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Fermi problems and b***sh*t filtration

By Robert Merkel - posted Wednesday, 9 January 2008


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.

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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:

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  • 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.

And how many pianos can a tuner service in a year? A tuner might tune five pianos a day; tuners work five days a week, 50 weeks a year (yes, 48 is more realistic, but you may as well round off); therefore one tuner would tune about 1,250 pianos a year.

Therefore, all we need to do divide the number of tunings required, by the number one tuner can do each year: 60,000÷1,250, do a bit more rounding-off, and get an estimate of about 50 piano tuners.

As it turns out, a check of the Yellow Pages website suggests that there are 137 piano tuners and repairers in Melbourne; clearly, my Fermi problem skills are not as good as Fermi’s! However, through a little bit of calculation, it was possible to get an estimate that was within a factor of three of the more precise figure - and, for many purposes, it’s good enough.

If somebody came to me with a proposal to produce a full-color glossy magazine aimed specifically at the piano tuners of Melbourne, I’d be extremely sceptical of the feasibility of making money from it, just on the basis of my Fermi, or “back-of-the-envelope”, calculation.

In my own particular line of work, the quantities I am most interested in estimating are the amount of time it will take for a computer to do something, and how much RAM or disk storage will be required along the way.

Quite often, I just want to know whether I have to worry any further about the time taken or not. So this leads a particular kind of back-of-the-envelope calculation, the upper bound - where I calculate a figure that must be higher than the actual quantity (of course, sometimes you want the opposite - a lower bound). For instance, if I know that my computer program will take 0.001 seconds, at most, to get an answer, I don’t care whether the actual time is 0.0001 or 0.00001 seconds.

Of all the skills I learned in my years of education, the ability to tackle Fermi problems is one of the most useful in daily life; it’s also damn handy in blog arguments. More seriously, I don’t see how one can tackle a lot of public policy debates without it.

Numbers ain’t everything, but for a lot of contemporary issues - particularly environmental ones - they are one of the most important things. Should we throw out appliances before the end of their useful life to replace them with more energy-efficient ones? Does drinking French wine, shipped half way round the world, instead of the local variety make a significant difference to global warming? Can we cut the demands on our dams significantly with water tanks?

I wouldn’t for a moment claim that back-of-the-envelope calculations are a substitute for detailed quantitive studies. Nor, would I make the ridiculous assertion that everything can be reduced to numbers. But, to me, Fermi-style estimation is an essential bullshit-filtering tool.

Which brings me to something I’ve always wondered about. How in the hell do people who haven’t done first-year uni physics cope without the immense utility of envelope backs?

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First published at Larvatus Prodeo on June 26, 2007.



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About the Author

Robert has a blog called The View from Benambra. He is a postdoctoral research fellow (in software engineering) at Swinburne University of Technology.

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