I know he’s a professor of math who specializes in combinatorics.
So, in lay-men’s terms (or as close as you can manage), how is he doing this?
The end of his talk there, where he takes you through the mental process of one of these calculations, is probably the best explanation you’re going to get. But I’ll take a stab at it: He breaks each multiplication up into sums of smaller multiplications, and uses code words to help remember the intermediate results. Add to that many years of practice and he makes it look easy.
It’s not a secret. He has written a book called Mathemagic that explains and teaches his methods, and that’s really your best source. It’s not hard to explain his methods in layman’s terms - this is exactly what he does in his book. But it is hard to provide a succinct summary of his methods here, because there’s a lot to explain.
I’ll give it a try.
First of all, he has memorised more useful tables than most people ever do. Most of us learn mutliplication tables up to 12 x 12, but human calculators like Benjamin tend to memorise much more information, such as multiplication tables up to 99 x 99, plus tables of squares, cubes (and roots) up to whatever range they find useful.
Secondly, Benjamin will have his own best algortithms for whatever calculations he wants to feature in his act, such as multiplying four digit numbers. ‘Short cut’ methods for mental arithmetic have been around for a long time - the Trachtenberg system achieved some popularity in the 1950s. These and other systems tend to share this characteristic: they do allow for faster mental arithmetic than most of us can manage using our ‘traditional’ school-taught methods, but only if you are willing to put in a lot of time practicing. In other words, these methods are not short cuts to begin with, but become short cuts if you have the time and inclination to put in a lot of practice.
Thirdly, Benjamin and others with some aptitude for rapid calculation tend to be good at spotting how to break a complex problem down into ones that are smaller and easier to solve. They are also good at spotting opportunities to use known shortcuts and ways of translating a seemingly complex, ‘new’ challenge into one that is more easily handled, or one for which there are known solutions. If I ask you to add up all the numbers from 1 to 11 (1 + 2 + 3 + 4… … +11) in your head, you will probably either decide you just can’t do it, or struggle with it in a laborious way. However, if you know the trick, it will take you about two seconds to tell me the answer is 66 (less time than it takes to add the numbers up on a desktop calculator). Likewise, if I give you a long number and ask you if it’s divisable by 11, you probably wouldn’t have a clue, whereas again experienced calculators know there is a little test you can apply that gives you the answer quite quickly.
The fourth factor is natural aptitude (some people just enjoy greater facility with numbers than others).
The fifth is lots and lots and lots of very diligent practice, so that the intermediate steps and sub-routines of calculation become ‘chunked’ into single mental operations. There’s no mystery about ‘chunking’ - we all experience this all the time. When you first attempt a task that involves multiple steps and operations (e.g. driving a car) you can only perfomr it in a slow and faltering way, paying conscious attention to each individual step. Later, with lots of practice and experience, you only consciously need to think ‘drive this car’ and you can more or less forget the rest, because all the learned behaviours have been built into a single subroutine. It is really very strange and wonderful that the brain manages to do this.
That’s about as much as I think I can reasonably cover in a short answer like this. If you want to know more, buy the book!
I went to College with him. He was quite good at these “tricks” back then, and used to do presentations.
That was excellent, thank you!
I just watched and in the 3 digit number part, two of his answers were wrong, but were confirmed by the people with the calculators.
For 457^2, he said 205,849. The correct answer is 208,849.
For 722^2, he said 513,284. The correct answer is 521,284.
Neither of his answers have an even square root.
I’m not trying to say that what he does isn’t completely amazing and he was pretty damn close on the two that he missed, but either the people with the calculators are incompetent or they were planted and were going to agree with him whether he got it right or not.
It’s interesting that he was off by exactly 3000 in both cases. This probably shows something about how the procedure works.
You have an interesting definition of 3000…
I agree that he’s probably memorized his multiplication tables out to 99 x 99; and memorized perfect squares up to 1000. Or at least tried.
For the digits trick, he conveniently picked a number that was a multiple of 9. If you take such a number and multiply it by a 3 digit number, the result is still a multiple of 9. Which means that the sum of all the digits will be a multiple of 9. So if somebody tells you all the digits except 1, and you can add reasonably quickly in your head, you can figure out the last digit.
:smack: OK, 8000 in that case. Still, big multiples of thousands could tell us something about the process. It’s very likely all in his book.
As far as the calendar trick goes, I would say it’s just more memorization.
There are 14 and only 14 possible calendars. You learn each one pretty well, and then memorize which years between 1800 and 2000 go with which of the 14 possible calendars. It would be a decent amount of work, but not an overwhelming amount.
There’s also a formula for figuring out what day of the week a given date falls on:
He’s on YouTube also, but the clip I saw ironically included the warning about not to republish copyrighted material. So let your conscience be your guide.
There are other methods that can be used, too, which are more amenable to human memorization. I read about one once, and was surprised to find the next day that I had retained it, and could use it without pencil and paper or other aids.