Fermi Problems: How To Deal With Huge Numbers

Fermi problems are a special category of estimation problems that demand quick back-of-the-envelope calculations. What makes them special is the Fermi estimate — the typical solution approach to Fermi problems. It enables us to tackle uncertainty as well as huge numbers (really enormous numbers) quite effortlessly.

You would think that we do not have to deal with huge numbers often. But Fermi problems are more common than we intuitively expect them to be. And knowing how to tackle them not only saves us a lot of time, but also enables us to operate very cleverly and efficiently.

In this essay, I will be starting with my personal experience with tackling Fermi problems; this will serve as a motivation to learn more about this topic.

Following this, I will touch upon the historical significance of Fermi problems. Then, I will get into the nitty-gritty details of how to solve Fermi problems. But not to worry. This topic is about simple back-of-the-envelope calculations after all.

Finally, I will demonstrate the Fermi estimation method using a practical example. By the end of this essay, you would be able to recognise Fermi problems in the wild more easily and start tackling them as well. Without any further ado, let us begin.

This essay is supported by Generatebg

A Walk Down the Memory Lane

Until a few years back, I used to develop race cars. Motorsport is a fiercely competitive field where teams try to gain performance advantages via every single vehicle component. Teams typically have gains in milliseconds mapped to millions of dollars in revenue gain. That is how fierce the field is.

More often than not, innovative engineers lay new ideas bare on the table. Teams have the capacity to design and manufacture entirely new car components (even complex carbon fibre components; my specialisation back then) in a matter of hours. But all this costs a lot of money.

One of my responsibilities was to vet the ideas using quick back-of-the-envelope calculations. This is because even computer-aided design and virtual simulation hours are too expensive to test the sheer number of ideas engineers often come up with.

For example, let us say that a design geometry features a sharp edge in a particular location. Based on the geometry and load assumptions, I could ‘guesstimate’ if the corresponding stress tensor component would stay within the elastic range or slip into the plastic range in that region.

For the uninitiated, plasticity is bad in most contexts. Once stresses exceed the plastic limit, the material deforms permanently. In any case, I share this story to give you a practical picture of how Fermi problems pop up.

More often than not, we just don’t recognise them. Don’t get me wrong. We could proceed with any generic back-of-the-envelope estimation method, but it is just that the Fermi estimation method is particularly geared for efficiency when tackling Fermi problems.

Before we proceed further with the method, let us take a look at who invented it.

The Origin of Fermi Problems

The term ‘Fermi’ in Fermi problems takes after Enrico Fermi, a world-renowned Italian-American physicist who helped create the world’s first nuclear reactor.

His work in the field of nuclear physics was so crucial that he is often praised as the “architect of the nuclear age”. He eventually won the Nobel Prize for his pioneering work in nuclear fission.

Not only that, but from all of the people he mentored or influenced, eight people went on to win Nobel Prizes as well. In short, he was really good in his area of expertise.

The first ever atomic bomb test was codenamed Trinity and Fermi was on site. As the explosive went off, Fermi dropped some pieces of paper and watched them blow away from the force of the blast wave.

Based on how far these pieces of paper travelled, Fermi estimated that the blast was roughly equivalent to 10 kilotons of TNT. Later on, the precise number turned out to be 21 kilotons.

Although the estimate was off, the error was within one order of magnitude of the precise number (10¹ Kilotons of TNT) — something that could be useful to make quick decisions.

Fermi became famous for his useful back-of-the-envelope estimations and his technique later became famously known as the Fermi estimation method.

But note that his technique only works for problems that have a large amount of uncertainty and involve huge numbers.

Look at it this way. Fermi problems are problems for which the answer estimate (Fermi estimate) is acceptable if the error is within one order of magnitude of the precise answer.

There is a trick to this that lets you use this approach to solve any estimation problem — more on this later in the essay. For now, let us take a look at a few examples.

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Examples of Fermi Problems

Here are some interesting examples of Fermi problems/questions that I managed to scavenge from various sources:

1. What is the volume of the moon in cubic metres?

2. How many kilograms of onion does the city of Sydney, Australia consume in a year?

3. Between two blinks of your eye, how far does light travel in a vacuum?

4. How many blades of grass does a football field contain?

5. What is the mass in grams of an average mature tree?

At first glance, you would think that we cannot answer most of these questions. But here’s the thing. Fermi estimation allows us to make remarkably good educated guesses of the answer’s order of magnitude.

If the guess indeed lands us in the right ballpark, we know that we probably made the right assumptions.

But if not, then, we need to reconsider our assumptions. Either way, the Fermi estimation method allows us to approach the correct answer at a very fast rate.

Let us take a look at how to actually solve Fermi problems.

