The idea in one line
What is a Fermi estimation game?
Fermi is an estimation game: every question has a real number for an answer, nobody knows it offhand, and the skill is reasoning your way close.
An estimation game is not a trivia quiz. Trivia rewards what you remember. Here you are asked something you could not have memorised, like how many trees grow on Earth, and you build the answer from things you do know. Physicists call this a Fermi problem, a Fermi question or a Fermi estimate, and a good one can be answered on the back of an envelope.
Fermi Unlimited is an educated guess game built on that habit. You are never asked to be exact. You are asked to land within 10 times the answer, too high or too low, which is what scientists mean by getting the order of magnitude right. That is the whole of the Fermi game: make an educated guess, see how close it was, and go again.
- how far off a guess can be and still survive
- 10×
- off or less counts as close
- 2×
- off or less is a bullseye
- 10%
Where the name comes from
Why is it called the Fermi game?
It is named after Enrico Fermi, the physicist who made a habit of answering impossible-sounding questions with a rough sum.
Enrico Fermi was born in Rome in 1901 and won the 1938 Nobel Prize in Physics for his work on neutron irradiation and slow neutrons. He left Italy for the United States that December. On 2 December 1942, in a squash court under the west stands of the University of Chicago's Stagg Field, the reactor built under his direction, Chicago Pile-1, produced the first human-made, self-sustaining nuclear chain reaction.
Among physicists he was just as famous for the way he estimated: he would take a question that seemed to need a laboratory and answer it from a handful of rough figures. The name stuck after his death: in August 1963 the physicist Philip Morrison, a Manhattan Project colleague, published a letter in the American Journal of Physics titled "Fermi Questions", and teachers have used the label ever since.
The name carries further than the classroom. Element 100 is called fermium, the particles that obey the statistics he worked out are called fermions, and the question he asked over lunch at Los Alamos, "Where is everybody?", became the Fermi paradox.
16 July 1945
How Fermi measured an atomic bomb with scraps of paper
The most famous educated guess in science took about forty seconds and a handful of torn paper.
Fermi watched the Trinity test, the first nuclear explosion, from Base Camp about ten miles away. His own declassified note records what he did. The explosion came at about 5:30 in the morning. As the blast wave arrived, about 40 seconds later, he dropped small pieces of paper from about six feet, before, during and after it passed. The blast carried the falling scraps about two and a half metres. From that shift he estimated the explosion at ten thousand tons of TNT.
The United States Department of Energy's long-standing figure for the yield is 21 kilotons, and a 2021 Los Alamos re-evaluation of the debris puts it at 24.8 kilotons, give or take 2. Either way, scraps of paper put Fermi within a factor of about two and a half of the measured yield. He never wrote down how he turned the distance into a yield, and a 2021 Los Alamos paper had to reconstruct how he might have done it.
- Fermi's estimate, made on the spot
- 10 kt
- the measured yield, then and now
- 21 to 24.8 kt
- how far off he was, well inside the 10 times this game allows
- about 2.5×
The method
How to make an educated guess at a number nobody knows
A Fermi estimate is four moves, and the classic example, the piano tuners of Chicago, uses all of them.
- 01
Break it into pieces
Turn one number you cannot know into several you nearly do: how many people, how many of them own the thing, how often it needs doing, how much one person can do.
- 02
Guess each piece to a power of ten
Carry one significant figure at most. Deciding between 100,000 and a million matters; deciding between 3 and 4 does not.
- 03
Trap it between two bounds
When a piece is a mystery, pick a number it is surely above and one it is surely below, then take the geometric mean: the square root of one times the other. Between 1 and 100, that is 10, not 50.
- 04
Multiply, then sanity check
Chain the pieces together and ask whether the result could be ten times bigger or ten times smaller. If neither is believable, you are done.
| People in Chicago | about 3,000,000 |
|---|---|
| Families, at 4 people each | 750,000 |
| Pianos, if one family in five owns one | 150,000 |
| Pianos one tuner can service in a year: 4 a day, 5 days a week, 50 weeks | 1,000 |
| Tuners needed, at one tuning a year | about 150 |
This is the version NASA's Glenn Research Center teaches. Its point is the confidence you can place in the result: the answer should be right within a factor of 2 or 3, and certainly within a factor of 10, so neither 15 tuners nor 1,500 is believable. Every question in this estimation game can be worked the same way, and the calculator in the game is there for the multiplying.
Why it works
Why a rough guess lands so close
Multiplying a chain of shaky numbers sounds like a way to multiply the error. In practice the opposite happens.
Errors cancel
If a result is ten factors multiplied together and each is only known within a factor of 2, the pessimistic range from lowest to highest spans a factor of about a million. It almost never happens: some pieces are guessed high, others low, and the misses offset. Sanjoy Mahajan's MIT Press textbook on estimation shows the uncertainties adding far more slowly than they multiply.
Judgment works in ratios
People sense size as a ratio, not a gap: twice as heavy, ten times as far. That is why the geometric mean is the fair midpoint between two bounds, and why a guess twice too big is the same size of mistake as one half too small.
