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Piano tuners (a Fermi problem)

SP-Piano Tuners

As a lecturer 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 (then) 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.
  • Assume that 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 s/he would service 1000 pianos (4 pianos x 5 days x 50 weeks). 150000/1000= 150, so that there would be about 150 piano tuners in Chicago.

This method does not provide exact results; but it does establish an estimate which might be off by no more than a factor of 2 or 3 – certainly well within one order of magnitude (say a factor of 10). We know, for example, that we should not expect 15 piano tuners, or 1500 piano tuners. 

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