Answering Unanswerable Questions: Fermi Problems in Product Management Interviews

“Tell us about your greatest weakness”. This is hands down the worst, laziest interview question in the world. It reeks of an interviewer who has done zero work in preparation for the interview and is a big red flag for me.

Fortunately interview questions have come a long way since this dud — particularly interviews for product managers. A good product management interview is an interactive conversation that allows a candidate to showcase their skills and experience through case studies and problem solving exercises.

And chief in these interview exercises is the Fermi Problem.

Who was Enrico Fermi and what’s his problem!?

Enrico Fermi was an Italian-American physicist who worked on the Manhattan Project during World War II. In addition to his breakthroughs in nuclear science Fermi was known for making incredibly accurate estimates with very little data.

Because of Fermi’s uncanny estimation abilities, mathematical estimation problems meant to answer a question are often referred to as a Fermi Problem.

Example Fermi Problems

The internet is full of example fermi problems. Here are a few of my favorite:

• How long would it take you to count to one million?
• How many jelly beans fit in a bucket?
• How many beats will your heart make in a lifetime?
• How many donkeys could fit in a 747?

How many jelly beans are in this jar? You could count. Or could you estimate ala Enrico Fermi.

How Do I Solve Them?

Simple answer: you don’t.
At least not truly.
Because in some cases there really isn’t a definitive answer to these questions (…that is until I get my Series A funding then I will find out how many donkeys fit in a 747.).
Fermi Problems are about demonstrating your critical thinking and reasoning skills. The actual “answer” to the problem is far less relevant than the process of getting to an approximate answer.

Why do Tech Companies Love Fermi Problems?

Tech and software companies are often working towards an unknown solution — so being able to logically think through a problem is a vital skill.
Fermi Problems can be a great way to understand how a candidate thinks and deals with a difficult, vague problem set.

Fermi Problem Solving Tips

• Think Out Loud — This entire exercise is about seeing how you think and problem solve. So talk out loud and provide commentary to your interviewer on your process.
• Write Things Down — A good ol’ pen and notebook are your friends. You will be estimating and trying to keep track of multiple variables so do yourself a favor and jot things down. There are no bonus points for keeping everything in your head.
• Make Assertions — Whether you’re determining the number of tulips in Denmark or calculating the weight of the Empire State Building it is your job to make assertions that narrows down the problem in order to make it solvable.
  For example you might say: “For this estimate we will say the Empire State Building has 100 floors and that they’re all equal size and shape, does that work for you?”. (Nevermind that the real answer is 102 floors and that the size and shape vary from floor to floor.)
  Your interviewer will almost certainly agree to this and you have now narrowed your work to a more solvable problem, ie “How much does one floor of the Empire State Building weigh?”
• Round Your Numbers — Fermi problems are not meant to produce specific fine-tuned results. Rounded numbers will do just fine and are MUCH easier to work with (particularly if you are feeling nervous during the interview and don’t want to botch your math).
• Keep it Simple — Don’t add unnecessary challenges to your problem. With each assertion you make you should be simplifying the problem not creating new problems to solve.
  For example: In our Empire State Building problem avoid pitfalls like “Do I include furniture weight?” or “What about the weight of people inside?”.
  These types of variables are unnecessary distractions and will just make your work more difficult.
  Use your assertions to keep the problem simple. “For this example I am going to concentrate only on the building itself and nothing else.”
• Practice Makes Perfect — There are any number of Fermi problems on online forums and YouTube. Review the questions and take a few minutes to work through to your answer. The more you train your brain to think in this manner the easier it will become during an interview.

Example Problem: How Many Piano Tuners Are There in Chicago?

Calculating the number of piano tuners in various cities is, for some reason, a VERY popular question in product management interviews.

Here’s my thoughts on finding an answer to the question “How many piano tuners are there in Chicago?”

1. First let’s define the total population of Chicago. For our purposes we will round the population up to 3 million people.
2. Next, let’s convert that population into “families” to give us a more refined number. In this case we will say a family is a group of four people.
   By that definition our fictional Chicago has 750,000 families. (3,000,000/4 = 750,000).
   Never mind that Chicago is not evenly divided into groups of four people. Again, making assertions are key to finding a fermi answer.
3. Of our 750K families, how many do we think have pianos?
   Half feels like too many.
   One it ten feels like too few.
   Our answer: “One in five Chicago families has a piano. (Another assertion)
   This means that Chicago has 150,000 pianos (750,000/5 = 150,000).
4. Now how often do those 150,000 pianos need to be tuned? In reality this number probably varies based on any number of factors but again we are making assertions and not getting bogged down in details. So we will say “Pianos are tuned once per year”.
5. Now let’s define our piano tuners: “Chicago Piano Tuners work five days a week and can tune 2 pianos per day.”
   Working every weekday our tuner would tune 520 pianos in a year.
   2x5x52 = 520
   (Pianos Tuned Per Day)x(Five Days a Week)x(Weeks in a Year) = 520
   But let’s give our hardworking piano tuners (and ourselves) a break by saying they only service 500 pianos in a year. (Piano tuners go on vacation too — and 500 is an easier number for us to manipulate).
6. Let’s review what we “know”.
   We know how many pianos are in Chicago: 150,000.
   And we know how many pianos a single piano tuner can tune in a year: 500.
7. The answer: There are 300 piano tuners in Chicago.
   150,000/500=300
   (Pianos that Need Tuning in a Year)/(Pianos Able to Be Tuned By a Tuner in a Year) = The number of required tuners.

No Wrong Answers

Fermi problems are a great opportunity to understand how a person approaches a large, ambiguous problem.

And though they can seem a bit daunting at first with a bit of practice they can be a lot of fun to solve.