A Fermi problem is a quick estimate of the order of magnitude of a quantity, often obtained starting from very little data, or even just common sense. The idea is to break down an apparently impossible question into a chain of simpler estimates, each of which we can reasonably guess, and then combine them. The physicist Enrico Fermi was famous for these “on the fly” estimates, so much so that they now bear his name.
Historical context — Fermi and the Alamogordo atomic bomb
On 16 July 1945, during the first atomic test (Trinity) in the New Mexico desert, Enrico Fermi dropped a few scraps of paper right after the explosion. By observing how far the shock wave pushed them, he estimated the bomb’s yield on the spot at about 10 kilotons: a remarkably close estimate to the true value (18 kilotons). He had used only paper and a mental stopwatch (Rhodes 1986, ch. 8; see Riferimenti bibliografici).
How many piano tuners are there in Milan?
This is the classic Fermi problem. We do not know the answer, but we can estimate it step by step:
- Inhabitants of Milan:
- Pianos in the city: roughly 1 per 100 households, i.e.
- Tunings a tuner does in a year:
- Number of tuners:
The result is of the order of 10: it could be 5 or 20, but certainly not 1 nor 100. And that is exactly what interests us — the order of magnitude.
Why do Fermi estimates work?
The idea is that, when several quantities are combined by multiplication or division, the relative errors tend to cancel out. If you underestimate one factor by 30% and overestimate another by 30%, the final result remains roughly correct. This is why the order-of-magnitude estimate usually “hits” the true value to within a factor of 10, and often much better.
Collegamenti
Argomenti: Grandezze fisiche e misura Competenze: Stime di Fermi
Esercizi collegati: Esercizio svolto — Il rubinetto che perde · Esercizio svolto — Area di un rettangolo misurato · Problema — Volume di un parallelepipedo