The appraisal questions they ask at interviews: what Fermi problems are and how to deal with them

At job interviews, very specific questions are often asked in which you are asked to estimate specific quantitiessuch as: “How many golf balls would it take to fill a room?” Such a question, at first glance, may seem absurd or impossible to solve, but in reality it is a very useful logical exercise and which clearly shows the reasoning skills of an individual. In fact, whoever has to solve the problem must be able to make an approximation of the quantity requested by dividing the question into small sub-problems easy to resolve. This type of problems are called “Fermi problems” and due to their characteristics they are often used in job interviews. Let’s see in this article how they work and how to deal with them.

Fermi problems: estimating quantities

How much we are capable of estimate the quantities? For example: how much money does an average smoker spend to buy cigarettes over his lifetime? It seems like a banal question, but in truth knowing how to estimate the order of magnitude of a phenomenon is a fundamental tool for knowing how to read the world around us. The smoker, for example, he might understand How much money would you save if you didn’t smoke?but not only. Often when we read the news in the newspaper we find ourselves faced with numbers that we have to interpret: how many people were at a given demonstration, how much money was allocated by the government for a certain resource, or on the contrary what cuts to funds were made in a certain sector, and so on… Understanding large or small numbers in fact it is not simple. If I told you – for example – that this year the Italian public debt increased by a billion, could you tell me if it is a lot or a little?

If we do not know how to interpret these numbers, the news we read may not tell us anything, or worse: we may not be able to recognize fake news. This is why it is very important, in life as well as in science, to be able to estimate the order of magnitude of a problem before even starting to solve it. This is how Fermi’s problems arise: questions that confront us with the need to know establish an estimate of the response that we are looking for, even if the data we have is limited.

These types of problems are called “Fermi” problems in honor of the Italian physicist Enrico Fermi, who was said to be extremely skilled in approximate calculation. Legend has it that, when he was a professor, he used to ask his students the following question: “How many piano tuners are there in Chicago?”. Clearly, it’s not a question we really care about the answer to, none of us need to tune a piano, let alone live in Chicago. What is interesting is that, through reasoning in a few precise steps, we can arrive at a sufficiently accurate approximate answer. And this type of reasoning can be repeated in any field, for any question and for any size. But How do you solve a Fermi problem?

How to solve Fermi problems: the example of piano tuners

As we said, any Fermi problem asks us how to estimate a certain quantity with little information to help us. To understand how to solve this type of problem, let’s solve it together classic Fermi questionrevisiting it a bit:

How many piano tuners are there in Milan?

On the spot it may seem like truly impossible information to obtain. And if we look at the request as a whole, it is. But the trick is right here: not to be assailed by the difficulty of not knowing a single piece of information, but divide the request instead in many sub-problems of which we are able to give an estimate. What we need to do, therefore, is divide the question into simpler problems and proceed by approximations:

1. First, to know how many piano tuners there are in Milan, we have to ask ourselves how many people might need a tuner in Milan. We must therefore start from number of people living in the citywhich is easily available information! To date, the population of the Lombardy capital is 1.5 million people.
2. To this we can ask ourselves how many possible houses there are that could have a piano, so we look for the number of members per family. The average in Italy is 2.2 people per family, so let’s make 1,500,000/2 = 750,000 families in Milan.
3. Now we have to make a logical estimate. How many of these families own a piano? Suppose it is 1 family out of every 20. We therefore have 750,000/20 = 37,500 pianos in Milan.
4. If tuned with the right regularity, a piano comes granted once a yearand let’s say it takes 2 hours to tune a single piano.
5. This means that every tunerworking 8 hours a day, manages to tune a maximum of 4 pianos a day, therefore 4×5 = 20 per week, and therefore 20×50 = 1000 pianos per year.
6. At this point, given that each of the 37,500 pianos in Milan is tuned once every year, we divide the number of pianos in Milan by the number of pianos that a single tuner manages to tune in a year: 37,500/1,000 = 37.5 tuners. And here is our estimate!

If we instead wanted to solve the example at the beginning, that is, how much a person spends per year smoker, we could set up the reasoning like this:

1. ask us on average how many smokers are there in Italy;
2. how many packs of cigarettes do you smoke per day/week an average smoker, and therefore per year;
3. how much does it cost to date on average one package of cigarettes;
4. then calculate with a simple multiplication how much does an average smoker spend per year;
5. at this point, estimating that each smoker consumes cigarettes per about 40 years old of one’s life, derive the total expense.

So as you can see, our estimate is obtained through small stepsi.e. considering simple sub-problems for which it is easy to find an estimate or approximation. For example, the estimate of the number of pianos per family may be incorrect, but what remains consistent is the order of measurement we are expressing.