How To Solve Any Problem Using Enrico Fermi's Back-Of-The-Envelope Math (And Some Common Sense)

In 1945, as the first atomic bomb exploded in the New Mexico desert, Enrico Fermi stood miles away, holding a few scraps of paper. As the shockwave rolled toward him, he dropped the papers and watched how far they traveled. With this simple act, he estimated the bomb’s yield. His guess? About 10 kilotons of TNT. The actual yield? 20 kilotons. Not bad for a back-of-the-envelope calculation.

Fermi, the Nobel Prize-winning physicist responsible for the world’s first nuclear reactor, was one of the foremost geniuses of his time. He could take the barest scraps of information and turn them into surprisingly accurate answers. Today, part of his legacy lives on in what we call Fermi problems — questions that seem impossible to answer but can be tackled with a little logic, a few assumptions, and a lot of common sense.

What Is a Fermi Problem?

The amazing Enrico Fermi. Credit: Atomic Heritage Foundation – Nuclear Museum.

A Fermi problem is a question designed to be solved through estimation and reasoning rather than precise calculation. Think: How many piano tuners are there in Chicago? Or How many jelly beans fit in a bucket? Solving these problems is never about getting the “right” answer. They’re about the reasoning process — breaking down a big, messy question into smaller, manageable pieces.

“It’s about being able to step back and look at your result. Does it make sense?” says Stefan Funk, a particle physicist who teaches Fermi problem-solving at Stanford University. “If you’re only off by a factor of two or three, that’s no big deal. If you’re off by, say, 10²⁰ — that’s a big problem.”

Take the classic piano tuner question. To solve it, you might start by estimating Chicago’s population (say, 3 million). Then, you’d guess how many households own pianos (maybe 1 in 5). Next, you’d figure out how often pianos need tuning (once a year?) and how many tunings one person can do in a year (500?). Crunch the numbers, and you’ll land on an estimate — say, 300 piano tuners in Chicago.

Is it exact? No. But it’s close enough to be useful. And that’s the point.

Why Fermi Problems Matter

Fermi problems aren’t just for physicists or math whizzes. They’re for anyone who wants to think more clearly about the world. Tech companies love to use them in interviews because they reveal how candidates approach complex, ambiguous problems. Scientists use them to sanity-check their experiments. And teachers use them to help students develop critical thinking skills.

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“Breaking a problem down to what you know and what you need to guess at can give you an intuition into the uncertainty of the answer,” Funk says. “And seeing how the pieces fit together gives you a better understanding of what parts of the problem are important.”

Consider this: How many cups of water are in an Olympic-sized swimming pool? You might start by estimating the pool’s dimensions (50 meters long, 25 meters wide, 2 meters deep). Then, calculate its volume (2,500 cubic meters). Convert that to liters (2.5 million), and then to cups (about 10 million). Is it perfect? Hell no. But it’s good enough.

How to Solve a Fermi Problem

Think Out Loud: The process is more important than the answer. Talk through your reasoning, even if you’re just talking to yourself. It’s actually fun when you do this with another person. (Okay, maybe not during interviews with annoying HR people).
Write Things Down: Jot down your assumptions and calculations. It’s easier to track your steps on paper than in your head.
Make Assertions: Simplify the problem by making reasonable assumptions. For example, “Let’s assume every household has two people.”
Round Your Numbers: Fermi problems thrive on approximations. Use round numbers to make the math easier.
Keep It Simple: Don’t overcomplicate things. Focus on the big picture, not the tiny details.

The real magic of Fermi problems lies in their imperfection. They remind us that it’s okay to be wrong — as long as you’re thoughtfully wrong. “There are no wrong answers,” says Funk. “It’s about the process.”

No single formula exists. Yet each problem invites the same approach: break it down, make realistic (or at least consistent) assumptions, and trust your critical thinking. “No Wrong Answers” is a common Fermi problem refrain because even if your math arrives at a slightly off result, you’ve shown how you reason. And that, ultimately, is the real answer.

So, the next time you’re faced with a seemingly impossible question — whether it’s How many grains of sand are on all the world’s beaches? or How long would it take to drive to the moon? — grab a napkin and a pen. Start breaking it down. Make some guesses. Do some math. You might just surprise yourself with how close you can get.