What's a Fermi Question

Fermi Questions is a Science Olympiad event that asks you to approximate a quantity that's hard or impossible to measure directly. For example:

• How many golf balls does it take to fill a Boeing 747?

To answer these questions, you need to learn some basic facts about the world — but you can't memorize everything so you also need to become comfortable with making reasonable approximations. Rounding For Fermi Questions, you're expected to give your answer to the nearest power of 10 — a whole number. For example, the value 2000 has a Fermi Answer of 3.

Note that when the first digit is 5 or larger, you are to round the exponent up. Though it matters only rarely, this rounding behavior is tricky! To see why, ask yourself: On a logarithmic scale, what's the halfway point between 10 and 100?

• wrong: (101 + 102) / 2 = 50 ≈ 101.7
• right: 10(1 + 2) / 2 = 101.5 ≈ 31.6

Nonetheless, a value of 50 has Fermi answer 2. Possibly this doesn't make much sense if you're just starting out, but don't say I didn't warn you :) Scoring If your Fermi Answer is spot on, you get 5 points. If it's off by 1 (in either direction), you get 3 points. If it's off by 2, you get 1 point. Otherwise you get nada. Keep in mind that since you have to express your answer as a Fermi Answer, "off by 1" actually means off by a factor of 10! Even if you are off by a factor of 100 you still get a point. (This is no excuse to be sloppy, but it does give me some hope when I face a particularly tough problem.) In my experience, there are 30 questions given during a 45–minute test. You'll be part of a two person team. Here are team dynamics that worked for me and my partner Narayan:

• Each person solves the problems independently.
• The average of your two answers becomes your team answer. Favor the stronger teammate in the case that the average isn't a whole number.
• Train to solve 1 question per minute. That way you'll have time to double check your work and briefly discuss a few of the trickiest problems.

Collaborating more closely is tempting, and may even work well for you. My opinion is that there unfortunately just isn't enough time for that, but I'd love to hear from you if you find differently. (Note that this is merely a rough sketch of the rules. For the official rules you need to buy, or wait for your team to buy, the manual.) E Notation As you solve more and more Fermi Questions, you'll probably get tired of writing 2 × 105, and if your handwriting is average like mine you'll start getting confused about what's in the base and what's in the exponent. So use E notation: 2 × 105 becomes 2E5, 5 × 10-3 becomes 5E-3, and so on. The scioly wiki page on Fermi Questions has an interesting bit about dropping the coefficients (e.g. the 2 and the 5 above) and just working with exponents. I've never tried it, but maybe I will. Dimensional Analysis The core tool for solving Fermi Questions is dimensional analysis, which is just a fancy way of saying "unit conversion." I first learned about this simple technique from my high school chemistry teacher Mr. Kennedy. Here's the basic idea: You know how if the same number multiplies the numerator and denominator, the number cancels? 507×350=37 Well, the same thing applies to units: 50 milessecond×5280 feetmile=264000 feetsecond Many Fermi Questions are solvable by guessing at some starting value, then using unit conversions to get to the desired answer. For example:

• If you walked a mile in another person's shoes for every other person currently living on earth, how many millimeters would you have walked?

∼7e9 people×1 mile person×∼5e3 feetmile×∼3e2 mm foot =1.05e16 mm⟶16 You can also solve more creative Fermi Questions via dimensional analysis:

• How many handcuffs would it take in order to create an interlocked chain from Paris to Beijing?

Let's start with the radius of the earth (6E6 meters) and use the approximation circumference = 6 × r, dimensional analysis, and estimated conversion factors to get to the answer: 6e6 Earth radius meters×6 circumference metersradius meter ×∼1 Paris to Beijing meter3 Earth circumference meters×∼5 handcuffsmeter =6e7 handcuffs⟶8 Finally, let's conclude with the most famous example of using dimensional analysis to solve a Fermi Question:

• How many civilizations in our galaxy might it be possible to communicate with?

