Fermi estimates or problems are quick heuristic solutions to apparently insoluble quantitative problems rewarding clever use of realworld knowledge and critical thinking; bibliography of some examples.
A short discussion of “Fermi calculations”: quickanddirty approximate answers to quantitative questions which prize cleverness in exploiting implications of common knowledge or basic principles in given reasonable answers to apparently unanswerable questions.
Links to discussions of Fermi estimates, and a list of some Fermi estimates I’ve done.
I enjoy Fermi problems (LessWrong)—it’s like dimensional analysis for everything outside of physics^{1}.
Not only are they fun to think about, they can be amazingly accurate, and are extremely cheap to do—because they are so easy, you do them in all sorts of situations you wouldn’t do a ‘real’ estimate for, and are a fun part of a physics education. The common distaste for them baffles me; even if you never work through Hubbard’s How to Measure Anything (some strategies) or StreetFighting Mathematics or read Douglas Hofstadter’s 1982_{42ya} essay “On Number Numbness” (collected in Metamagical Themas), it’s something you can teach yourself by asking, what information is public available, what can I compare this too, how can I put various boundaries around the true answers^{2}
An entire “estimation” subreddit is devoted to working through questions like these (it can be quite fun), and of course, there are the memorable “what if?” Xkcd columns.
Timothy Gowers suggests a number of problems which might help children really learn how to think with & apply the math they learn.
To look further afield, here’s a quick and nifty application by investor John Hempton to the Sino Forestry fraud: “Risk management and sounding crazy”. What I found most interesting about this post was not the overall theme that the whistleblowers were discounted before and after they were proven right (we see this in many bubbles, for example, the housing bubble), but how one could use a sort of Outside View/Fermi calculation to sanitycheck the claims. If Sino Forestry was really causing 17m cubic meters of wood to be processed a year, where was all the processing? This simple question tells us a lot. With medicine, there is one simple question one can always ask too—where is the increased longevity/allcause mortality reduction? (This is an important question to ask of studies, such as a recent caloric restriction study.)
Simple questions and reasoning can tell us a lot.
My Fermi Examples
You especially want to do Fermi calculations in areas where the data is unavailable; I wind up pondering such areas frequently:

how many women dye their hair blonde

how many people does Folding@home kill

or how big a computer is needed to compute the universe

how much do Girl Scouts cookies cost and earn them

how many people have used modafinil, and what is the cheapest we can expect to find modafinil for

quickly assessing the rough probability of some event as I made one of my thousands of predictions

how many sellers are there on Silk Road 1

checking whether posthumous organ donation justifies the Chinese justice system (no)

how many tourists to the Egyptian pyramids there have been compared to the workers who build them

is tiger skin more profitable than tiger bone wine?

the plausibility of Oreimo’s Kirino Kosaka character

how many integers one to a million do you see over a lifetime? (R simulation + Benford’s law)

See Also: Fake Journal Club: Teaching Critical Reading, Little’s law; Frink

This is a little misleading; dimensional analysis is much more like typechecking a program in a language with a good type system like Haskell. Given certain data types as inputs and certain allowed transformations on those data types, what data types must be the resulting output? But the analogy is still useful.↩︎

eg. if someone asks you how many piano tuners there are in Chicago, don’t look blank, start thinking! ‘Well, there must be fewer than 7 billion, because the human race isn’t made of piano tuners, and likewise fewer than 300 million (the population of the United States), and heck, Wikipedia says Chicago has only 2.6 million people and piano tuners are rare, so there must be many fewer than that…’ You always know something, and have a universe of beliefs which imply constraints on everything and equilibria.↩︎