quick(er) calculations [book review]

Upon my request, Oxford University Press sent me this book for review in CHANCE. With the extended title How to add, subtract, multiply, divide, square, and square root more swiftly. This short (173 pages) book is written by Trevor Davis Lipscombe, currently Director of the Catholic University of America Press (which are apparently not suited for his books, since his former Physics of Rugby got published by Nottingham University Press). The concept of the book is to list tricks and shortcuts to handle seemingly tough operations on a list of numbers. Illustrated by short anecdotes mostly related to religion, sports (including the Vatican cricket team!), and history, albeit not necessarily related with the computation at hand and not providing an in-depth coverage of calculation across the ages and the cultures. While the topic is rather dry, as illustrated by the section titles, e.g., “Multiply two numbers that differ by 2, 4, 6, or 20” or “Multiply or divide by 66 or 67, 666 or 667” (!), the exposition is somewhat facilitated by the (classics) culture of the author. (I have to confess I got lost by the date chapter, i.e., finding which day of the week was December 18, 1981, for instance. Especially by the concept of Doomsday which I thought was a special day of the year in the UK. Or in the USA.) Still, while recognising some simple decompositions I also used for additions and subtractions, and acknowledging the validity of the many tricks I had never though of, I wonder at the relevance of learning those dozens of approaches beyond maintaining a particular type of mental agility… Or preparing for party show-time. Especially for the operations that do not enjoy exact solutions, like dividing by √3 or multiplying by π… The book reminded me of a physics professor in Caen, Henri Eyraud, who used to approximate powers and roots faster than it took us to get a slide rule out of our bags! But Guesstimation, which I reviewed several years ago, seemed more far-reaching that Quick(er) calculations, in that I had tried to teach my kids (with limited success) how to reach the right order of magnitude of a quantity, but never insisted [beyond primary school] on quick mental calculations. (The Interlude V chapter connects with this idea.)

[Disclaimer about potential self-plagiarism: this post or an edited version should eventually appear in my Books Review section in CHANCE.]

dial e for Buffon

The use of Buffon’s needle to approximate π by a (slow) Monte Carlo estimate is a well-known Monte Carlo illustration. But that a similar experiment can be used for approximating e seems less known, if judging from the 08 January riddle from The Riddler. When considering a sequence of length n exchangeable random variables, the probability of a particuliar ordering of the sequence is 1/n!. Thus, counting how many darts need be thrown on a target until the distance to the centre increases produces a random number N≥2 with pmf 1/n!-1/(n+1)! and with expectation equal to e. Which can be checked as follows

p=diff(c(0,1+which(diff(rt(1e5))>0)))
sum((p>1)*((p+1)*(p+2)/2-1)+2*(p==1))

which recycles simulations by using every one as starting point (codegolfers welcome!).

An earlier post on the ‘Og essentially covered the same notion, also linking it to Forsythe’s method and to Gnedenko. (Rényi could also be involved!) Paradoxically, the extra-credit given to the case when the target is divided into equal distance tori is much less exciting…