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Fermi estimates

Heartbeats in a day, litres in a cube, seconds for sunlight to reach us: estimate to the nearest power of ten by breaking it into factors.

  • Number sense
  • Untimed
  • Endless levels

How it plays

A quantity no one knows offhand, and you pick it to the nearest power of ten — within a factor of about 3. Break it into things you can guess: beats a minute × minutes an hour × hours a day. It starts with two or three factors; more are chained as you climb, and then the numbers the question gives grow without end.

Every factor comes from a table that says where its number comes from: definitions (60 s in a minute, the speed of light) are exact; typical values (a resting heart rate, a population) are approximate, with the range they span. A question is asked only when every figure in those ranges gives the same answer, and only when the true value is not near the line between two powers of ten.

The wrong options are the neighbouring powers of ten. The working multiplies the factors out and names the source of each estimate.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This trains breaking a quantity into factors and keeping track of powers of ten. Its figures are approximate; it claims nothing wider.

Learn

How to get better at fermi estimates: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Break the question into a chain of things you can guess or know: beats a minute × minutes an hour × hours a day. Say whether each one multiplies or divides.
  2. Write every factor as a number between 1 and 10 times a power of ten: 70 is 7 × 10¹, 1,440 is 1.44 × 10³, 8.2 billion is 8.2 × 10⁹.
  3. Add the powers of ten (subtract them for a division), and multiply the front numbers roughly, pairing them into tens where you can.
  4. Round the result on a log scale: if the front number is 3.16 or more, go up to the next power of ten; below 3.16, stay. 8.64 × 10⁴ is nearest to 10⁵.

Techniques and tricks

  • Add the exponents, keep the front numbers

    Multiplying numbers adds their powers of ten. Keep a running exponent in one hand and a running front number in the other, and fold each factor into both.

    e.g. Heartbeats in a day: 7 × 10¹ × 6 × 10¹ × 2.4 × 10¹ — fronts 7 × 6 × 2.4 ≈ 100 = 10², powers 10³, so about 10⁵ (it is 100,800)

  • The √10 line

    Halfway between two powers of ten, on the scale that matters here, is √10 ≈ 3.16, not 5. A front number from 3.16 up rounds to the next power.

    e.g. Sunlight takes 499 s = 4.99 × 10² s. 4.99 is above 3.16, so the answer is 10³, not 10².

  • Pair and cancel the front numbers

    Look for pairs that make about 10 — 2 × 5, 4 × 2.5, 3 × 3 — and for a top and a bottom that nearly match. Each pair becomes one more power of ten, or vanishes.

    e.g. Food in kWh a year: 2 × 10³ kcal × 4.184 × 10³ J × 3.65 × 10² days ÷ 3.6 × 10⁶ J. The 3.65 and 3.6 cancel; 2 × 4.184 ≈ 8.4; powers 3 + 3 + 2 − 6 = 2. So 8.4 × 10², nearest 10³.

  • Carry a few anchors

    Some products come up again and again; know their size by heart.

    e.g. A day ≈ 8.6 × 10⁴ s; a week ≈ 6 × 10⁵ s; a year ≈ 8.8 × 10³ h, 5.3 × 10⁵ min and 3.2 × 10⁷ s; light ≈ 3 × 10⁸ m/s; Earth to Sun ≈ 1.5 × 10¹¹ m

  • Divide by subtracting powers

    For “how long does it take”, divide the distance by the speed: subtract the powers of ten, divide the front numbers.

    e.g. 1.5 × 10¹¹ m ÷ 3 × 10⁸ m/s = 0.5 × 10³ = 5 × 10² s, which rounds to 10³

  • Cube the side, then the litres

    A cube’s volume is side × side × side in cubic metres; each cubic metre holds 1,000 litres, and a litre of water is about a kilogram.

    e.g. A 4 m cube: 4³ = 64 m³ = 64,000 L = 6.4 × 10⁴, so 10⁵ bottles

  • Any fair figure will do

    The game only asks when every figure in the stated range gives the same power of ten. Pick a round, defensible value — 70 beats a minute, 2,000 kcal — and move on; do not agonise over it.

As the levels climb

  • It starts with two factors. A third, a fourth and then a fifth are chained on as you climb — people × beats × minutes × hours × days. Write the chain down in order before you multiply, so no 60 or 24 goes missing.
  • Then the numbers the question states grow, a power of ten at a time and without end: years, kilometres and people get more zeros, and the cubes get bigger sides. Turn each stated number into a power of ten first and only then multiply.
  • The clock is 15 seconds plus 4 for each factor, and it closes to 55%. With five factors that leaves about 19 seconds: keep the running exponent in your head and multiply the front numbers only roughly.

Traps to avoid

  • Reading the power of ten off the written form. 8.64 × 10⁴ is not “about 10⁴”: 8.64 is above 3.16, so it is nearest 10⁵. The neighbouring powers are what the wrong options are made of.
  • Treating 60 or 24 as one power of ten. 60 is 6 × 10¹ and 24 is 2.4 × 10¹, so 60 × 24 is 1.44 × 10³, not 10² or 10⁴.
  • Forgetting the hidden 1,000: kilometres to metres, cubic metres to litres. Each is three powers of ten, and it is easy to drop when the question gives the unit in passing.
  • Multiplying by a speed when you should divide. A time is a distance divided by a speed: more speed means fewer seconds.
  • Losing zeros in “billion”. 8.2 billion is 8.2 × 10⁹, and 1.45 billion is 1.45 × 10⁹; count them in threes.

How to practise

  • Use “Answer by” set to “Typing the power of ten” once choosing is easy: typing n means you cannot lean on the shape of the options.
  • Raise “Options shown” to 6 or 8 so the neighbouring powers on both sides are all there, and your rounding has to be exact.
  • Read the working after every answer, misses especially. It multiplies the chain out and names the source of each figure, so you see which factor carried you off.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More number sense

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