A book about mental arithmetic? By Rob Eastaway? Count me in! In my fuzzy mental Euler diagram of topics and authors, Maths On The Back Of An Envelope lies in the intersection of several ‘favourite’ circles.
Perhaps paradoxically, this meant I was expecting to be a little disappointed: how can a book, by an author I admire, on a topic I both love and have Strong Opinions about, live up to what I’d like it to be? Luckily, Eastaway’s writing is excellent, even taking into account that you expect it to be excellent.
Aperiodipal and MathsJam regular Rob Eastaway organised an inter-MathsJam competition for last month’s events, challenging Jams to make Fermi estimates on the back of an envelope. The prize was a copy of his new book, Maths on the Back of an Envelope. Here Rob gives a summary of the entries he received, and shares his favourites.
Regular attendees of MathsJam will know that in September, Katie Steckles kindly allowed me to hijack the evening (in the nicest possible sense) by posing some envelope-related challenges, in celebration of the publication of my new book Maths On The Back Of An Envelope. In addition to some envelope-related puzzles, there was also an open challenge to Maths Jam groups to come up with their own back-of-an-envelope problems, with the chance to win the book as a prize.
The cosmos is rich beyond measure. The number of stars in the universe is larger than all the grains of sand on all the beaches of the planet Earth.
More or Less come to a fairly standard answer, that Sagan was correct. This sort of problem, which involves approximating unknowable numbers based on a series of estimates, is called a Fermi problem. I’ve written about Fermi problems here before. The More or Less approach to answering this raised a question from a reader of this blog.
But that's less than a factor of 3 difference! For Fermi estimates of numbers of that size, those two answers are essentially the same. It wouldn't take much of an error in either estimate to push sand ahead of stars…
I gave a talk on Fermi problems and a method for approaching them using the approximate geometric mean at the Maths Jam gathering in 2017. This post is a write up of that talk with some extras added in from useful discussion afterwards.
Enrico Fermi apparently had a knack for making rough estimates with very little data. Fermi problems are problems which ask for estimations for which very little data is available. Some standard Fermi problems:
How many piano tuners are there in New York City?
How many hairs are there on a bear?
How many miles does a person walk in a lifetime?
How many people in the world are talking on their mobile phones right now?
Hopefully you get the idea. These are problems for which little data is available, but for which intelligent guesses can be made. I have used problems of this type with students as an exercise in estimation and making assumptions. Inspired by a tweet from Alison Kiddle, I have set these up as a comparison of which is bigger from two unknowable things. Are there more cats in Sheffield or train carriages passing through Sheffield station every day? That sort of thing.