The game
A game of reasoning, not recall
The game is built on Fermi estimation: finding an approximate number by breaking a hard question into smaller quantities you can estimate.
A Fermi problem, also called a Fermi question, asks for a number when you do not have all the data. To estimate how many trees grow on Earth, for example, you could combine the area covered by forests with an estimate of trees per unit of area. State your assumptions, check the units and use the result as a rough estimate rather than a precise count.
Fermi turns that method into an educated guess game. Each question has an answer built from published figures, so you can compare your reasoning with the calculation after you guess. Land within 10 times the answer, too high or too low, to keep your run going. The practice questions below show how to build and check an estimate before trying more questions in the game.
- how far off a guess can be and still survive
- 10×
- off or less counts as close
- 2×
- off or less is a bullseye
- 1.1×
The name
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.
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
Use convenient rounded numbers
One or two significant figures are usually enough. Distinguish thousands from millions before worrying about small differences, and keep the units next to each estimate.
03
Try low and high assumptions
Give uncertain quantities plausible lower and upper values. Calculate both scenarios to see how much the answer could change, then improve the input with the biggest effect.
04
Multiply, then sanity check
Combine the pieces with the right arithmetic and check the units. Ask whether the result could be ten times bigger or smaller, then revisit whichever assumption has the most influence.
| 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 |
These are illustrative assumptions from NASA Glenn Research Center's piano-tuner lesson, not a count of Chicago's current piano tuners. The model assumes every piano is tuned once a year. If only one family in twenty owned a piano instead of one in five, the estimate would fall to about 38 tuners. Writing down the assumptions makes it possible to see which ones matter and improve them.
Practice
Practice questions, worked through
Try each question yourself, then reveal the assumptions, calculation and checks behind a worked answer.
Each question asks for a quantity you cannot look up in your head. Make an educated guess by choosing reasonable assumptions, breaking the question into smaller parts and checking the scale of the result. The examples move from a simple volume estimate to planning enough capacity for a crowd.
These are authored practice exercises. All numerical inputs are illustrative assumptions, not measured facts, and the worked answers follow those assumptions: they are not answers from the live Fermi game.
Write down your own assumptions and estimate before opening each worked answer. Follow the method above if you need a starting point. Different defensible assumptions can produce different useful answers.
01 · Volume and unit conversion
How many mugs of water would fill a bath?
Picture a bath filled to a comfortable level. Estimate how many full mugs of water it would take to reach that level.
Show one worked answerHide the worked answer: How many mugs of water would fill a bath?
Illustrative assumptions
- Assume the chosen fill level holds 150 litres of water.
- Assume each mug holds 300 millilitres and no water is spilled.
Work through the estimate
- Convert the bath volume: 150 litres = 150,000 millilitres.
- Divide by the mug volume: 150,000 ÷ 300 = 500 mugs.
The estimate
About 500 mugs.
Check the estimate
Keep the mug size fixed. If you instead assume a bath volume of 120–180 litres, the estimate becomes 400–600 mugs. A larger bath must need more mugs; a larger mug must mean fewer refills.
What this one teaches
Convert both volumes to the same unit before dividing. The result describes the assumed fill level, not the capacity of every bath.
02 · Distance and time
How long would it take to walk across a town?
Choose an imaginary route across a small town. Estimate the walking time before checking a map or a clock.
Show one worked answerHide the worked answer: How long would it take to walk across a town?
Illustrative assumptions
- Assume the route is 4 kilometres long, including its turns.
- Assume an average walking speed of 4 kilometres per hour, with no extra stops.
Work through the estimate
- Use time = distance ÷ speed.
- 4 kilometres ÷ 4 kilometres per hour = 1 hour, or 60 minutes.
The estimate
About an hour.
Check the estimate
With the same assumed route, a speed of 3 kilometres per hour gives 80 minutes; 5 kilometres per hour gives 48 minutes. Under those alternative speed assumptions, a range of roughly 50–80 minutes is more informative than a precise arrival time.
What this one teaches
Route distance can differ from straight-line distance. If your route includes waiting at crossings, include that time separately or account for it in the average speed.
03 · Breaking a total into parts
How many pages of reading fit on one bookshelf?
Imagine one fully occupied shelf of books. Estimate the total number of pages without counting each book or opening it.
Show one worked answerHide the worked answer: How many pages of reading fit on one bookshelf?