A factor of ten is enough
Weinstein and Adam, the physicists behind the book Guesstimation, argue that knowing a number within a factor of ten settles most real decisions. Merriam-Webster defines an order of magnitude as the range from a value to ten times that value, a term in print since 1875.
A short history
Fermi estimation, from Rome to the search for aliens
1901
Born in Rome
Enrico Fermi is born on 29 September.
1938
The Nobel Prize, then America
He receives the Nobel Prize in Physics for his work on neutron irradiation and slow neutrons, and leaves Italy for the United States that December.
1942
Chicago Pile-1
On 2 December the reactor built under his direction sustains the first human-made nuclear chain reaction, under the stands of Stagg Field in Chicago.
1945
The scraps of paper
At the Trinity test on 16 July he estimates the blast at ten thousand tons of TNT from how far it moves falling paper.
1950
"Where is everybody?"
Over lunch at Los Alamos, most probably that summer, he asks Emil Konopinski, Edward Teller and Herbert York why no visitors from other stars have turned up. York recalled him following it with a chain of probability estimates.
1954
Dies in Chicago
Fermi dies on 28 November, aged 53.
1961
The Drake equation
Frank Drake writes the same kind of estimate as a formula, seven factors multiplied together, as the agenda for the first meeting on the search for extraterrestrial intelligence at Green Bank, West Virginia.
1963
The label "Fermi questions"
Philip Morrison's letter of that title in the American Journal of Physics gives the habit the name physics teachers still use.
Beyond the game
Who uses Fermi estimation
The same skill turns up wherever a decision cannot wait for exact data.
Physics classrooms
OpenStax's University Physics treats guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations and Fermi calculations as one skill, and teaches it in its first chapter. Its example: a moose weighs more than a person and less than a car, so call it 300 kilograms.
Science Olympiad
The American school competition has run an event called Fermi Questions. Answers are written as a power of ten and nothing else: 5 points for the right exponent, 3 for one power out, 1 for two. One official sample asks how many drops of water there are in Lake Erie.
Job interviews
Consulting, finance and technology interviews call them guesstimates or market-sizing questions. Georgetown University's career centre lists them as a standard case-question type, with the example "How many gas stations are there in the U.S.?"
Vocabulary
Estimation terms, defined
- Educated guess
- Cambridge Dictionary: a guess that is made using judgment and a particular level of knowledge, and is therefore more likely to be correct.
- Fermi problem
- A question whose answer is a number you could not look up in your head but can reason towards. Also called a Fermi question or a Fermi estimate.
- Order of magnitude
- A factor of ten. Getting the order of magnitude right means landing within ten times the true value.
- Guesstimate
- A blend of guess and estimate. The Oxford English Dictionary finds it in print in 1904, nearly sixty years before the label Fermi question.
- Back-of-the-envelope
- Merriam-Webster: done quickly to provide a rough estimate, not exact. The sum fits on whatever paper is to hand.
- Estimation game
- A game scored on how close a reasoned guess lands, not on what you remember. Estimation games reward method over memory.
- Fermi game
- Any game built on Fermi problems. The best known is fermi gg, the daily one with three questions a day; Fermi Unlimited is the same idea with no daily limit.
- Ballpark figure
- Merriam-Webster: approximately correct. From the phrase in the ballpark; the adjective dates from 1957.
In this game
How Fermi Unlimited turns estimation into a score
The rules follow the science. A guess survives if it is within 10 times the answer in either direction, one order of magnitude, the standard NASA's piano tuner lesson calls certain for a careful estimate. Within 2 times is close, and within 10% is a bullseye. Error is measured as a ratio, so a guess of 50 and a guess of 5,000 are equally far from 500.
After every guess the answer screen shows the published figures the answer was built from and the arithmetic that joins them, so each question doubles as a worked Fermi estimate. Your score is how many you survive in a row, and unlike fermi gg, the daily Fermi game, there is no limit on how many you can play: make an educated guess, then another.
Sources: NobelPrize.org, Enrico Fermi, Facts and Biographical; U.S. Department of Energy, The Manhattan Project: an interactive history; Enrico Fermi, "My Observations During the Explosion at Trinity on July 16, 1945", U.S. National Archives; Selby and others, and J. I. Katz, Nuclear Technology 207 (2021), Los Alamos National Laboratory; Philip Morrison, "Fermi Questions", American Journal of Physics 31 (1963); NASA Glenn Research Center, Fermi's Piano Tuner Problem; Sanjoy Mahajan, The Art of Insight in Science and Engineering (MIT Press, 2014); Lawrence Weinstein and John A. Adam, Guesstimation (Princeton University Press, 2008); OpenStax, University Physics Volume 1; Science Olympiad, Division C Rules Manual 2024; Eric M. Jones, "Where Is Everybody?" (Los Alamos National Laboratory, 1985); SETI Institute, The Drake Equation; Oxford English Dictionary, Merriam-Webster and Cambridge Dictionary.