In his great book * Range*, David Epstein talks about how he had a chemistry professor who would throw in some oddball questions on an exam such as, “How many piano tuners are there in New York?”

Epstein notes that the ultimate lesson behind using such questions was to show that detailed prior knowledge was less important than a way of thinking. Epstein states that in his current job as a reporter, he regularly uses the thinking process he developed over the years as a result of working in such problems so that he can leverage what little he does know to start investigating what he doesn’t know.

Formally, the question about the piano tuners is referred to as a Fermi problem.

**Enrico Fermi** is the father of “solving math problems we will never know the exact answer to.” Such as how many leaves are on all the trees in Central Park?” They are great for getting people to think mathematically and use problem-solving skills.

Fermi questions often require people to make reasonable assumptions and estimates about the situation in order to come up with an approximate answer.

Perhaps the most famous Fermi problem is just a minor variation of the one Epstein highlights; the only difference is that instead of New York, it uses Chicago.

**From Wikipedia**, a typical solution to this problem involves multiplying a series of estimates that yield the correct answer if the estimates are correct. For example, we might make the following assumptions:

- There are approximately 9,000,000 people living in the Chicago metropolitan area.
- On average, there are two people in each household in Chicago.
- Roughly one household in twenty has a piano that is tuned regularly.
- Pianos that are tuned regularly are tuned on average about once per year.
- It takes a piano tuner about two hours to tune a piano, including travel time.
- Each piano tuner works eight hours in a day, five days in a week, and 50 weeks in a year.

From these assumptions, we can compute that the number of piano tunings in a single year in Chicago is

- (9,000,000 persons in Chicago) ÷ (2 persons/household) × (1 piano/20 households) × (1 piano tuning per piano per year) = 225,000 piano tunings per year in Chicago.

We can similarly calculate that the average piano tuner performs

- (50 weeks/year) × (5 days/week) × (8 hours/day) ÷ (2 hours to tune a piano) = 1000 piano tunings per year.

Dividing gives

- (225,000 piano tunings per year in Chicago) ÷ (1000 piano tunings per year per piano tuner) = 225 piano tuners in Chicago.

In 2009, the actual number of piano tuners in Chicago was about 290.

So not too bad an estimate, and it’s certainly closer, and more easily defended, than throwing out a random number like 1,000 or 20,000.

So if you’ve got nothing planned this weekend, here are a few fun Fermi problems for you to sit and ponder; no answers provided… (from **site one** and **site two**)

1) How many people could you fit into your house or apartment? How many soccer balls?

2) How old are you if you are a million seconds old? A million hours old? A million days old?

3) Could you fit $1,000,000 worth of $1 coins in your kitchen? What about a billion dollars worth of $1 coins?

4) If all the people in your country joined hands and stretched themselves out in a straight line, how long would it reach?

5) How long would it take to count to a million?

6) How many cups of water are there in a bathtub? What about in an Olympic pool?

7) How many grains of rice are in a 10kg bag?

8) How many 10-year old children are needed to have a mass the same as an elephant?

9) How long would it take to drive to the moon (if you could!)?

10) What is the total mass in kilograms of all the people who live in your city?

11) What is the weight of garbage thrown away by each family every year?

12) How many pizzas are eaten in the United States each year?

13) How far could you walk in one year?

14) How much water does your household use each week? Can you answer this without using a water bill?

15) How many blades of grass on a soccer field?

16) How many beats will your heart make in a lifetime?

17) How many bricks are there in one wall of the classroom? The whole school?

18) How many songs are bought on iTunes in a year?

19) How many times could you say the alphabet in 24 hours?

20) How many hairs on your head?

By the way, if you answer all of these questions this weekend, all I can say is, you may want to look for a hobby…

*image from **The Australian Association of Mathematics Teachers, Inc**

This was a very entertaining post. Also a bit sneaky, as you somehow tricked me into doing math on my weekend off!

nothing like a little math to get your weekend off to a good start!

Thought provoking post as usual, but as one of your students might say, “Uh, I didn’t know there would be any math on this test.” 🙂

B/t/w and off topic, you raised Enrico Fermi and it led me to the Fermi Paradox about intelligent extraterrestrial life and our failure to find any. That proposition ties in with a post I’ve been contemplating. Thanks.

Thanks!

I also read about Fermi’s Paradox, so I look forward to reading what you have to say about it.

math and I have never been close, ti was more like an arranged situation. I am happy just reading these and will wait to see the answers. meanwhile, I love the random questions thrown in and the theory behind it.

I’m sure you do estimating problems with your 4-year olds, so these are really no different. And as for the answers, I don’t have any! 🙂

Absolutely

Interesting post and great questions, Jim. Sadly, I have other things demanding my time this weekend. 😂

more demanding than figuring out how many times you could say the alphabet in 24 hours?! 🙂

😂🤣😉

Omg this is such an interesting way to learn new things!

it’s a nice way to think about things…

Yeah and I’ve noticed it helps remember things better too.