Below is a picture of the Fermi problems names sake, Enrico Fermi.
The typical Fermi Problem
Fermi problems, or back of the envelope calculations, are a method of estimating an answer to a question where there is seemingly not enough information to answer the question.
The prototype Fermi Problem is as follows:
How many piano tuners are in the city of New York?
One might think there is not enough information to answer this question.
However, by using a bit of common sense and a few reasonable guesses, one can estimate the answer.
For instance, one could go about tackling this question as follows:
First, how many people are in NYC?
We can estimate this as about 8 million.
How many people have a piano? 1 in 10? 1 in 20? Lets use 1 in 50.
That gives us the number of pianos in NYC as 160000.
Lets suppose a piano needs tuned once every 4 years. This means that each year about 40000 pianos need tuned.
Suppose an average piano tuner can tune 4 pianos a day, and works 200 days a year, this means that the average piano tuner tunes 800 pianos a year.
Therefore there are about 40000/800 = 50 piano tuners in NYC.
This seems to be a reasonable number, as 5 would be far to low and 500 would be too many.
A Fermi problem solution is also known as an order of magnitude answer, meaning it is accurate to a power of 10, or at best a factor of 2 or 3.
Try this problem on your home town, then afterward check the phone book. You will find that your answer will be remarkably close.