How the physicist Enrico Fermi guessed big numbers from small ones, the classic "piano tuners in Chicago" problem worked step by step, why rough guesses often land close, and three party-sized practice problems.
How many piano tuners work in Chicago? How many jelly beans are in that jar at the office party? Nobody at the table knows. But with a pencil, a few honest guesses and some multiplication, you can get surprisingly close. It's called Fermi estimation, and it's the closest thing our calculator has to a scientific method.
\*Very scientific. Not really. But the method in this guide is real, and physicists, engineers and teachers use it all the time.
Enrico Fermi was one of the great physicists of the 20th century. According to his Nobel Prize biography1, he was born in Rome on September 29, 1901. He won the 1938 Nobel Prize in Physics for his work on radioactivity produced by neutrons, then emigrated to the United States that same year, "primarily to escape Mussolini's fascist dictatorship."
In America he led the experiments behind the first controlled nuclear chain reaction, in Chicago on December 2, 1942, on a squash court beneath a stadium. He then taught at the University of Chicago until his death on November 28, 1954.
The classic Fermi problem is "How many piano tuners are there in Chicago?" Accounts agree Fermi asked it but differ on where. A William & Mary physics course page says it was during graduate oral exams2. The American Institute of Physics says he asked his introductory course students3. The University of Houston's Engines of Our Ingenuity says he asked his students at the University of Chicago4 to work it out "making full use of what they already knew."
Here's how it goes. Every number except the first is a guess, and that's fine.
| Step | Our guess | Running total |
|---|---|---|
| People in Chicago | About 2.7 million | 2,700,000 people |
| People per household | About 2.5 | 1,100,000 households |
| Households with a piano | About 1 in 20 | 55,000 pianos |
| Tunings per piano per year | About 1 | 55,000 tunings a year |
| Tunings one tuner can do | 4 a day, 5 days a week, 50 weeks | 1,000 per tuner per year |
| Tuners needed | 55,000 ÷ 1,000 | about 55 |
The only number we looked up is the first: the U.S. Census Bureau counted 2,746,388 people5 in Chicago in 2020. The rest is everyday common sense.
So is 55 right? The William & Mary page does its own version, lands on about 80, and notes that when its author counted in 2004, "there were 60." Engines of Our Ingenuity says the answer "varies but it hangs around 130 or so." Different guesses, same ballpark: dozens, maybe a hundred-odd. Not five, and not five thousand. That's the whole goal.
1. Decompose. Break the scary question into small ones you can actually guess: people, households, pianos, tunings, a tuner's workload.
2. Guess with bounds. For each piece, ask what's clearly too low and clearly too high, then pick something in between. The William & Mary page does exactly this: more than 1% of families have pianos, "certainly less than 20%."
3. Multiply (and divide). Chain the pieces together, and write each assumption down. The AIP article says to state them "explicitly so that I can go back and change them later if I need to."
4. Sanity-check the order of magnitude. That means the power of ten: is the answer in the tens, hundreds or thousands? Forty thousand tuners for 55,000 pianos? Something blew up. Go find the step. The William & Mary page adds a nice house rule: never more than two significant figures. "About 55" is honest. "54.93" is pretending.
Here's the surprising part: every guess could be way off, yet the answer often lands close. The William & Mary page explains why: "sometimes the uncertainty is in one direction, sometimes in the other, and the uncertainties do tend to balance."
Say the real share of homes with a piano is 1 in 10, not 1 in 20, so we guessed half as many pianos as there are. But say pianos really get tuned every two years, so we guessed twice as many tunings. Those two mistakes cancel completely.
You can see the effect with simple coin-flip math. Imagine five guesses, each off by exactly a factor of two, randomly too high or too low. If all five miss in the same direction, the answer is off by 2 × 2 × 2 × 2 × 2 = 32 times. But that only happens 2 times out of 32. Most of the time, 20 times out of 32, the misses mostly cancel and the final answer is off by only a factor of two. The other 10 times out of 32, it's off by a factor of eight. That's why rough guesses so often get the order of magnitude right.
Cancellation is a tendency, not a promise. It fails in a few predictable ways:
Try these before peeking. No phones allowed (that's the fun part).
1. How many times will your heart beat this year? The American Heart Association says a normal resting heart rate6 for adults is between 60 and 100 beats per minute. Pick 70. That's 70 × 60 minutes = 4,200 an hour, × 24 hours = about 100,000 a day, × 365 days = about 37 million a year. Using 60 or 100 instead gives roughly 32 to 53 million. Heart rate rises with exercise and, the AHA notes, may be lower during sleep, so "a few tens of millions" is the honest answer.
2. How long will you spend brushing your teeth this year? The American Dental Association recommends brushing twice a day for two minutes7. That's 4 minutes a day, × 365 = 1,460 minutes, or about 24 hours a year. Lovely sanity check: the same ADA page estimates brushing plus cleaning between your teeth at "around 30 hours per year." Over 50 years, our 24 hours a year adds up to about 50 full days of brushing.
3. How many jelly beans are in a 1-liter jar? A liter is 1,000 cubic centimeters. Guess one bean takes up about 2 of them. That fits 500 beans, but beans don't pack perfectly, so guess two-thirds of the jar is beans and the rest is air: roughly 330 beans. Write down "a few hundred," let someone count, then eat the evidence.
CountMyBits works a lot like a Fermi estimate. As How it works explains, it turns a how-often answer into a number per year ("about once a week" counts as 52), multiplies by your years, then by a small ballpark average, and adds your history answers on top. Fifty-two a year for ten years is already 520 before the last multiplier. Small guesses, multiplied, get big fast.
One big difference: a good Fermi estimate keeps its guesses balanced so errors cancel. Our calculator is generous on purpose, so its errors all lean the same way, upward. Your roses and eggplants come out impressively high. That's the joke, and it's why no statistician would sign off on any of it.
Ready to watch a few small guesses turn into one big number? Take the quiz (18+). Want the full method behind the math? Read How it works.
Quiz culture
What survey research says about social desirability bias, why people estimate instead of counting, and why self-reported numbers pile up at 5, 10 and 20, plus why CountMyBits never needs your name.
Party games
Eleven low-prep, laugh-first party games for a girls' night or bachelorette weekend, with simple opt-in rules so everyone has a good time.
Quiz culture
How a ten-question survey in a 1924 student newspaper became a campus ritual, and how "Which ___ are you?" quizzes took over the internet in 2014.
One minute, one gloriously inflated number, drawn as roses and eggplants.
Count my bits (18+)
