The Fermi estimation guide
What is the Fermi game?
The Fermi game is a daily puzzle about numbers nobody knows. Each day it publishes three questions with enormous, unglamorous answers: how many paper bus tickets London prints in a year, how many pet snakes live in England, how many steps a mail carrier takes between January and December. You are not expected to know. You are expected to think.
That distinction is the whole Fermi game. A trivia quiz rewards memory and punishes everyone else. An estimation puzzle rewards a method, and the method can be learned in an afternoon. You take a quantity that feels unknowable, split it into two or three quantities that feel merely uncertain, put a rough number on each, multiply, and check whether the result is sensible. Do that three times a day and within a couple of weeks your average score drops noticeably, not because you learned facts but because you learned to reason about scale.
The Fermi game shows its hand after every guess. You see the real answer, your multiple, whether you were too high or too low, how everyone else guessed on a log-scale chart, and a short worked example of napkin math that lands close. Even a bad miss teaches you something, which is rare in a daily puzzle.
Where Fermi questions come from
Enrico Fermi was an Italian-born physicist who won the Nobel Prize in 1938 and later worked on the first nuclear reactor and the Manhattan Project. Among colleagues he was known for a particular habit: when a question came up that nobody could answer precisely, he would produce a rough answer almost immediately, and it would turn out to be close.
The most repeated story is from the Trinity test in July 1945. As the shock wave from the first nuclear explosion reached the observation post, Fermi dropped small pieces of paper and watched how far the blast displaced them. From that displacement he estimated the yield at around ten kilotons of TNT. The instruments later put it at roughly twice that. Being within a factor of two from a handful of paper scraps is the standard Fermi estimation has aspired to ever since.
The teaching version of the technique is the Chicago piano tuner problem. How many piano tuners work in Chicago? Nobody knows, but you know roughly how many people live there, can guess what share of households own a piano, how often a piano is tuned, how many tunings one tuner can do in a year, and you can multiply. The answer you reach is wrong in detail and right in scale, and that is a genuinely useful thing to be able to do. Physics departments, engineering schools and consulting interviews have used problems like this for decades. The daily game simply puts a score on it.
Why the score is a multiple, not a difference
If a question's answer is 1.8 million and you guess 1.2 million, you are 600,000 off. If the answer is 18 and you guess 12, you are 6 off. Measured by difference, the first miss looks a hundred thousand times worse. Measured as a multiple, both are exactly 1.5×, and that is the honest comparison. You were equally good at both questions.
Scoring by multiple also makes the two directions of error symmetric. Guessing half the answer and guessing double the answer are both 2×. This matters because estimation errors naturally live on a log scale: when you are unsure whether something is in the thousands or the tens of thousands, you are unsure about a factor, not an amount.
The practical consequence is that the Fermi game punishes disasters far more than it punishes mild imprecision. A day with scores of 1.4×, 1.6× and 1.5× averages 1.5× and lands in the top bracket. A day with 1.1×, 1.2× and 12× averages about 4.8×, and two near-perfect answers cannot save it. The best players are not the ones who occasionally nail a question. They are the ones who never lose touch with the right order of magnitude.
Why your first instinct is usually off by ten
Human intuition is poor at large numbers for well-documented reasons, and a week of Fermi questions exposes all of them.
The first is anchoring. Whatever number appears in the question, or whatever number you happened to think of first, pulls your estimate toward it. A question that mentions 2007 makes people think in terms of years rather than units, and a question that mentions a school bus makes them picture a small space rather than count volume.
The second is scope insensitivity. People feel roughly the same about ten thousand and ten million because neither is a quantity anyone has experienced directly. If you cannot picture the difference, you cannot estimate it, and your guess collapses toward whatever large number feels large.
The third is confusing a rate with a total. Oreos eaten per day, bus tickets printed per year, steps walked per year: every one of these is a rate multiplied by a time, and forgetting the time, or using the wrong one, is the fastest way to be off by 365×.
None of this is a character flaw. It is the default state of a brain that evolved to count sheep, not smartphones. The Fermi method below exists because intuition alone does not work, and the game gives you a daily reminder of exactly how it fails.
A method for any Fermi question in sixty seconds
Step 1: Decide what kind of quantity it is
Is the answer a count of things that exist right now, a rate per unit of time, or a cumulative total over a period? Oreos per day is a rate. iPhones since 2007 is a cumulative total. Pet snakes in England is a stock. Naming the type tells you which factors you need and which time period you must not forget.
Step 2: Write the chain of factors
Express the answer as a product of two to four quantities, each of which you can put a rough number on. For Oreos eaten per day in the US: population, times share of people who eat Oreos in a given week, times Oreos per eater per week, divided by seven. You do not need the factors to be accurate. You need them to be independent enough that errors cancel rather than compound.
Step 3: Put bounds on each factor, then take the geometric mean
For each factor, name a number you are sure is too low and one you are sure is too high. Then choose the geometric mean of the two, not the midpoint. If a factor is somewhere between 1,000 and 100,000, the midpoint is about 50,000 and the geometric mean is 10,000. On a log scale, 10,000 is the choice that minimises your expected multiple, and the multiple is what you are scored on. The built-in calculator handles the square root.
