1. Fermi Questions
Enrico Fermi

2. Who is This Fermi and Why Does He Ask Questions?
Fermi questions receive their name from Enrico Fermi ( ). He was an Italian physicist known for his participation in the Los Alamos atomic bomb project (code named the “Manhattan Project”) during World War II. Fermi is best known for his contributions to nuclear physics and the development of quantum theory. In addition to his contribution to theory, he is also noted as an experimentalist.

3. Who is This Fermi and Why Does He Ask Questions?
Fermi was awarded the Nobel Prize for his physics work on the nuclear process. Shortly after receiving the Nobel Prize, Fermi was forced to flee Italy. He settled in the United States, first at Columbia University and then later at the University of Chicago.

4. Who is This Fermi and Why Does He Ask Questions?
Throughout his work, Fermi was legendary for being able to figure out things in his head, using information that initially seems too meager for a quantitative result. He used a process of "zeroing in" on problems by saying that the value in question was certainly larger than one number and less than some other amount. He would proceed through a problem in that fashion and, in the end, have a quantified answer within identified limits.

5. What is a Fermi Question?
Fermi questions emphasize estimation, numerical reasoning, communicating in mathematics, and questioning skills. So often, we believe that "word problems" have one exact answer and that the answer is derived in a unique manner. Fermi questions encourage multiple approaches, emphasize process rather than "the answer", and promote non-traditional problem solving strategies.

6. Fermi Questions
About 50 years ago, Enrico Fermi asked his physics students at the University of Chicago
"How many piano tuners are there in Chicago?"
his idea was to encourage the students to think about the process of estimating the answer without any specialised knowledge
a Fermi question requires estimation of physical quantities to arrive at an answer.
Throughout his work, Fermi was legendary for being able to figure out things in his head, using information that initially seems too meager for a quantitative result
he used a process of "zeroing in" on problems by saying that the value in question was certainly larger than one number and less than some other amount. He would proceed through a problem in that fashion and, in the end, have a quantified answer within identified limits.
In a Fermi question, the goal is to get an answer to an order of magnitude (typically a power of ten) by making reasonable assumptions about the situation, not necessarily relying upon definite knowledge for an "exact" answer.
R. Hornsey

7. A Fermi question is posed with limited information given.
how many water balloons would it take to fill the school gymnasium?
how many piano tuners are there in New York City?
what is the mass in kilograms of the student body in your school?
A Fermi question requires that students ask many more questions.
how big is a water balloon?
what are the approximate dimensitons dimensions of the gym?
what measurment must be estimated using the dimensions of the gym?
... and the list goes on.
A Fermi question demands communication.
A Fermi question utilizes estimation.
A Fermi question emphasizes process rather than "the" answer.
R. Hornsey

8. How many piano tuners are there in New York City?

9. How many piano tuners in NYC?
Approximately how many people are in New York City?
10,000,000
Does every individual own a piano? No
Would it be reasonable to assert that "individuals don't tend to own pianos; families do?
Yes.
About how many families are there in a city of 10 million people?
Perhaps there are 2,000,000 families in NYC.
Does every family own a piano?
No.
Perhaps one out of every five does.
That would mean there are about 400,000 pianos in NYC.
R. Hornsey

10. How many piano tuners are needed for 400,000 pianos?
Some people never get around to tuning their piano; some people tune their piano every month. If we assume that "on the average" every piano gets tuned once a year, then there are 400,000 "piano tunings" every year.
How many piano tunings can one piano tuner do?
Let's assume that the average piano tuner can tune four pianos a day. Also assume that there are 200 working days per year. That means that every tuner can tune about 800 pianos per year.
How many piano tuners are needed in NYC?
The number of tuners is approximately 400,000/800 or 500 piano tuners.
R. Hornsey

11. General principles
When you use back-of-the-envelope calculations, be sure to recall Einstein's famous advice.
“everything should be made as simple as possible, but no simpler”
Don’t worry about factors of 2, π, etc.
round to the nearest 0, 5, 10
corollary: don’t make numbers more precise than is necessary
Guess numbers you don’t know
but try to make your guesses good ones and within the bounds of common sense
common sense requires some education – the accuracy of common sense increases with experience
Adjust geometry etc. to suit you
assume a human is spherical if it helps
Extrapolate from what you do know
e.g. use ratios
assume unknown value is same as a similar known quantity
R. Hornsey

12. General principles ctd …
Use the principle of conservation
what goes in must either come out or stay inside
things are not generally destroyed, so work out where they have gone
Ensure formulas are dimensionally correct
i.e. an expression to tell you the length of something must have overall dimensions of metres
Apply a ‘plausibility’ filter
if an answer seems unbelievable, it probably is
you can usually set a range of possible/reasonable values for a quantity that will indicate a major mistake (e.g. speed cannot be faster than speed of light!)
R. Hornsey

13. Fermi Problem Example “What is the length of the equator?”
Fermi problems are solved by assembling simple facts that combine to give the answer:
The distance from Los Angeles to New York is about 3000 miles.
These cities are three time zones apart.
So each time zone is about 1000 miles wide.
There are 24 time zones around the world.
So the length of the equator must be about 24,000 miles
The exact answer is 24,901 miles.