Fermi Problems Part 1: Envelopes at the Ready!

Becca Tanner

What if I told you that you could solve some of the most seemingly complex mathematical problems with little more than, well, the back of an envelope (and a healthy bit of critical thinking)? I wouldn’t blame you for feeling slightly sceptical, but stick with me, because what lies ahead is the opportunity to dive into the prodigious realm of problem solving: from the strength of atomic bombs to the mass of the ocean, and ultimately, to the quest for extraterrestrial life.

So what are Fermi Problems? They are often termed “back of the envelope problems” as they can literally be completed on a scrap piece of paper. Fundamentally, they involve the process of breaking down a large complex question into smaller problems which can individually be solved through a process of educated guesses and simple calculations. By making a set of estimations and guesses however, there is bound to be a degree of error, so your final answer probably won’t be exactly right – but that doesn’t matter! The point is that your estimate should hopefully be at the same order of magnitude (i.e. within the same power of ten) as the correct answer. The reason for this is that by making a series of estimations, there will be both overestimates and underestimates. If the spread of these overestimates and underestimates from the true value is roughly equal – they should cancel each other out. Thus leaving a pretty accurate estimate of the true answer which can be computed in a fraction of the time.

Enrico Fermi (1901-1954) was the pioneer of such puzzles. His most renowned puzzle is his Piano Tuners in Chicago problem, which he used as a method of demonstrating that seemingly impossible questions, like how many Piano Tuners there are in Chicago, could be answered through a few estimations.

For a fully worked through solution see the link above, but here is the basic idea (in case you wish to try it for yourself and compare with Fermi’s solution). We start by estimating how many families there are in Chicago, and then the proportion of those families who own pianos. From this we can estimate how often these pianos need tuning, and therefore how my piano tuners are needed to meet this demand based on how many a single tuner can visit in a day. The end result is an estimate for the number of piano tuners in Chicago!

Another famous example of Fermi’s method occurred as he watched the detonation of a plutonium bomb at the Trinity Test site in New Mexico in 1945. Before the blast he tore some tissue paper into small pieces, dropped them from a height of six feet and estimated their displacement. He then repeated this as the blast wave reached the observers, and then again after the blast had passed. Due to the lack of wind he was able to observe an accurate displacement shift of ~2.5 metres as a result of the blast. Using this value and some further calculations, he arrived at a figure of 10 kilotons of T.N.T for the strength of the atomic bomb. The now accepted value, which was calculated later using proper measurements, is 21 kilotons, hence Fermi’s estimate was well within the correct order of magnitude. This is pretty astounding and an impressive feat of estimation; with little more than a few scraps of paper and no mechanical measuring devices, Fermi estimated, to an acceptable degree of accuracy, the strength of an atomic bomb. 

Fermi Problems are of course not limited to the fields of nuclear physics and piano tuners. They can be applied to a whole manner of disciplines from engineering, to computer science, through to demographic analysis and, well, any field that involves numerical problem solving! In the next article in this series we will begin to discover this range of applications; starting with calculating the mass of the ocean… so – as the title suggests – get your envelopes at the ready!

Fermi Problems Part 2: Don’t sweat the small stuff

Fermi Problems Part 3: Where is everybody?


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