Fermi Problems

Enrico Fermi was a scientist who was part of the Manhattan Project, which was the research and development team of scientists that developed the first nuclear weapons during World War II. Fermi was present at the time of the Trinity Test - the first test detonation of an nuclear bomb, which took place in New Mexico.

Before the test detonation, Fermi took some paper and shredded it into small pieces, which he then dropped from a height of 6 feet, during various phases of the test process. His goal was to use the displacement of the paper as a way of making an estimate of the detonation energy release. In his own words:

"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 2 1/2 meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T."

This was an impressive feat of estimation, considering he used no mechanical measuring devices, and had to extrapolate a great deal of information based on a very small sample of data. In the end, he was only off by an order of 2 (the actual detonation was probably in the vicinity of 20 thousand tons of T.N.T.).

This has led to a whole class of problems called "Fermi Problems" in which estimates have to be made, and sometimes those estimates must involve very large (or very small) numbers.

Often, when dealing with large quantities we must estimate, we will simplify the process of estimating by limiting ourselves to powers of 10. For example, suppose I was asked to estimate the population of my home town. I would ask myself, "Is the population of my town closer to 1, 10, 100, 1,000, 10,000 or 100,000?" Now, I don't live in a very big town, but I'm quite sure it's got at least 1,000 people. So I will eliminate 1, 10, 100, and 100,000 as possible answers. Now I've just got to decide between 1,000 and 10,000. Based on a gut feeling about the size of the town and the density of houses in the downtown village, I feel comfortable guessing 10,000.

It might seem crazy to limit ourselves to powers of ten, but there are a couple of advantages to doing this: one is that it makes the calculations simpler, and the other is that if we're stringing together a series of such estimates, some will be high and some will be low, and on average, those differences will likely help to cancel each other out. We won't get super close to the actual answer, but we'll get in the ballpark.

Fermi problems are a fun excercise in estimation. Let's take a very simple example to start with.

Question: How many chairs are there in the state of Maine?

The way I will approach this problem is to consider it as a function of the population of the state of Maine. Now, I don't know Maine's population, but I'm pretty sure it's between 1 million and 10 million. I happen to know that the population of New York City is pushing 10 million, and I feel fairly confident that Maine's population isn't near that, so I'll make an estimate of 1,000,000.

Now I need to ask how many chairs there are per person. My instinctive guess is: There are three chairs per person. Every person has one chair at their work/school, and every person has two chairs at home (one in the dining room, and one in the living room). This may not be a terribly accurate estimate, and I have some doubts about it - for example, I think to myself, "What about all those chairs at stores like Office Max and Staples, and furniture stores?" But then I argue back to myself, "But those chairs are going to be sold largely to people who need to replace existing chairs, and they're not a significant source of error." I also ask myself, "What about little children who aren't in school/work?" But this is a change in the opposite direction of my previous issue (the furniture stores), so I feel even better about my initial guess. I think I'll stick with it.

You might have come up with a different guess, and that's probably okay. You may have also considered whether chairs in theaters, churches, auditoriums, etc. will significantly change the estimate. But if you came up with a guess of 30, you're probably off by an order of magnitude, and you're not going to come up with a good result.

And if you really want to, you can ask yourself, "Is the number of chairs per person closer to 1 or 10?" and then use one of those two numbers. Some people will insist that you always use powers of ten when performing estimates in Fermi problems, but others aren't sticklers for that. I'm one of the non-sticklers; I'm happy to use 3 instead of 1 or 10!

So, since I've said there are 1,000,000 people in Maine, and each one has 3 chairs, that's a total of 3,000,000 chairs.

Do I think my answer is super close? Not really, but I feel reasonably good that it's within an order of magnitude of the correct answer. In other words, I think it's closer to 3 million than it is to 300,000 or 30 million. And that's really the goal of a Fermi problem.

For each of the questions below, explain the estimates you made, and give your final estimate. Use powers of ten whenever you think it makes sense to do so.