Illustrative assumptions
- Assume 120 centimetres of usable shelf length.
- Assume books average 3 centimetres thick and contain 250 pages each.
- Assume the books stand upright in one row with no gaps or bookends.
Work through the estimate
- Estimate the book count: 120 ÷ 3 = 40 books.
- Multiply by pages per book: 40 × 250 = 10,000 pages.
The estimate
About 10,000 pages.
Check the estimate
Holding the assumed page count fixed, an average thickness of 2–4 centimetres gives 60–30 books, or 15,000–7,500 pages. This is a sensitivity check, not a measured range for real books.
What this one teaches
Related assumptions can move together. Thicker books may also have more pages, so changing thickness while fixing page count is useful for isolating an input but may not describe a realistic collection.
04 · Geometry and empty space
How many marbles fit inside a box?
Imagine pouring identical marbles into a cube-shaped box. Estimate the number of marbles, remembering the gaps between them.
Show one worked answerHide the worked answer: How many marbles fit inside a box?
Illustrative assumptions
- Assume the box measures 20 centimetres along each inside edge.
- Assume spherical marbles with a diameter of 2 centimetres, so their radius is 1 centimetre.
- Assume the marbles themselves occupy 60% of the box volume; the rest is empty space.
Work through the estimate
- Box volume = 20 × 20 × 20 = 8,000 cubic centimetres.
- Using sphere volume = 4πr³ ÷ 3, one marble occupies approximately 4.2 cubic centimetres.
- Volume occupied by marbles = 8,000 × 0.60 = 4,800 cubic centimetres.
- Number of marbles ≈ 4,800 ÷ 4.2 ≈ 1,140.
The estimate
Roughly 1,100 marbles.
Check the estimate
If the assumed occupied fraction changes to 50–70%, the calculation gives approximately 950–1,330 marbles. Ignoring all gaps would give about 1,900, which should be an upper bound for this volume model.
What this one teaches
Dividing container volume by object volume is only a starting point. Packing, object shape and the container edges affect the usable space.
05 · Population and demand
How much drinking water does a school need at lunch?
Estimate the water served to students during a single lunch period. Define the scope as drinking water, excluding cooking and washing up.
Show one worked answerHide the worked answer: How much drinking water does a school need at lunch?
Illustrative assumptions
- Assume 600 students attend lunch.
- Assume 75% choose water and each of those students drinks 0.4 litres.
- Assume no additional water is needed for staff, spills or unused servings.
Work through the estimate
- Water drinkers = 600 × 0.75 = 450 students.
- Water served = 450 × 0.4 = 180 litres.
The estimate
About 180 litres of drinking water.
Check the estimate
If the assumed share choosing water were 50–100%, with everything else fixed, demand would be 120–240 litres. Check that the number of water drinkers never exceeds the number of students.
What this one teaches
A population is not the same as a customer count. Estimate the fraction participating before multiplying by consumption per person.
06 · People versus journeys
How many bicycle commuting trips happen in a town each weekday?
Estimate one-way bicycle trips made by residents travelling to work or school and back. Leave leisure rides and visitors outside the estimate.
Show one worked answerHide the worked answer: How many bicycle commuting trips happen in a town each weekday?
Illustrative assumptions
- Assume a town has 100,000 residents.
- Assume 60% of residents travel to work or school on the chosen weekday.
- Assume 10% of those travellers cycle, each making two one-way trips.
Work through the estimate
- Travellers = 100,000 × 0.60 = 60,000 people.
- Cyclists = 60,000 × 0.10 = 6,000 people.
- One-way bicycle trips = 6,000 × 2 = 12,000 trips per weekday.
The estimate
About 12,000 one-way commuting trips.
Check the estimate
If the assumed cycling share is 5–15%, the estimate becomes 6,000–18,000 trips. An answer of 6,000 under the original assumptions counts cyclists, not their outbound and return journeys.
What this one teaches
State exactly what you count. People, return journeys and one-way trips are different units, even when they describe the same activity.
07 · Area, depth and volume
How much rainwater could a roof collect in one shower?
Imagine a roof draining into an empty tank. Estimate the water reaching the tank during a rain shower.
Show one worked answerHide the worked answer: How much rainwater could a roof collect in one shower?
Illustrative assumptions
- Assume the roof has a horizontal footprint of 10 metres by 8 metres.
