This post is a sequel to the post How Mr. Bidder calculated logarithms published a few days ago. As with that post, this post is based on an excerpt from The Great Mental Calculators by Steven B. Smith.
Smith’s book says Arthur Benjamin squares large numbers using the formula
n² = (n + a)(n − a) + a²
where a is chosen to make the multiplication easier, i.e. to make n + a or n – a a round number. The method is then applied recursively to compute a², and the process terminates when you get to a square you have memorized. There are nuances to using this method in practice, but that’s the core idea.
The Great Mental Calculators was written in 1983 when Benjamin was still a student. He is now a mathematics professor, working in combinatorics, and is also well known as a mathemagician.
Major system
Smith quotes Benjamin giving an example of how he would square 4273. Along the way he needs to remember 184 as an intermediate result. He says
The way I remember it is by converting 184 to the word ‘dover’ using the phonetic code.
I found this interesting because I had not heard of anyone using the Major system (“the phonetic code”) in real time. This system is commonly used to commit numbers to long-term memory, but you’d need to be very fluent in the system to encode and decode a number in the middle of a calculation.
Maybe a lot of mental calculators use the Major system, or some variation on it, during calculations. Most calculators are not as candid as Benjamin in explaining how they think.
In memory sports it is common to define fixed encodings from numbers 0 to 99 or even 0 to 999 as words that can easily be visualized, based on the major system.
So Benjamin might not have to come up with the encoding on the fly either.
The encoding system he’s using (or was using 40 years ago!) is standard — e.g. DoVeR for 184 — so I’m sure he didn’t come up with the encoding on his own. The book I quoted says that he did come up with the squaring trick on his own, though it had been discovered before, probably many times.
I’m familiar with memory sports and with mental calculation, but I haven’t seen much about their overlap.