Wednesday, May 27, 2009

Fermi problems

Princeton University Press sent me a copy of Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, by professors Lawrence Weinstein and John A. Adam. The book is a compendium of Fermi problems -- that is, problems which are simply stated and whose answers can be estimated at the order of magnitude level through simple logic from a few factual inputs.

The classic Fermi problem is: How many piano tuners are there in Chicago?

When I took my oral exam as a first year graduate student at Berkeley, theoretician Geoff Chew (a former student of Fermi's) asked me:

1. How many blades of grass are on your front lawn?

2. What is the ratio of paved to unpaved surface area in Iowa? (He had earlier asked where I grew up.)

Luckily I got them both right. The experimentalist in the examining pair, Paul Richards, held up a cylindrical metal device of some sort and asked me what it was. He let me hold it; it was heavy. I stared at him blankly. To this day, I still don't know what the gizmo was :-) I suppose I was destined to be a theorist!

Physicists are constantly solving Fermi problems in the course of their work, because it's the first step in sizing up any potential project, theoretical or experimental. When I talk about entrepreneurship I emphasize the same kinds of problem solving in business or technology: how many servers will we need? how fast will sales grow? how much capital should we raise? ...

Watching someone work out a Fermi problem in real time reveals a lot about their brainpower. Wall Street firms, consultancies like McKinsey, Microsoft, and even small startups have been known to ask these kinds of questions of job applicants. This book discusses similar problems in a business context.

The difficulty of most Fermi problems is limited, unless the problem requires some specialized knowledge. But I like them slightly better than puzzles or brain teasers which rely on esoteric tricks that the solver either gets or doesn't get. A former collaborator of mine came up with the following (slightly broadening the genre) one evening while I was visiting U Chicago:

1. If the sun stopped radiating energy, what temperature would the surface of the Earth cool to?

2. In the above scenario, could humans survive using current technology if given enough time to prepare?

Weinstein and Adam's book is a nice collection. None of the problems require any specialized knowledge of math or physics; the level is appropriate for a bright kid or moderately technical reader.


kurtosis said...

Also noteworthy are the E Purcell's column in the American Journal of Physics "The back of the envelope"

Weisskopf's search for simplicity in the same journal is along similar lines.

Nameless said...

Interesting questions.

The obvious answer to (1) is 3 K, but it's wrong, because, according to Wikipedia, total energy flux due to the radioactive decay of elements within the Earth is 10^-4 of solar. In the absence of the Sun, this flux would balance Stefan-Boltzmann cooling when surface temperature is around 30 K.

Living without the Sun would be tough ... however, relocating everyone to the depth of, say, 1 km below the surface should buy us a few millennia before it starts getting really cold. (We'd have to solve some PDEs to find out exactly how long it would take.)

It would be difficult to grow food without sunlight, we'd have to use electric lighting. Even if we mine all known reserves of uranium ore and relocate them into the "ark", that only amounts to 50,000 kwh of energy per capita, which obviously wouldn't last very long (equal to the amount of solar energy received by a 100 m^2 garden patch in 2 months). Two solutions would be a selective culling of the population (say, a lottery) or an effort to develop unconventional sources of energy. If we could extract and store all seawater uranium before oceans freeze, that could be enough to support human population at current levels for a few hundred years. If we want to last much longer without reducing the population, it is necessary to solve the deuterium fusion problem.

Anonymous said...

Scott Aaronson had a fun discussion on the possible size of a prospective black hole at the center of the earth.

Donald Pretari said...


Is this a Fermi Problem?

How much wood would a woodchuck chuck if a woodchuck could chuck wood?

Don the libertarian Democrat

Ian Smith said...

How many gas stations are there in the US?

Unknown said...

I didn't know you were allowed to use Wikipedia for Fermi problems.

gcochran said...

I'm supposed to write a science fiction book about this. I think that the population culling would be more like the Raft of the Medusa.

Anonymous said...

I must admit, that I looked at the question of dark energy and dark matter, as a Fermi-like problem.

If we are trying to identify what we believe is 74% or the universe (and 22% dark matter) that seems like a lot of the universe.

I postulated that if we are looking for 96% of the universe, then perhaps we should consider what exists everywhere in the universe. Time and space, when you start looking at that, the ramifications are quite interesting. Never got any feedback on the idea, but spend a lot of time looking into it and thinking about it and it did seem to answer some of the big questions... maybe just in my mind.

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