How to Get Better at Fermi: Estimating Numbers Nobody Knows

The first few questions in Fermi feel like trivia: either you know how many trees grow on Earth or you do not. After a few runs it becomes clear that nobody knows, and that the players with long streaks are not the ones with the most facts. They are the ones who build each answer from two or three things they do know. The numbers below are the rules of Fermi Unlimited, the version with no daily limit, and the method carries over to any Fermi game.

Fermi does not care how far your guess was from the answer in units. It cares how many times too high or too low you were, which is why a guess at the population of a city is marked exactly like a guess at the weight of a coin. Within 10% of the answer is a bullseye and within twice is close. Anything up to ten times off, either way, survives, and the run goes on. Beyond ten times, the run ends. Your score is how many questions you survive in a row.

Ten times sounds generous, and it is, on purpose. The game is asking whether you got the number of zeros right, not the first digit. A guess of 50 and a guess of 5,000 both survive an answer of 500; a guess of 40 does not. The whole skill is landing in the right power of ten, and everything below is a way of doing that reliably.

Enrico Fermi's famous classroom question was how many piano tuners work in Chicago, and his point was that nobody needs to know. You need the city's population, how many people share a piano, how often a piano is tuned, and how many tunings one tuner can do in a year. Each of those you can guess within a factor of two or three, and the product lands within a factor of ten of the real figure almost every time.

The same move works on every question in the game. How many times does a resting fan's heart beat during a soccer match? A resting heart is about 70 beats a minute and a match is 90 minutes, so a little over six thousand. How many laps of a running track make a marathon? A lap is 400 metres and a marathon is a little over 42 kilometres, so about 105. Neither is a fact you knew. Both are two facts you did know, multiplied.

Splitting also makes errors cancel. If one piece is guessed too high and another too low, the product is closer than either. Three pieces each within a factor of two rarely compound past a factor of ten, and a run that ends usually ends on a question that was guessed in one piece.

When a question refuses to split, bracket it instead. Find a number that is certainly too low and one that is certainly too high, then take the middle. The trap is that the middle of an estimate is not the average. Between 1,000 and 100,000 the right guess is 10,000, the geometric mean, not 50,500. The average would leave you five times too high against a true answer of 10,000; the geometric mean is exact. Multiply the two bounds and take the square root, or just count the zeros and split the difference.

A run does not draw questions at random from one pile. The first three come from the easiest band, questions four to eight from the middle one, and everything from the ninth on from the hardest. The steps are where runs end. A streak of three means nothing yet; a streak of nine means you have cleared both steps, and the questions from there stay hard rather than getting harder still.

So the habit to build is slowing down at the fourth and ninth questions specifically. The early ones can be answered on instinct. From four onwards, split every question, and from nine onwards, bracket the ones that will not split.

The guess box and the calculator read numbers the way people write them: 1.2 million, 3e6, 12k and 365 * 24 are all accepted, and the calculator does the arithmetic. That matters more than it sounds, because the method above produces a product of pieces, not a final number, and typing 8.2 billion * 0.6 * 2 is far safer than multiplying it in your head with a streak on the line.

A surprising number of questions come down to a handful of quantities. Knowing these to one significant figure turns a cold guess into a split one.

The world has about 8.2 billion people, the United States about 340 million, and a large city a few million.

Earth is about 40,000 km around, the Moon about 384,000 km away, and the Sun about 150 million km.

A year is about 31.5 million seconds, a lifetime about 4,000 weeks, and a resting heart beats roughly 100,000 times a day.

A car is about 4.5 m long, a football field about 100 m, Everest 8,849 m tall, and a marathon 42.2 km.

An Olympic pool holds about 2,500 cubic metres, which is 2.5 million litres, and a litre is 1,000 cubic centimetres.

The answer screen shows the figures the answer was built from and where each was published. Reading it for the questions you missed is the fastest way to grow the list above, because the figures repeat: populations, distances, volumes and rates come up across everyday life, sport, space and the human body alike.

Watch your direction, too. People judging quantities pull toward the middle of whatever range they have in mind, so very large numbers get guessed too small and very small ones too big. The reveal says which side you landed on. If it is the same side for most of a run, correct for it before you commit the next guess.

A streak of ten means you have cleared both difficulty steps, and that is a good run by any measure. Most runs that end early do not end on ignorance. They end on a unit: beats per minute multiplied by seconds, metres read as feet, a per-day figure for a per-year question. The word under the guess box says what is being counted. Read it before you type.

Calibration comes from volume, which is the case for the Fermi game with no daily limit while you are learning: question after question, across everyday life, sport, space, history, food and the human body, for as long as the streak lasts. If judging size by eye is the weak spot, Size It Up Unlimited asks the same kind of question visually, and we have written up how to get better at Size It Up as well.

Wikipedia, Fermi problem.

Hollingworth, H. L., The Central Tendency of Judgment (Journal of Philosophy, Psychology and Scientific Methods, 1910).