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Feb 25

Fermi’s Piano Tuner Problem

As a lecturer, Enrico Fermi used to challenge his classes with problems that, at first glance, seemed impossible. One such problem was that of estimating the number of piano tuners in Chicago given only the population of the city. When the class returned a blank stare at their esteemed professor, he would proceed along these lines: From the almanac, we know that Chicago has a population of about 3 million people. Now, assume that an average family contains four members so that the number of families in Chicago must be about 750,000. If one in five families owns a piano, there will be 150,000 pianos in Chicago. If the average piano tuner serviced four pianos every day of the week for five days rested on weekends, and had a two week vacation during the summer, then in one year (52 weeks) he would service 1,000 pianos. 150,000/(4 x 5 x 50) = 150, so that there must be about 150 piano tuners in Chicago. This method does not guarantee correct results; but it does establish a first estimate which might be off by no more than a factor of 2 or 3—certainly well within a factor of, say, 10. We know, for example, that we should not expect 15 piano tuners, or 1,500 piano tuners. (A factor of 10 error, by the way, is referred to as being ‘to within cosmological accuracy.’ Cosmologists are a somewhat different breed from physicists, evidently!!!)


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