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.]
Xi'an's Og 
July 5, 2021 at 6:10 am
This is a very useful algorithm. I use it all the time.
First, I keep in mind the Doomsday for the current year. In 2021 it is Sunday. So 4/4, 6/6, 8/8, 10/10 and 12/12 all fall on Sunday this year, as do 9/5, 7/11, 5/9, and 11/7. Also 3/0 and (since 2021 isn’t a leap year) 2/0 and 1/3. (The zeroth day of a month is the last day of the previous month; in a leap year the last two would be 2/1 and 1/4).
The hardest part is calculating the Doomsday for the year in question. The general rule is this (this isn’t John Conway’s original rule, but it is easier to apply because it doesn’t require remembering a remainder that has to be used several steps later):
For any year in the 21st century, take the last two digits. This year, they are 21. If odd, add 11, if even leave it alone. It’s odd this year, so 21+11= 32.
Take that number, whatever it is, and divide by 2. If the result is odd, add 11. If even, leave it alone. 32 is even so I leave it alone. Divide the result by 2. I divide 32 by 2 to get 16.
Take that number, divide by 7 and determine the remainder. 16=2*7+2 so the remainder is 2.
In this century, count backwards that number of days from Tuesday. The remainder was 2 so I count backwards 2 days from Tuesday. That gives you Sunday. So Doomsday this year is Sunday, as I asserted above.
For other centuries, there is a 400-year cycle. In the 20th century, instead of counting backwards from Tuesday, use Wednesday. In the 19th century, Friday. In the 18th century, Sunday. In the 17th century, Tuesday again (it’s a 400 year cycle).
The reason for the advancement only one day in century years divisible by 400 is that those century years actually have a leap year, whereas (for example) 1700, 1800 and 1900 are NOT leap years, due to the 16th century Gregorian calendar reformation. The year 2000 was a leap year.
So, December 18, 1981: 81 is odd, add 11 to get 92. Divide by 2 to get 46. That’s even so I don’t have to change it by adding 11 again, I leave it alone. 46=6*7+4 so the remainder is 4. Count backwards 4 days from WEDNESDAY (this is a 20th century date) to get Saturday. So Doomsday for 1981 is Saturday. So 12/12/1981 was Saturday. So 12/19/1981 was Saturday and December 18, the day before, was a Friday.
I explain all this in a page I developed for a class I taught on “Time”. You can find it at http://billandsue.net/BillInfo/doomsday.html
It also gives information on how to calculate the day of the week when the date is given on the old Julian calendar. It’s not that difficult and could be useful when interpreting historical documents.