## Enrico Fermi Approach

Enrico Fermi developed a problem solving technique with which his name has been associated ever since whilst working on the production of the atomic bomb during the Second World War. The technique involves making educated guesses about quantities that otherwise would be difficult to calculate given limited available information. For example, how many molecules are there in the letter L written in ink on a piece of writing paper? The crucial estimate in the problem is the thickness of the layer of ink deposited on the paper.

Fermi put his approach into practice when estimating the strength of the atomic bomb detonated in the first nuclear weapons test on July 16th 1945. He estimated the strength of the bomb by observing the distance travelled by pieces of paper dropped from his hand during the blast.

Enrico Fermi's approach to estimating the number of piano tuners in Chicago given only the population of the city involved making a series of reasonable assumptions as follows:

- There are approximately 6 million people living in Chicago according to the almanac of the time.
- There are two persons on average in each household in Chicago.
- One household in twenty has a piano that is tuned regularly.
- Pianos tuned regularly are tuned on average about once per year.
- Each piano tuning consumes about two hours of a piano tuner's time including time taken travelling between jobs.
- Each piano tuner works a 50 week year for eight hours per day and for five days a week.

From these assumptions the number of piano tunings in a single year in Chicago can be estimated as follows:

number of households x the fraction with a piano x number of tunings in a year of that piano =

(6,000,000 persons in Chicago) / (2 persons/household) x (1 piano/20 households) x (1 piano tuning per piano per year) = 150,000

The number of piano tunings per year performed by the average piano tuner is calculated as follows:

Number of piano tunings performed per day by a tuner x number of working days in a week x number of working weeks in a year =

(8 hours/day)/(2 hours to tune a piano) x (5 days/week) Ã— (50 weeks/year) = 1000 piano tunings per year per piano tuner.

This requires the following number of piano tuners:

(150,000 piano tunings per year in Chicago) / (1000 piano tunings per year per piano tuner) = 150 piano tuners in Chicago.

This method does not guarantee correct results but it does establish a first estimate which might be out by no more than a factor of 2 or 3 and certainly well within a factor of, say, 10. Common sense tells us that we should not expect 15 piano tuners, or 1,500 piano tuners.