- Assume 10 millimetres of rain falls uniformly across that footprint.
- Assume 75% of that water reaches the tank and the tank has enough spare capacity.
Work through the estimate
- Catchment area = 10 × 8 = 80 square metres.
- Rain depth = 10 millimetres = 0.01 metres.
- Water falling on the footprint = 80 × 0.01 = 0.8 cubic metres = 800 litres.
- Water collected = 800 × 0.75 = 600 litres.
The estimate
About 600 litres collected.
Check the estimate
With assumed rainfall of 5–20 millimetres and the same roof and collection fraction, the result is 300–1,200 litres. Doubling rain depth should double the estimate.
What this one teaches
Use the horizontal catchment footprint for this model, rather than the sloping surface area. Area multiplied by depth becomes volume only when the length units agree.
08 · Demand, service time and capacity
How many drink counters would a concert need during an interval?
Estimate the number of staffed counters needed to serve interval drink orders. This is a rough capacity model; arrival patterns can still create queues.
Show one worked answerHide the worked answer: How many drink counters would a concert need during an interval?
Illustrative assumptions
- Assume an audience of 4,000 people, with 30% placing one order each.
- Assume the interval lasts 20 minutes and each counter serves one order every 20 seconds.
- Assume a planning target of 75% of theoretical counter capacity to leave room for delays.
Work through the estimate
- Demand = 4,000 × 0.30 = 1,200 orders.
- The interval lasts 20 × 60 = 1,200 seconds.
- Theoretical capacity per counter = 1,200 ÷ 20 = 60 orders.
- Planned capacity per counter = 60 × 0.75 = 45 orders.
- Counters needed = 1,200 ÷ 45 ≈ 26.7; round up to 27 staffed counters.
The estimate
About 27 counters under these assumptions.
Check the estimate
Changing the assumed buying share to 20–40% gives 800–1,600 orders. At the same planned capacity, round 800 ÷ 45 and 1,600 ÷ 45 upward to get 18–36 counters.
What this one teaches
Capacity estimates need both a time window and a service rate. Round up when a fraction of a counter cannot operate, and remember that sufficient total capacity does not guarantee short queues.
Checking
How to check your own answers
Compare the reasoning as well as the final number: most large misses come from a unit, a definition or a missing fraction.
Record a central estimate, a lower and upper estimate, and the assumption you are least sure about. When you reveal an example, compare the reasoning as well as the final number. The ranges in the worked answers change selected assumptions; they are not statistical confidence intervals.
If your answer differs by a large factor, check units, the definition of what you counted and any repeated trips or missing fractions. If those are consistent, the difference may come from your model rather than an arithmetic error. Explain which assumptions would bring the estimates closer together.
To write another question of your own, keep a useful structure and change the setting: container volume divided by object volume, people multiplied by demand per person, or work divided by a service rate. The skill is building and checking the estimate, rather than remembering these particular answers.
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×
Why it works
Why a rough guess lands so close
A rough model makes your assumptions visible, so you can check which estimates matter most.
Errors can offset
Some assumptions may be high and others low, so their errors can partly offset. That is not guaranteed: several estimates based on the same mistaken premise can reinforce one another. Check your most uncertain inputs and try plausible low and high values.
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
From Rome to the search for aliens
Fermi's life, and the kind of estimate that kept his name, in eight dates.
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
Where educated guesses get used
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.?"
In this game
How the score works, and how it differs from fermi gg
Every guess is judged by how many times off it is, and a run lasts as long as each guess stays within 10 times.
The game measures how close your estimate is as a ratio. A guess survives if it is within 10 times the answer in either direction. Within 2 times is close, and within a factor of 1.1 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: make an educated guess, then another.
fermi gg, a well-known daily game of this kind, publishes a few questions a day and scores each guess by how many times off it was. This game measures the same ratio but turns it into a run: stay within 10 times and the next question comes, miss by more and the run ends.
Vocabulary
Estimation terms, defined
The words people use for a rough number, and where each one comes from.
- 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.
- Educated guess game
- A game scored on how close a reasoned guess lands, not on what you remember. It rewards method over memory.
- Fermi game
- Any game built on Fermi problems. A well-known one is fermi gg, a daily game on another site; this one is a different game with no daily limit.
- Ballpark figure
- Merriam-Webster: approximately correct. From the phrase in the ballpark; the adjective dates from 1957.