Fermi Problems: How You Think About A Problem Is More Important Than The Solution

The US food stamps program, also known as SNAP, helps many low-income households afford nutritious food. In December 2016, Fox ran a story about food stamp fraud. “According to the USDA, $70M of taxpayer money was wasted in 2016 due to food stamp fraud.” $70M is a lot of money! Sounds like a disastrously run program, right?

Before we answer that, let’s take another classic question: “How many piano tuners are there in Chicago?” You first instinct would be to check on Google, but back when the Italian American physicist Enrico Fermi—who created the first nuclear reactor beneath the University of Chicago football field—concocted this little brainteaser, it was decades before the invention of the Internet. The challenge was to get to an estimate just by reasoning, instead of “looking it up” somewhere.

These are known as “Fermi Problems”. It was characteristic of Enrico Fermi’s to constantly make back-of-the-envelope estimates to help him approach problems. During the first atomic bomb test he dropped pieces of paper before, during, and after the passage of the blast wave, and used the distance the paper travelled to estimate the explosion strength.

At first glance, this seems very difficult to answer. Rather than guessing a random answer, we start to Fermi-ize it, i.e., we start to break down the problem into smaller chunks, and put them into buckets of knowable and unknowable. By working at a problem this way, we expose what we don’t know and flush ignorance into the open.

If a question is asked in a school exam, you know that the answer exists. You also know where to look it up. But in a wicked world, you are never sure if you have the right answers. Often there’s no clear indication that you are approaching the correct solution. Therefore how you think about solving problems becomes more important than the solutions themselves. In wicked domains, prior knowledge is less important than way of thinking.

So what would we need to know to calculate the number of piano tuners in Chicago, or Mumbai? The number of piano tuners depends on how much piano-tuning work there is, and how much work it takes to employ one piano tuner. So we could nail this question if we know the following:

  1. The number of pianos in Mumbai
  2. How often pianos are tuned every year
  3. How long it takes to tune a piano
  4. How many hours a year an average piano tuner works

With the first three facts, we can figure out the total amount of piano-tuning work in Mumbai. Then we can divide it by the last and get a pretty good idea of how many piano tuners there are.

The funny thing is that we’ll still have to rely on some guesswork. Breaking down a problem into granular pieces doesn’t eliminate guesswork, but instead of guessing one big random number, we can intelligently guess smaller chunks that can be investigated individually. This way the final result tends to be a more accurate estimate.

Let’s look at each of the items separately now. How many pianos are there in Mumbai? We have no idea. But we can break this further down by asking the golden question: what would we need to know in order to answer it? Perhaps, the number of people living in Mumbai, which is 18.4M as per Wikipedia. Let’s make it 20M for the sake of simplicity.

Now, what percentage of people own a piano? Lots of poor families live in Mumbai. Dharavi, one of Asia’s largest slums houses around 1M people. Pianos are too expensive even for affluent families—and most who can afford one don’t really need one. So we can estimate it as 1 in 10,000. That’s a wild guess but it’s the best we can do right now.

Apart from homes, pianos would be present in music schools, bands, concert halls, and bars. Again, we don’t know how many. But we can assume that music schools would own many pianos. We’ll again make a black-box guess and say that it’s enough to double the per person number of pianos to roughly 2 in 10,000. With those guesses, we can do some simple math and conclude that there are 4,000 pianos in Mumbai.

This brings us to the second question: how often are pianos tuned? I know for a fact that pianos are used to tune other instruments, such as a guitar. They don’t go out of tune for a long time. So let’s say, it needs to be tuned once a year.

Next, how long does it take to tune a piano? It’s a big instrument with lots of keys. Would definitely take some time. Let’s go with 4 hours.

How many hours a year does the average piano tuner work? Again, we can break it down. An average workweek for a workman is ~80 hours in India. Let’s say a piano tuner takes a two week vacation every year to visit their native place. So we can multiply 80 hours by 50 weeks to come up with 4,000 hours a year.

The numbers suggest that there’s a good chance the piano tuner tunes other instruments as well. Since there are only 4,000 pianos in Mumbai, there isn’t enough work for piano tuning to be a primary job. Let’s say a workman spends 10% of their time in tuning pianos. So it comes down to 400 hours a year.

Since the pianos cannot be brought to them, they would have to spend some of that time commuting between pianos, so we reduce the total by that much. How much time do they spend commuting? Let’s say 20% of their work hours. So we conclude that the average piano tuner works 320 hours a year.

Now we’ll assemble all the guesses to make a final calculation. If 4,000 pianos need tuning once a year, and it takes 4 hours to tune one piano, that’s 16,000 total piano-tuning hours. Divide that by the annual number of hours worked by one piano tuner, and you get 50 piano tuners in Mumbai. Voila!

How accurate am I? I dunno! The correct answer might be way off. But this is a fair back-of-the-envelope estimate. Fermi-izing dares us to be wrong, and overcome our deep-rooted fear of looking dumb. And accuracy isn’t the point here. More than accuracy, we need good reasoning and strong problem solving skills to do fair estimates. That’s what matters.

Problems like these require thinking on a large scale between different systems. Learning to reason in this manner emphasises general understanding over getting a “correct” answer.

This brings us back to the original question—the $70M food stamp fraud. We may not know how expansive the food stamp program is, but we do know that this is given to only those who are in need. Let’s assume that about 10% of Americans are on food stamps. There are roughly 300M people in the country, so around 30M people are on food stamps.

We aren’t sure of the annual benefits of the program, so let’s go with $1,000. Therefore, the US invests approximately $30B annually in its food stamp program. That means that the fraction lost to fraud is around 0.2 percent. Even if there is substantive inefficiency in the food stamp program, it certainly isn’t a disaster. Fox got it wrong.

The power of estimation is amazing. The ability to construct close-to-accurate measurements with limited information is a useful skill to have. A little training in broad thinking strategies like Fermi-izing can go a long way, and can be applied across domains.

The good news is that this type of thinking can be taught and practised, and can be nurtured in kids from a very young age. It is an increasingly important skill in a technology-driven world with untold unknowns.

You don’t need right answers. You need opportunities to exercise your ingenuity, find new connections, and to allow experimentation to become a normal part of your thinking. This will upgrade your problem solving skills.

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