Fermi: the estimation game

How many photos fit on a phone? How long would it take to drive to the Moon? Fermi asks big questions you cannot look up in your head, and rewards a good educated guess. You get five questions a game, each answer comes with the arithmetic behind it, and you can play as many games as you like for free.

Scoring: how close is close enough?

Each question is worth up to 100 points. Fermi scores on a multiplying scale, because in estimation being twice too high is as wrong as being half too low. A guess within 10% of the answer earns full marks. After that, every factor of ten costs about half your points.

So the first goal is the right power of ten. If the answer is about 300 and you say 1,000, you still keep most of your points. Say 100,000 and you get nothing.

What is a Fermi problem?

A Fermi problem is a question that seems impossible to answer without data, but becomes manageable once you break it into parts you can estimate. The name comes from Enrico Fermi (1901–1954), the Nobel Prize-winning physicist who was famous for quick, rough calculations that landed surprisingly close to the truth. At the first atomic test in 1945, he famously estimated the strength of the blast by watching how far scraps of paper drifted as the shock wave passed.

The classic classroom example is “How many piano tuners work in Chicago?” Nobody knows that number offhand. But you can estimate how many people live in the city, how many households own a piano, how often a piano needs tuning, and how many pianos one tuner can service in a year. Multiply and divide those guesses and you get a defensible answer. That process, not a memorized fact, is what a Fermi estimation is about.

How to make a Fermi estimate

1. Pin down the unit. Are you counting people or trips, liters or bottles, hours or days? Many wrong answers are right answers in the wrong unit.
2. Write the chain. Turn the question into a short product: rate × time, length × width × depth, people × share × amount each.
3. Round every factor. Use friendly numbers like 1, 2, 3, 5 or 10, times a power of ten. Rounding errors tend to cancel out.
4. Multiply the powers of ten separately. 3 thousand × 2 hundred is 6 × 100,000, or 600,000.
5. Sanity-check with bounds. Ask what would be surprisingly low and surprisingly high. If your answer is outside that range, look for a slipped unit.

Two worked examples

How many seconds are in a year?

A year has 365 days, each day has 24 hours, and each hour has 3,600 seconds. Rounded: 400 × 25 × 4,000 = 40,000,000. The exact product is 365 × 24 × 3,600 = 31,536,000. The rough version is within a factor of 1.3, comfortably in the high-scoring range, and took seconds to compute.

How long would it take to read a million words?

Assume a steady 250 words a minute. One million divided by 250 is 4,000 minutes. Divide by 60 and you get about 67 hours, or close to three full days of nonstop reading. Notice how the assumption, 250 words a minute, defines the answer. Fermi questions always state their assumptions for exactly this reason.

Getting the power of ten right

When you can only narrow an answer to a range, pick the middle on a multiplying scale, not an adding one. If you think the answer is somewhere between 100 and 10,000, guess 1,000: it is ten times away from both ends. Guessing 5,000, the midpoint you would get by adding, is fifty times higher than your low end and only twice as low as your high end.

The ruler in the game is built on this idea. Each labeled tick is ten times the one before it, so sliding the same distance always multiplies or divides your guess by the same amount.

Common mistakes

• Mixing units. Milliliters and liters differ by a factor of 1,000. So do meters and kilometers. Megabits and megabytes differ by a factor of eight.
• Forgetting squares and cubes. Doubling the side of a square tile quadruples its area. Doubling every side of a box multiplies its volume by eight.
• Applying a rate to the wrong span. A rate per second becomes 3,600 times bigger per hour and 86,400 times bigger per day.
• Counting the wrong thing. A bus that makes ten trips counts as ten departures, but it is still one bus. Read what the question counts.

Scenario questions and reference questions

Every question is labeled Scenario or Reference above its title. Scenario questions are original exercises with every assumption written out, like a fixed heart rate or a dripping tap at a set speed. Their answers follow from those assumptions and do not claim to describe every real case. Reference questions use published figures, such as NASA’s planetary fact sheets, and show the rounded calculation. Our answer sources explain how we choose reference data.

Fermi in the classroom

Fermi works well as a warm-up for lessons on estimation, unit conversion and scientific notation. Everyone who plays the daily challenge on the same date gets the same five questions, so a class can play individually and then compare their reasoning, not just their scores. Good prompts for discussion: which factor in your chain was the most uncertain? Which unit conversion changed the answer the most? What would the answer be if one assumption doubled?

Questions about Fermi

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