# Ballparking with Enrico Fermi

We’re all familiar with ‘ballparking’ – making rough estimates or vague guesses. Today I want to talk about a particular kind of ballparking, namely how we solve a type of mental puzzle called “Fermi questions”. They were employed by physicist Enrico Fermi during the Manhattan project as fun puzzles to keep up morale (at least that’s what he thought he was doing). Nowadays, they’re used by employers like Google and Microsoft in interviews to test candidates’ problem-solving skills. They point to some interesting things about the mind, but I also think they’re a great exercise in testing the limits of thinking. Here are some typical examples of Fermi questions.

How many playgrounds are in New York?

How long would it take to bike from Seattle to Los Angeles?

How many mice could you fit in an average sized SUV?

Obviously, no-one knows the answer to these off the cuff (unless your job requires you specifically to cram mice in SUVs, in which case – stop). The goal isn’t to get the right answer exactly. Instead, the goal is come up with a method that gives you an answer in the right ballpark (say, one order of magnitude).

But you don’t have a clue about these questions, right? Well, that’s not true. You know it wouldn’t take a year to bike from Seattle to LA, and you know it wouldn’t take 2 hours. You have some kind of sense of the right answer, so it’s just a matter of refining that feeling. You can do this by making estimates about other variables.

Take the first question. You might think something like this to yourself. “Well, there are eight million people in NYC, and the city seems to have a reasonable number of playgrounds (after all, I don’t typically see the playgrounds overflowing, but I don’t see them consistently empty). Let’s say ten percent of them are young enough to use a playground. That’s 800,000 young kids. Some of them will almost never use a playground or use one only rarely. Other kids will use them almost every day, and some will be in between. So let’s there’s five visits per year on average per child, which means 4 million playground visits per year. How many visits does the average playground get every day? Well, again it depends. In summer the playgrounds probably have hundreds of visits, in winter or in rain, way less. So let’s say 10 visits per day, which gives us 3650 visits per playground per year. So now we divide the total 4 million visits that we expect to happen by the 3650 visits per playground, which gives 1095 playgrounds.”

How close were we? Surprisingly close! The actual answer is just under a thousand. I have no idea if my other figures were correct, but it doesn’t matter if one of them was too high or too low, because I’m just as likely to overestimate as underestimate, so the highs and lows will probably cancel out (this is related to the wisdom of crowds effect).

Other than being fun puzzles, and maybe helping you get a job, Fermi problems raise a couple of interesting philosophical issues.