Introduction to

Fermi Problems

Enrico Fermi (September 29, 1901 – November 28, 1954)

An Italian-born physicist

Known for his work on the development of the first nuclear reactor, and contributions to quantum theory, nuclear and particle physics

Fermi was known for his ability to make good approximate calculations

Fermi Problems (aka Fermi Estimates)Named after Enrico Fermi, they are problems

that require making educated guesses using estimation skills and prior knowledge

They are solved by breaking what seems like an impossible problem into smaller/simpler problems, and using estimation skills to continuously refine your answer

These problems often involve making justified guesses about quantities that seem impossible to compute given limited available information

How do I solve a Fermi Problem?Most of the time, you will need to ask

questions!A Fermi question demands communicationA Fermi question utilizes estimation

Write these down!A Fermi question might require you to

make assumptionsWrite these down!

A Fermi question emphasizes process rather than “the” answer.

Show how you got to your answer

The Classic Fermi Problem“How many piano tuners are there in

Chicago?”

Sample Solution to the Classic ProblemTypical solution involves multiplying a bunch of

estimates that leads to the correct answer if the estimates are correct

Sample solution:There are ~ 5 000 000 people living in ChicagoOn average, there are 2 people in each household

in ChicagoRoughly one household in 20 has a piano that is

tuned regularlyPianos that are tuned regularly are tuned once a

yearEach piano tuner works 8 hours a day, 5 days a

week, and 50 weeks a year

Sample Solution to the Classic ProblemTypical solution involves multiplying a bunch of estimates

that leads to the correct answer if the estimates are correct

Sample solution:There are ~ 5 000 000 people living in ChicagoOn average, there are 2 people in each household in ChicagoRoughly one household in 20 has a piano that is tuned

regularlyPianos that are tuned regularly are tuned once a yearEach piano tuner works 8 hours a day, 5 days a week, and 50

weeks a yearWith all of these assumptions…we get 125 piano tuners

in Chicago.See SAMPLE WRITE UP

Why do these ‘work’?Fermi estimates generally work because…

The estimations of the individual terms are often close to correct

Over- and under- estimates help cancel each other out

So,if we can’t get the answer, what’s the point?

Looking for Fermi estimates of the answer to a problem before turning to calculations provides you with a useful check on your results

Although Fermi estimates are often not accurate (it depends on how good your estimates are), this type of analysis does tell us what to look for in order to get a better answer (e.g. is one of your estimates really off?)

Fermi estimates give us a rough idea of the answer… which may be enough in some cases

What’s the point (continued)…They are good brain exercise

They often don’t have an exact answer, forcing you to…Use your imaginationThink outside the box (sometimes)

Your task…Work in groups to tackle a fermi problemDo a written report (see sample report)Present your solution to the class