Why I Love Fermi Problems

The power of estimation is amazing. The ability to construct close-to-accurate measurements with limited information is such a useful skill to have because it’s encountered so often in the real world.

And that’s why I love Fermi problems. They were named after Enrico Fermi, an early physicist who helped develop the atomic bomb. He was able to predict the TNT equivalent of an atomic bomb, with limited information and simple mental math, to an accurate measure.

The explosion took place at about 5:30 A.M. After a few seconds the rising flames lost their brightness and appeared as a huge pillar of smoke with an expanded head like a gigantic mushroom that rose rapidly beyond the clouds probably to a height of 30,000 feet. After reaching its full height, the smoke stayed stationary for a while before the wind started dissipating it. About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during, and after the passage of the blast wave…The shift was about 2 1/2 meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T.

— Enrico Fermi, Trinity Test

The calculated value turned out to be the equivalent of 18.6 kilotons of TNT, extremely close to his estimated value. These calculations are meant to be less accurate than mathematical equations and precise measurements, but much more accurate than a Scientific wild-ass guess. Think of it like a back-of-the-napkin estimation on roids. Randall Munroe, creator of xkcd, has given a great talk about his estimation process in really obscure settings (thanks Freia Lisa Lobo for the reminder!):

Problems like these require thinking on a large scale between different systems. Learning to reason in this manner emphasizes general understanding over getting a “correct” answer. This problem solving method allows you to easily sanity check complex calculations.

I value the process of persevering through the various conversions and estimations much more than the final result. Getting an exact answer to the decimal point doesn’t actually matter that much in the long run. The most important part is the magnitude of your answer — you answer questions in factors of 10, and the closer you get to the actual exponent, the more points you receive. You receive 5 points for the correct exponent, 3 points for an exponent ± 1, and 1 point for an exponent ± 2.

Last year, before school got hectic, my roommates used to stay up and solve 5 problems a night, and we progressively got better. I really wish I could take a class in school on this. Interestingly enough, MIT used to run a class called the Art of approximation in science and engineering, which is available through their OpenCourseWare (Already on my to-do list!). It teaches many great problem-solving techniques surrounding approximation (divide and conquer, dimensional analysis, continuity, etc.) with real life-examples.

Some problems are easy (“As a factor, how much more is the price of an average new car than the price of a Nike men’s Dual Fusion sneaker?”), and some are just plain hard (“Determine the distance from Pluto to the Sun divided by the width of a single HIV virus.”). Determining which problems are hard is entirely dependent on your background. I’ve solved problems as small as one step and some as complicated as twelve.

Here are some tips for solving a Fermi problem:

1. Always keep units in mind, and try to keep the units familiar. Most of the time, the final answer doesn’t require a unit, but rather a ratio. If you keep the units familiar, you can easily weed out any answers that don’t seem rational.
2. Round! The scoring is based on scientific notation and powers (big picture), rather than small constants. Make your calculations as simple as possible, so you don’t confuse yourself.
3. Look at the subject material and see if it really makes sense in its respective context. Do you really think Americans eat 4 trillion pizzas every year?
4. Learn the Metric system and physical unit conversion tables. If you’ve recently taken a science class, you’re in luck. The conversions you memorized can finally come in handy!
5. Be obsessed with learning about random things. I’ve spent many late nights reading about the randomest of topics on Wikipedia. It turns out that this is super useful for Fermi Problems. It’s not really about memorizing these facts (although it helps), but more about recognizing scale and estimation in different fields.

And with these tips — A relatively straightforward problem with steps to achieve the solution:

From Fermi Questions

Steps:

1. Let’s look at this problem backwards. We know the final result, and we need to find the original value.
2. The weight of a standard car is about 2 tons (1 ton = 2000 pounds). For a school bus, I’d imagine 2–3 cars would equal the weight of one school bus. I put the estimate right in the middle — 5 tons = 10,000 pounds.
3. Now to convert to a familiar unit. Using conversion factors, 10,000 pounds = 160,000 ounces ≈ 1.5 * 10⁵ (1 lb = 16 oz, then we simplify). Now we need to see how many dollar bills fit in an ounce. I’d imagine 1 dollar bill is about one gram. Given there are about 3.5 grams in an 1/8th of an ounce (don’t ask how I know that…), there are 28 grams in an ounce. This means 28 dollar bills per ounce!
4. Given we have 1.5 * 10⁵ ounces, we multiply by 30 (≈28) to get the number of grams (and dollar bills). 1.5 * 10⁵ * 30≈ 4480000 = 10⁶.
5. The answer seemed a bit low (dollar bills might even be lighter than a gram), so I bumped up the magnitude by one fudge factor => 10⁷.

Final answer

Pretty cool huh? The problems do tend to get harder, but following a similar procedure will set you in the right direction.

The time is now. Go solve some problems!

I love chatting about new ideas and experiences! Feel free to message me on twitter @niraj!