Once you've tried to solve this problem by breaking it down into a starting quantity and a set of conversion factors, you can visit the Wikipedia page for the Drake equation and compare your approach with the one taken by Frank Drake. Making Assumptions Dimensional analysis is the core method of computation for Fermi Questions. But where do we get the data that goes into those computations? We need to make assumptions — we need to guess the input values if we don't know them (which is usually the case). In short

1. While solving a Fermi Question, do not look up quantities you don't know.
2. Guess them using one of the techniques described below.
3. After solving a Fermi Question, do look up any information you need to compute the "true" answer, and your skills will quickly improve.
4. This is why I provide a list of assumptions and sources for every problem.
5. Here's an example of how to carefully research the "true" answer.

In Detail Here are some ways you can get better at making assumptions:

1. Practice estimating length, weight, distance, speed, etc. You can do this throughout the day, and check your results:
2. How many calories are in a bag of chips?
3. How much does the bag weigh?
4. How many keys are on a computer keyboard?
5. How many words are on a web page?
6. How many words are in your essay?
7. How fast is a car … a bicycle … a softball pitch?
8. What's the population of your home town?
9. Compile a list of commonly appearing facts and memorize them. I still have the list I compiled; I'll update the values and share them here by 11/13.
10. What's the population of your state … the US … the Earth?
11. What's the distance from the Earth to the Sun … the Moon?
12. What's the volume of the Atlantic ocean?
13. Be creative! Use facts that you memorized, or have known for a long time, to figure out things you don't know. Example: Fermi computes the Earth's circumference.

Problem Types Exponentiation This section will help you solve problems of the form:

• What is pi to the power of 25?

There are a few tricks, but the basic approach is straightforward:

So in general, to solve exponentiation problems we need to compute

• exponent × log base

Memorizing the following logarithm values is a good place to start:

If you haven't memorized the value for the base, you need to approximate it based on what you have memorized. (Review the properties of logarithms & exponents!) For example:

Then, since 25 * 0.488 = 12.2, the Fermi Answer to "What is pi to the power of 25?" is 12. Compare this to the non-Fermi answer of approximately 2.68 e 12. Finally, a few notes of caution:

1. Since (exponent × log base) is a log value, do not round at 0.5! Round at (log 5 = log 10 - log 2), which is approximately (1 - 0.301 = 0.699). For example, to solve 225, compute _(25 * 0.301 = 7.525)_which rounds down to give Fermi Answer 7.
2. The more digits there are in the base and exponent, the more accuracy you need in your approximation of the log value. Using 3 digits of precision is good enough for most problems that will arise.

Electricity Every once in a while you'll run into problems like:

• How many electrons pass through a 100-Watt lightbulb in an hour?
• How many electrons pass through a MacBook Air during the time it takes the laptop to consume a million Joules?

Fortunately, although these problems might seem complicated at first, they only require a basic familiarity with a small set of units and equations.

These units can be quite a lot to take in, especially if you're a physics newbie like me. The good news is that once you know the units, the equations are easy to derive. For example:

Using these equations, we can solve the problems mentioned above:

• How many electrons pass through a 100-Watt lightbulb in an hour?
  • P = 100 W
  • F = 120 V — assuming it's a US electrical outlet
  • I = P ÷ V = (100 ÷ 120) A
  • t = 1 h × 60 minute/h × 60 s/minute = 3600 s
  • electrons = I × t × 6.24E18 = (100 ÷ 120) A × 3600 s × 6.24E18 = 1.8E22
  • Fermi Answer 22
• How many electrons pass through a MacBook Air during the time it takes the laptop to consume a million Joules?
  • Similar to the previous question, except we need to compute the time interval from the fact that the laptop consumes a million Joules.
  • t = E ÷ P = (1E6 ÷ 45) s
  • electrons = 45 W ÷ 120 V × (1E6 ÷ 45) s × 6.24E18 = 5.2E22
  • Fermi Answer 23

So as you can see, with just a little memorization and a bit of practice, the common electricity problems become much easier to do.