07 January 2008

Arthur Benjamin's mental arithmetic

Arthur Benjamin is a research mathematician (I've actually mentioned him before, although I didn't realize that until I looked at his web page and saw that the title of one his papers looked familiar...) and also a "mathemagician" -- he has a stage show in which he does mental calculations. See this 15-minute video of his show at ted.com.

He starts out by squaring some two-digit numbers... this didn't impress me much, because I could almost keep up with him. (And 37 squared is especially easy for me. One of my favorite coffeehouses in Cambridge was the 1369 Coffee House, and at some point I noticed that that was 37 squared. So I'll always remember that one.) Squaring two-digit numbers is just a feat of memory. Three- and four-digit numbers, though... that's a bit more impressive. And of course I'm harder to impress in this area than the average person.

One trick that might not be obvious how it works: he asks four people to each find a number 8649x (the 8649 was 93 squared, from the number-squaring part of the show) for some three-digit integer x, and give him six of the seven digits in any order; he says which digit is left out. How does this work? 8649 is divisible by 9. So the sum of the digits of 8649x must be divisible by 9. So, for example, say he gets handed 2, 2, 2, 7, 9, 3; these add up to 25, so the missing digit must be 2, to make 27? (How could he tell apart a missing zero and a missing nine? I suspect there's a workaround but I don't know what it is; the number 93 was given by someone in the audience, so I don't think it's just memory.)

He also asks people for the year, month, and day which they were born and gives the date; I found myself trying to play along but I can't do the Doomsday algorithm quite that fast... and I suspect he uses something similar. (I noticed that he asked three separate questions: first the year, then the month, then the day. This gives some extra time. I know this trick well; when a student asks a question I haven't previously thought about, I repeat it. I suspect I'm not the only one.)

The impressive part, for me, is not the ability to do mental arithmetic -- I suspect most mathematicians could, if they practiced -- but the ability to keep up an engaging stage show at the same time.

(The video is on ted.com, which shows talks from an annual conference entitled "Technology, Entertainment, and Design"; there look to be quite a few other interesting videos on there as well.


Anonymous said...

I have had limited success in learning some of AB's tricks.
BTW, knowing that 38x38=1444 is helpful. Numbers a distance 12 away from
0,50, 100, etc, have their last two digits as 44. Since 60x60 is 3600, you can extrapolate via the mnemonic that 62x62=3844. That 88x88=7744 follows from considerations about 11.

The way I see the multiplication algorithm for proximate 2 digit numbers, one memorizes squares from say 1 to 50, and memorizes products of the form
n(n+1). Every product is the difference of squares. When two proximate numbers are of differing parity, then instead of squaring their mean you compute n(n+1) where the mean is n +1/2. The reason for this trick is that
(10n +5)(10n+5)=
100n(n+1)+25. When squaring anything plus a half there is a quarter tagged on.

It is possible to train really young children in these algorithms, and especially get them to think algebraically while they are computing arithmetically.

Here is an example computation. It doesn't look so flashy in writing, but I am working it out in real time as I go. OK, I want to compute the product 52x57. The mean is 54.5. The square of 54 is about 100 less than 3025, so it must be
2916. Now I want to
add 54 to this (to get 54x55) and also subtract 6 from that. So I need to add 48 to 2916. I think that 52x57 is
2964. The result I have is clearly divisible by 3, so I am pretty confident.

Now I went to pull down my (yuck) windows calculator and got the same result.

With practice, one remembers more products and implements the most convenient algorithm for the product at hand.

For example, for the problem above the numbers in question factor as 4x13 and 3x19. One could compute 12x13x19 instead. For example 13x19=16x16-9=256-9=247.

Again these calculations are being done as I type.

I started working on these in my spare time, while riding in the car to and from work. I turned off the misearbly terrible local radio, and started teaching one of my sons these techniques.

I can't do the calculations as quickly as I might with a pencil and paper, but I can get accurate results without pencil and paper and without calculator.

My belief is that such methods should be taught to grade school kids so that they have internalized algebra before working with it.

Anonymous said...

Listening to him got me curious to what day of the week I was born; Friday the 13th; which is funny because I was 13 on Friday the 13th

Michael Lugo said...


that can't be right. If you were born on Friday the 13th, then your 13th birthday would have been on either a Sunday or a Monday. A date falls one day later each year than in the year before, except two days later in leap years; thus the day of the week of your 13th birthday would be thirteen days after the day of the week of your birth, plus three or four extra days for the leap years.

Michael Lugo said...


although I haven't really tried to learn these tricks, the way I do mental arithmetic probably has more in common with such "tricks" than with the various grade-school algorithms. I haven't consciously memorized things, but the fact that I do a lot of arithmetic means that results which come up often in the sorts of calculations I do come to mind quickly.

Anonymous said...


In high school I had a close friend who could out calculate me quickly and accurately. I told him a few years ago that I was trying to learn how to compute products by the difference of squares. Since he had taught himself to multiply, he agreed that this was the method he used.

I am fairly bad at mental arithmetic (like many mathematicians), but the exercise helps sharpen my ability. The largest obstruction remains subtraction. I can't seem to do borrowing and keep focused on the problem. Sometimes I simply forget a given difference. I found that by subtracting squares from squares and n(n+1) from m(m+1) to be helpful. I mentioned the method in detail for your younger readers.

Maybe some of Vlorbik's fans can pick this thread up.

Anonymous said...

Yep I was really dumb.

I became curious about what happened June 13 '45 did a search; saw an article about a concert on June 13 written on MONDAY the 16th and BINGO Friday the 13!! Well I did not read the article, it was written APRIL 16th '45.