I’ve read more than a few books on mental arithmetic, and have found some that I like. Strangely enough, they are not common in libraries, despite the fact that one of them is written by an author virtually every one will recognize instantly.
I’ve been reading most recently Secrets of Mental Math by Arthur Benjamin and Michael Shermer. Arthur Benjamin recently gave a keynote speech at a national tech event (was it OSCON?). I wanted to see his book and what he had to offer that I hadn’t read recently. I didn’t like it as much as I thought – more of the book was dedicated to “tricks” (to amaze your friends!) than I would have liked. Most books of this sort focus on making daily calculations much easier. I also found his writing to be not as clear as the others I have read on this topic.
One interesting item put into Secrets of Mental Math is the method of finding the day of the week for any future year. This topic is also considered in The Memory Book by Harry Lorayne and Jerry Lucas, but in The Memory Book the method is to memorize a number representing the days that Monday falls on throughout the year (that is, a 12-digit number). Secrets of Mental Math proposes a mathematical formula, utilizing a code number for the month and a code number for the year. The code number for the month (and for the year) is never completely explained, but would probably be based on the day of the week that the month will have given that the year starts on Monday (or something like that) – plus a shift factor coded into the year code.
So your choice – choose the memorization method in The Memory Book or the calculation method found in Secrets of Mental Math. Of course, the latter handles every date in every year without memorization – but people would rarely use such a method for anything other than the current year, most likely.
The book I have in my collection that I prefer in this area is Calculator’s Cunning by Karl Menninger, translated from German by E. J. F. Primrose (with foreward by Martin Gardner). I am surprised that this book is not more widely available; it appears to be out of print. Karl’s writing (and Primrose’s translation) is very easy to read, understandable, and clear – and mathematical proofs are offered at the end of each section where the trick is introduced.
Another which is almost as hard to find in the library as the tome by Karl Menninger is a book by the venerable Isaac Asimov titled Quick and Easy Maths. This book may or may not be available at your local library, but chances are good you’ll have to go to interlibrary loan to get it, or even buy used. The book is out of print, just as Calculator’s Cunning is. Asimov is one of the clearest writers I’ve seen, and Quick and Easy Maths is no exception.
Also, the methods described in all three mental mathematical books are essentially the same – even so, the experience of reading of all three is all worthwhile.
Then next time you have to calculate the number of hosts in a subnet, you’ll be able to do it in your head – or adding up an invoice!