill for scientists and engineers, and one of the very first things taught in physics classes. But it’s not just restricted to strictly physics-related problems. Consider this classic Fermi problem, a generic variant of one usually attributed to Fermi himself: how many piano tuners are there in your state capital?

To tackle this, one approach would be first to estimate how many pianos there are in the appropriate city, guess how often they get tuned, estimate how many pianos a piano tuner can tune per year, and from there compute an estimate of how many tuners there would need to be. Each of these subsidiary quantities would need to be broken down and estimated in turn. For the first question:

• there are about 3.6 million people Melbourne;
• assume there’s roughly two people per household;
• making 1.8 million households; and
• maybe one in 30 households has a piano (though this is perhaps the weakest guess so far).

So, at a guess, there’s 60,000 pianos in Melbourne.

As far as piano tunings go: let’s take a guess and say that a piano will be tuned once a year. So there’s 60,000 piano tunings to be conducted annually in Melbourne.