Brainteasers & Logic

Lesson 4 of 4

Fermi Estimation and Mental Math

To estimate a number like the count of barbers in New York, split it into factors you can guess, give a range, and do the arithmetic in your head fast enough that the interviewer keeps listening.

Break the question into pieces

A Fermi problem asks for a number nobody has memorised, such as the number of barbers in New York City. You aren't expected to know the answer. What gets graded is a chain of factors, each small enough to guess, multiplied together out loud.

Start from demand. New York has about 8.5 million people. Say half of them (mostly men and boys) use a barber instead of a salon, and they go about every four weeks, so 12 times a year. That makes

8.5M×0.5×12≈51M haircuts per year.8.5\text{M} \times 0.5 \times 12 \approx 51\text{M haircuts per year}.

Now supply. A barber spends about 30 minutes per cut, and with gaps and slow hours manages about 12 a day. Over 250 working days that is 12×250=300012 \times 250 = 3000 cuts a year. Divide:

51,000,0003000=17,000.\frac{51{,}000{,}000}{3000} = 17{,}000.

Say "about 17,000, and I'd believe anything from 8,000 to 40,000." Every factor is written down, so if the interviewer thinks men get haircuts less often, you change one number and redo the division in five seconds.

The usual template is population, times the fraction that uses the thing, times how often, divided by how much one unit of supply handles. Piano tuners, gas stations and taxis all fit it. Pick round inputs on purpose: 8.5 million is fine, 8,336,817 wastes time and adds nothing.