Step 4: Multiply and sanity-check the units
Multiply the factors, then read the result back as a sentence. Three hundred million Oreos a day means roughly one per American per day, which feels high but not absurd. Thirty billion a day would mean a hundred each, which is nonsense. If the sentence sounds wrong, one of your factors has the wrong units or the wrong time period, and that is where to look before you lock in.
Step 5: Lock in and read the reveal
After the answer appears, look at the game's own napkin math. It often uses a different decomposition from yours. Comparing the two is how you learn which factors you systematically over- or underestimate, and that pattern is worth more than any single answer.
Anchor numbers worth knowing
A small set of memorised figures makes almost every daily question tractable. Round them aggressively; precision you do not have is precision you should not pretend to use.
| Anchor | Rough value |
|---|---|
| World population | 8 billion |
| United States population | 340 million |
| US households | 130 million |
| United Kingdom population | 68 million |
| Large city (London, New York) | 8 to 9 million |
| Days in a year | 365 |
| Hours in a year | about 8,800 |
| Seconds in a year | about 30 million |
| Average human lifespan | about 75 years |
| Steps per mile walked | about 2,000 |
| School bus interior volume | about 70 cubic metres |
| Tennis ball volume | about 150 cubic centimetres |
Notice how many Fermi questions reduce to these. Steps a mail carrier walks in a year is miles per day, times steps per mile, times working days. Tennis balls in a school bus is bus volume divided by ball volume, with a packing factor of about two thirds. Once you have the anchors, the question is only asking you to pick the right chain.
Mistakes that cost a factor of ten
Forgetting the time period. A per-year question answered with a per-day number is off by 365×. This single error accounts for a large share of the catastrophic scores you will see on the results chart.
Using the midpoint instead of the geometric mean. If you believe the answer is between 1,000 and 1,000,000, guessing 500,000 is a 500× miss if the truth is 1,000 and only a 2× miss if the truth is a million. Guessing 31,000, the geometric mean, caps your worst case at about 32× in either direction.
Anchoring on the biggest number in the question. Mentioning 2007 in a question about iPhones tempts people to estimate years rather than units sold per year. Write the chain of factors before you let any number in the question influence you.
Confusing stock and flow. How many pet snakes live in England is a stock. How many pet snakes are sold in England each year is a flow, and it is far smaller because a snake lives for many years. Decide which one the question asks before you estimate.
Refusing to commit. The game gives you one guess, and a guess of zero or a guess typed as a joke scores as badly as it deserves. An honest estimate from a bad chain of reasoning still usually lands within 10×. A non-estimate lands wherever.
Reading the results screen
After each question the Fermi game shows four things. The real answer, your guess and the resulting multiple. A direction, too high or too low, which over many days tells you whether you have a systematic bias. A percentile, phrased as closer than some share of players. And a histogram of everyone's guesses on a log scale, where you can see whether the crowd clustered near the truth or split into two camps that used different decompositions.
The histogram is the most instructive part. On a question like how many iPhones have been produced since 2007, you will often see one hump near the right answer and a second hump a factor of ten lower, which is the crowd that estimated a single year's sales and forgot to multiply by the years. Spotting those patterns in other people's guesses makes you better at spotting them in your own.
At the end of the day the game shows your average multiple and your overall rank, plus a share card. The card contains your three scores and your percentile but not the answers, so it is safe to post before friends have played.
Fermi questions beyond the game
Estimation of this kind is one of the few puzzle skills that transfers directly to ordinary life. Will this project take three weeks or three months? Is this charity's claim about meals served plausible? Is a headline figure about a market, a disease or a budget the right size? In every case the move is the same: decompose, anchor, multiply, sanity-check.
That is why Fermi problems appear in physics and engineering courses, where they teach students to check whether a calculated result is even the right order of magnitude before trusting it. It is why consulting and technology interviews have used them, sometimes to a fault, as a way of watching how a candidate reasons under uncertainty. And it is why forecasting researchers treat the ability to break a question into estimable parts as one of the strongest predictors of good judgement.
The daily Fermi game is a low-stakes way to practise exactly that. Three questions a day, a score you can track, and an archive to go back through when you want more. It will not make you Enrico Fermi, but it will make you noticeably harder to fool with a big number.
Playing the archive
The side menu of the Fermi game lists every previous day's puzzle. Playing the archive is the fastest way to improve, for two reasons. You can do five or ten puzzles in one sitting, which is enough repetition for the method to become habit. And because the questions are old, you can look up the real answers afterward and trace the full chain of reasoning, which the daily format discourages.
A good routine is to play today's puzzle first, then one or two archive puzzles in which you deliberately write out your factors before guessing. Within a few weeks you will notice that the question types repeat: a rate times a period, a volume divided by a volume, a population times a share. When a new question appears you will recognise the shape, and recognising the shape is most of the work.