How to Tackle Fermi Problems and Deal With Huge Numbers

Once you have determined that you are dealing with a Fermi problem, the first step is to break the problem down into two parts:

1. Gather all the information that you know about the problem as variables.

2. Come up with estimates for information that you don’t know about the problem also as variables.

Each of these variables interacts with another variable typically via multiplication or division.

To proceed, you perform one operation at a time between two variables and consider this as a step. Then, you take the intermittent solution and perform the next operation with the next variable as the next step and so on.

The more steps you have, the more variables you are dealing with. The reason why this method works boils down to two factors:

1. The information about the variables that you know are usually close to correct.

2. Your estimates for variables that you don’t know about, if unbiased, tend to err on both sides of the accurate numbers. So, the errors tend to cancel each other out.

All of this might sound too abstract now, but as soon as we cover a practical example, you would realise that the Fermi estimation method is remarkably simple.

Taking about examples, why don’t we tackle an example question from our list?

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Solution to an Example Fermi Problem

Let us try and estimate a solution for the first Fermi problem from the example list:

1. What is the volume of the moon in cubic metres?

Here are two things that I think are relevant to solve this problem:

1. I know that I can safely assume the moon to be a sphere. I know that its precise volume would not deviate more than one order of magnitude based on this assumption. I also know that I can compute a sphere’s volume using the formula: ((4/3)*π*r³), where r is the radius of the sphere.

2. I don’t know the radius of the moon in metres.

So, my task now is to come up with an estimate of the moon’s radius. Full disclosure here: I am not looking up any resources and am coming up with estimates purely based off my own knowledge; so cut me some slack as you read along.

I know that the moon is smaller than the Earth. Let me assume that the moon is one-third the size of the Earth.

But then, I don’t know how big the Earth is. What I do know is a continent on Earth is a few thousand kilometres long. Let me assume that the Earth’s diameter is the length of 5 continents side-by-side.

If one continent is 5000 kilometres long, the Earth’s diameter would be 25000 kilometres long. Consequently, its radius would be 12500 kilometres long. Converting this to metres, the volume of the Earth would be is as follows:

Estimated volume of the Earth = (4/3)*π * (12500 * 10³)³ cubic metres

The moon’s volume (based on my assumption) would be around one-third of this value:

My Fermi estimate of the moon’s volume = (4/9)*π*(12500 * 10³)³ cubic metres = 2.727 * 10²¹ cubic metres.

Now comes the moment of truth! I searched for the size of the moon, and this is the result that I got:

So, the actual answer turns out to be roughly equal to 1.103*10¹⁹ cubic metres.

As you can clearly see, I messed up on my attempt the keep the error within one order of magnitude (my estimate was off by two orders of magnitude).

But here’s the thing. I came pretty darn close to the actual answer than I really thought I would come. I was correct in my estimation about the size of the moon in relation to that of the Earth.

But my estimation of Earth’s diameter was off by a factor of 2. Since the radius is cubed in the sphere’s volume calculation, my error got amplified in the end. With more experience, I or you would be able to make better and better Fermi estimates.

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Final Comments

To round up, here is the step-by-step algorithm to come up with Fermi estimates for Fermi problems:

1. Break the problem down into two sets of variables: one with known numbers and one with estimates of unknown numbers.

2. Compute the Fermi estimate by performing one step (between two variables) at a time.

3. The variables are usually related by their units (like kilometre/hour), so pay attention to units and their relationships.

While we are on the topic of units, here is the ace in the sleeve when it comes to Fermi problems. You can transform ANY estimation problem into a Fermi problem by manipulating units.

The deciding conditions, then, are your final precision requirement (for the estimate) and your lack of knowledge (estimates). For instance, 1 kilometre is equal to 10¹² nanometres.

Another tip for getting good at Fermi estimates is to memorise ‘Landmark Numbers’. Examples of these could be the diameter of the Earth (I would have profited from this knowledge), the height of the Eiffel tower, the speed of light, etc.

When you have access to such landmark numbers, you can manoeuvre around them to get to more and more accurate/useful Fermi estimates.

Now, I urge you to pick up an example Fermi problem from the list and give it a try on your own. For your convenience, here is the example list of Fermi problems once again:

1. What is the volume of the moon in cubic metres?

2. How many kilograms of onion does the city of Sydney, Australia consume in a year?

3. Between two blinks of your eye, how far does light travel in a vacuum?

4. How many blades of grass does a football field contain?

5. What is the mass in grams of an average mature tree?

Post your Fermi estimate in the comments section and also include the precise answer (if you managed to compute or find it) so that readers can see how close you got. All the best!

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Further reading that might interest you: 

• How To Calculate Day Of The Week For Any Date?
• How To Quickly Calculate Percentages In The Head?
• Variance: The Reason Why Rich Get Richer And Poor Get Poorer

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