Fermi Estimation (Classic)

How it works

Classic mode deals a 10 question round drawn from a hand-written question bank plus a batch of 30 procedurally generated questions that is refreshed every time you return to the menu. You can filter the pool by category (everyday, physics, geography, economics, biology, engineering, tech, pop culture) and by difficulty (Warm-up, Interview, Hard) before starting.

Each question asks for a real-world quantity - platform activity, volumes, populations, physical scales. Instead of a single number you enter two: a LOW and a HIGH bound. You have 60 seconds per question; typing the bounds and submitting is the whole interaction, then the truth and a worked explanation are revealed before the next question.

The interval format is the point. A bare number hides whether you had any idea; a range you are willing to be held to is a real answer, and the scoring makes the tension explicit - the range must contain the truth, and the tighter it is, the more it is worth.

How scoring works

Width is measured in orders of magnitude: log10(high) - log10(low). If the truth falls inside your range, you score 4 points for width up to 0.5 orders of magnitude, 3 points up to 1, 2 points up to 2, and 1 point for anything wider - contained but too wide to be useful.

If the truth falls outside your range you score 0, whether you just missed or were wildly off - the feedback distinguishes the two, the points do not. An invalid range (non-positive bounds, or high below low) also scores 0.

Your best round total is stored locally and shown on the menu.

Anchor the order of magnitude first

Before any arithmetic, place the answer on the powers-of-ten ladder. Is this a thousands question, a millions question, a billions question? A handful of memorized anchors covers most of the pool: world population about 8 billion, US population about 330 million, a big city in the single-digit millions, a year is about 3 x 10^7 seconds, a paper bill about 0.1 mm thick. Getting the exponent right is worth more than any precision in the mantissa - the scoring is entirely in log space.

Once the exponent is anchored, the mantissa only needs one significant figure. The difference between 3 x 10^9 and 5 x 10^9 is 0.2 orders of magnitude - inside the top scoring band. The difference between 10^9 and 10^10 is a full band drop. Spend your 60 seconds on the exponent, not the digits.

Decompose into factors you actually know

Every Fermi question factors into pieces with known anchors: a population, a per-person rate, a timeframe, a unit conversion. "Messages sent on a platform per week" is users x messages per user per day x 7. Write the chain, put one-significant-figure numbers on each factor, and multiply the exponents mentally - add powers of ten, then multiply the leading digits.

Decomposition also tells you how wide to quote. If your chain has three factors and you trust each to within a factor of 2, the product is uncertain by roughly a factor of 8 - just under one order of magnitude, so quote a range about that wide centered on your estimate. A single-factor question you know cold deserves a half-order range and the 4 point band.

Errors in a factor chain partially cancel - some factors you overestimate, some you underestimate. This is why decomposed estimates beat gut numbers: the log-space errors of independent factors average out rather than compound in one direction.

Size the range to the scoring bands, not your anxiety

The bands are at 0.5, 1, and 2 orders of magnitude. A half-order range means high is about 3 times low (10^0.5 is roughly 3.16); a one-order range means high = 10 x low. So the practical widths are: confident answer, quote low to 3x low; normal answer, quote low to 10x low; genuinely unsure, quote two orders and take the 2 points rather than gamble on 0.

Center your range asymmetrically in log space, not linear space. If your point estimate is 4 x 10^7 and you want one order of width, quote about 1.3 x 10^7 to 1.3 x 10^8 - a factor of 3 on each side - rather than adding and subtracting the same linear amount. Linear-symmetric ranges around a big number are secretly lopsided on the log scale where you are being graded.

Never quote an unbounded hedge. A range from 1 to 10^12 always contains the truth and scores just 1 point; two of those cost more than one honest miss next to two tight hits. The scoring exists specifically to make the cowardly quote a losing strategy.

A worked example

Question: how tall would a stack of a $1.1B net worth in $100 bills be, in kilometers? Decompose: number of bills = 1.1 x 10^9 / 100 = 1.1 x 10^7 bills. A bill is about 0.1 mm = 10^-4 m thick. Height = 1.1 x 10^7 x 10^-4 = 1.1 x 10^3 m, so about 1.1 km.

Both factors are solid (the net worth is given, bill thickness is a standard anchor), so quote tight: low 0.7, high 2. That range is a factor of about 3 wide - just under 0.5 orders of magnitude - and contains 1.1, scoring the full 4 points.

Common mistakes

• Quoting a single-number-in-disguise range (low 40,000, high 42,000) on a question you have not decomposed - a tight miss scores exactly 0.

• Quoting the cowardly 1-to-a-trillion hedge - it always contains the truth and always scores just 1 point.

• Centering the range symmetrically in linear space around a large estimate, which makes it lopsided on the log scale where width is measured.

• Spending the 60 seconds refining the leading digit instead of checking the exponent - one wrong power of ten costs more than any mantissa precision recovers.

• Forgetting unit conversions in the factor chain - days to years and millimeters to kilometers are where clean decompositions silently lose three orders of magnitude.

Why interviews test this

Estimation questions are a staple of quant and consulting interviews because they expose how a candidate reasons with incomplete information: decomposition, sensible anchors, and honest uncertainty. Interviewers generally care about the structure of the estimate far more than the final number.

The range format matches how trading firms actually probe this: an interviewer will often ask for a market - a bid and an offer - on an uncertain quantity, then tighten it. Quoting wide when unsure and tight when confident, and knowing which is which, is the skill being tested.