Whenever I got stuck with math homework in my childhood, I looked for my mom. I often found her on the living room couch relaxing after work catching up on the news while the local Cantonese news station was booming on TV and The Economist was open on her lap.
“I don’t know how to do that,” I complained and sat on the carpet at her feet.
“Read me the question.”
I recited, “It takes Sarah six hours to paint a fence and it takes John 12 hours to paint the same fence. How long does it take to paint a fence twice as long if they work together? “
She wouldn’t even look at the side.
“How many hours do you think it will take them?”
“I don’t know, otherwise I wouldn’t ask you!”
“Single digit? Ten hours? Hundreds of hours?”
My mother just finished her Ph.D. Studied physics when she was unexpectedly diverted into running the family business, but never lost her love of scientific methods. One of her favorite books is “Powers of Ten”, a flip book that starts with a picture of the universe with speckled galaxies and then zooms in on the order of magnitude of our solar system, then the blue marble of our earth, until we arrive at a couple lying on a picnic blanket . The book dives on, to the ants in the grass, then smaller and smaller into the invisible world of atoms and subatomic particles. My mother’s brain worked like this book, moving up and down the ladder of powers of ten, always looking for a broad angle. She urged me to do the same, to pull my nose out of the formula I just copied from my textbook and to judge from a distance: “Does that make sense, Caroline? Look at your answer. How could the painters spend more hours painting the fence together than if they did it alone? “
If we put all the pasta we eat in a year together from end to end, what percentage of the circumference of the earth would that take up?
There is a name for the estimation problems my mother liked to pose: Fermi problems, named after the Italian physicist Enrico Fermi, who had an uncanny knack for making pinpoint approximations with little available data. One of the most famous examples is: How many piano tuners are there in Chicago?
Without looking, I could guess that Chicago’s population is anywhere from one to five million people. With 2.5 million at the beginning and assuming the average household has four people, we would have 625,000 households. For example, suppose one in five households has a piano; that brings us to around 125,000 pianos. Let’s say they are all tuned once a year. The question now arises as to how many pianos a tuner can maintain each year. I guess you can tune three pianos a day. Multiply it by five days a week for 50 weeks a year, and that makes about 750 pianos per tuner per year. Divide the number of pianos (125,000) by 750 and we get roughly 170 tuners across Chicago. The goal here is not to know the exact number, but to be able to estimate the correct magnitude only with common sense.
When I was young, I got annoyed by my mom’s questions because I was more busy doing my homework so I could play. But now that I have no real problems, I’ve found that I return to these problems to entertain myself, the more absurd the better.
At dinner with my husband I asked: If we put all the pasta we eat in a year together from end to end, what percentage of the circumference of the earth would that take up? Lying in bed together: how many penguins do you think would fit in our bedroom? When our cat jumps on my sternum and wakes me up in front of my alarm clock: How big is the volume of Polo in cubic inches in your opinion?
Only after we have made our best guesses do we turn to our phones for the internet’s answer. Actually, Google is usually not that helpful. “How many square meters does a penguin take up?” Does not provide satisfactory answers. Reddit suggests, in typical winking fashion, that the best way to calculate a cat’s volume is to stun the poor animal, throw it into a tub of water, and then measure the amount of water displaced. However, with a lot of giggling, we always learn funny things. Now I know that the world’s smallest penguin is called the fairy penguin. Spaghetti was traditionally up to 20 inches long before shrinking in half to fit modern packaging.
I could argue for the practical benefits of having a Fermi mentality, but that’s not why I love these questions. When I think about Fermi problems, I am curious about the world and how things relate to each other. The pasta question impressed me how big the earth was. (We thought that even our most passionate pasta food would only get us 0.01 percent around the equator.) If you’re interested in trying one, remember: if you don’t limit yourself to the questions you knew about, that Google has an answer ready, what do you want to know? It’s about imagining the infinite cosmos, not organizing it, naming it or conquering it.
One evening in the early days of the pandemic, when I was walking my husband, tired of the empty streets, I stopped to stare at the night sky.
“Hey, how many boxes of floss would it take to get to Alpha Centauri?”
And so we meandered, discussed light years and rolls of waxy strings, not looking for answers, content to just wonder.
Caroline Chen is an investigative health and science reporter for ProPublica.