Posted on Fri 08 November 2019

how to solve a Fermi problem

A lot of annoying irrelevant interview questions are Fermi questions. (Enrico Fermi was a famous nuclear physicist.) A Fermi question is one where there’s some real answer that you could get by carrying out an arduous and possibly ridiculous procedure, but making a good guess and showing your work will be just as good and be much less effort. Interviewers sometimes like to pose questions of this type “to see how you think”, but their actual motivation is a combination of finding out if you are reasonably numerate – and perhaps for the feeling of superiority they can derive from your failure.

Examples of Fermi questions include “How many piano tuners are there in Chicago?” and the one I’ll tackle here: “How many tennis balls will fit inside a 747?”

You need to have some background knowledge about tennis balls and aircraft. If you don’t know that a 747 is a large aircraft, you can’t solve this problem. If you’ve never seen a tennis ball, you can’t solve this problem. Does that make you an unfit candidate for a job? If the job has to do with sports or aircraft, maybe. Otherwise, I don’t think so.

The solution method for all Fermi questions is the same:

  • make extremely reductionist assumptions about the problem

  • write down what you’re going to guess at

  • write down the procedure to turn your guessed numbers into an answer

  • guess the numbers, pretending that you have some confidence in them

  • do the math

  • explain why you’re probably wrong

You can do the first four things in any order, or even jumble them up as they occur to you.

A 747 is a large passenger aircraft. How large? Wikipedia will tell you, but interviewers generally don’t let you look that up. So we’ll document our guesses and say that a 747 has a cylindrical body which is about the length of a football field (300 feet). How wide is it? One common seating arrangement is three seats, aisle, four seats, aisle, three seats. Each seat is perhaps 18" across, and the aisle is probably 24" across, maybe even 30“. So that’s 18 x 10 + 2 x 30 = 240” wide, or 20 feet. That’s probably the widest part of the airplane, so we’ll call it the diameter of the cylinder.

The volume of a cylinder is the area of the circular cross-section times the length, so we have a radius of 10 feet (half of 20): pi * r ^2 * l: 3.1 * 10 * 10 * 300. That gives us a volume of 90000 cubic feet.

How big is a tennis ball? I don’t know. I remember that one fits in my hand and two are a little hard to hold and three is right out, so I’ll claim that a tennis ball is 3 inches across. Conveniently, that means that I can pack four across in a foot, and so 4 x 4 x 4 (64) will fit in a cubic foot. 90000 * 64 gives 360000 + 5400000 = 5760000 tennis balls.

All of these assumptions are wrong in various ways. A 747 isn’t a cylinder. It’s probably not exactly 24 feet wide or 300 feet long. It has lots of internal fittings that just can’t be moved or removed. I don’t actually know how big a tennis ball is in terms of measurements. It’s easier to look up the size of a tennis ball than it is to look up the precise volume of a 747, and there are several sizes and configurations of Boeing’s jumbo jet anyway.

The business case, though, has to be even further off: why do you need the tennis balls moved so quickly? How are you going to onload and offload them on the other side? If you’re thinking ahead, maybe you can just get standard shipping containers full of tennis balls, which can be moved by ship (really cheap) and train (cheap) and truck (pretty cheap) instead of airplane (quite expensive). The industry produces something like a million tennis balls a day, so this is only a week’s production or so: enough to get you a discount, not enough to justify building a new factory. Or is your demand goint to be continuous? Have you considered what you’ll do if the plane crashes? What are your alternate supply chains?

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