Faster than a calculator | Arthur Benjamin | TEDxOxford

Benjamin makes numbers dance. In his day job, he's a professor of mathematics at Harvey Mudd College; in his other day job, he's a "Mathemagician," taking the stage to perform high-speed mental calculations, memorisations and other astounding mathematic stunts. It's part of his drive to teach math and mental agility in interesting ways, following in the footsteps of such heroes as Martin Gardner.
This talk was given at a TEDx event using the TED conference format but independently organized by a local community. Learn more at http://ted.com/tedx

Good afternoon, ladies and gentlemen,

my name is Art Benjamin
and I am a "mathemagician".

What that means
is I combine my loves of math -

or I should say maths -

and magic to do what I call "mathemagics".

But before I get started,

I've got a quick question
for the audience.

By any chance, did anyone
here in the audience

happen to bring with them
this afternoon a calculator?

If you have one, perhaps
on your phone, or somewhere,

and you're pretty comfortable
using it, raise your hand.

I'll need a couple of people
to help me out here.

I see one... two...
and perhaps one more... three.

With the three of you, bring out your
calculators and join me up here on stage,

and let's give these volunteers
a nice round of applause.

Come on up.

(Applause)

Great.

Over on this side, please.

Awesome.

Since I have not had the chance
to work with these calculators,

I need to make sure
that they are all working properly.

Would somebody get us started
by giving us a two-digit number, please?

How about a two-digit number?
Just yell one out.

Audience: 74.

Arthur Benjamin: Oh, okay. That's fine! 74

And another... how about
another two-digit number?

How about on this side?

Audience: 39.

AB: 39. Multiply 74 times 39
on the calculator.

Make sure you get 2,886,
or the calculators are not working.

Do each of you get 2,886?
Give them a round of applause!

(Sighs)

(Applause)

I noticed that took some of us
a little bit of time to get the answer.

That's ok.

I'll give you a shortcut for multiplying
even faster on the calculator.

There's something called
"the square of a number"

which most of you know is taking a number

and multiplying it by itself.

For instance 5 squared would be...

25.

6 squared would be...

36.

73 squared would be...

something else. Yeah.

(Laughter)

On most of these calculators,
they've little shortcut buttons

that allow you to square
numbers even faster.

What I'm going to try
and do is to square -

and you might test it - make sure
you could square 5 or 6 with it,

but what I'm going to try and do

is to square in my head,
three two-digit numbers

faster than they can do
on their calculators,

even using the shortcut method.

But I'll ask these three people
in the third row - one, two, three.

Each yell out a two-digit number,

and if you would square the first one,
the second, and the third one,

I will try and race you to the answer.

So quickly a two-digit number, please?

Audience: 24.

Arthur Benjamin: 24, great, next...

What was that?

Audience: 98.

AB: 98... and one more...

Audience: 26.

AB: 26. Would you call out
your answers, please?

First volunteer: 576.

AB: 576.

Second volunteer: 9,604.

AB: 9,604.

Third volunteer: 676.

AB: 676.

Give them a round of applause!

(Applause)

Let me try to take this one step further.

I'm going to try to square
some three-digit numbers.

This time, I won't even write these down.

I'll just call them out
as they're called out to me.

Anyone at all call out
a three-digit number,

anyone on our panel verify the answer.

If I get the answer right,
give me a big thumbs up.

If I make a mistake,
let me know, and I'll try and fix it.

A three-digit number, anyone?

Audience: 576.

AB: 576 is 331,776?

Second volunteer: Yes.

AB: Yes? Good!

How about another three-digit number,
sir! a three-digit number?

Audience: 103.

AB: 103 is 10,609. Way too easy!
Another three-digit number, please?

Audience: 125.

AB: 125 is 15,625, but that's 5
to the 6th power, so that was easy too.

How about another three-digit number, sir?

Audience: 985.

AB: 985 is 970,225.
Yes, thumbs up, if it's right.

One more three-digit number, sir?

Audience: 926.

AB: 926 is 857,476?

Thank you very much.

(Applause)

(Sighs)

(Applause)

Let me try to take this one step further.

I am going to try to square
a four-digit number this time,

I am not going to beat you
to the answer on this one,

but I will try to get the answer right.

To make this a little bit more random,

how about we use
the fourth row, four people.

Each of you calls out a single-digit
number between zero and nine,

that will be a four-digit number
that I will square.

One, Five, Seven, Seven.

1, 5, 7, 7, this will take me
a little bit of time, so bear with me -

3 million -

(Sighs)

(Laughter)

486,929?

No? OK, don’t tell me.

The number was 1, 5, 7, 7. Oh, wait!

3 million - so far so good?

Oh, is that were I went wrong?

I never make a mistake, wait.

Oh. 2 million, was everything else right?

Volunteer: Yes!

AB: Thank you very much.

(Laughter)

What's one million off? That's all I ask.

(Applause)

Now I would attempt to square
a five-digit number, and I can.

But unfortunately,
most calculators cannot.

So, since we have reached
the limits of our calculators,

although some of yours may go higher,

I'm going to try to conclude
the first part of my show,

by trying something even trickier.

Let's take the first number
on the board here: 576.

Would you each enter 576
on your calculator?

And instead of squaring it this time,
I'd like you to take that number

and multiply it by any
four-digit number that you want.

But don't make it too easy
like 1,000 or 1234,

but just some random four-digit number.

So you should have as an answer

either a six-digit or possibly
a seven-digit number.

How many digits are on your answer,
six or seven digits?

Seven.

Seven.

Six.

Is there any possible way
that I could know

what six or seven-digit numbers
they have? Say "No".

Audience: No.

AB: Good.

Then I shall attempt the impossible,
or at least the improbable.

What I'd like each of you to do
is to call out for me

any six of your seven digits,

or in your case five of your six digits

in any order you do like.

One digit at a time, I shall try
and determine the digit you've left out.

So starting with your six-digit number,
call out any five of them please.

First volunteer: 8, 0, 9, 3, 8.

AB: 8, 0, 9, 3, 8,
did you leave out the number 8?

Yes, that is one.

You have got seven-digit number.
Call out any six of yours, loud and clear.

Second volunteer: 4, 7, 2, 5, 8, 4.

AB: Did you leave out the number six?

AB: That's two.

The odds of me getting all three
of these right by pure guessing

would be one in 1,000:
10 to the third power.

OK, any six of your digits.

Really scramble them up this time.

Third volunteer: 9, 4, 4, 5, 4, 4.

AB: Did you also leave out the number six?

Volunteer: Yes.

AB: Great, and let's give all three
of these people a nice round of applause.

Thank you very much.

(Applause).

For my next number...

(Laughter)

I have another question for the audience.

By any chance, does anyone
here happen to know

the day of the week
that they were born on?

If you think you know your
actual birthday, raise your hand.

Starting with you. What year if I may?

Audience: 1992.

AB: 1992, and what month?

Audience: July.

AB: July what?

3rd. Was that a Friday?

Audience: Yes.

AB: Yes, excellent.

Somebody else? Yes sir, green shirt.

What year?

Audience: 1992

AB: 1992, and the month?

Audience: June.

AB: June what?

Audience: 30th.

AB: A Tuesday?

Audience: Yes.

AB: Excellent. Somebody else,
how about you? What year?

Audience: 1995.

AB: I am sorry sir, what year was it?

Audience: 1995.

AB: 1995 and what month?

Audience: June.

AB: June what?

Audience: 26th.

AB: 26th. Was that a Monday?

Audience: Yes.

AB: Excellent. I see a hand up
in the balcony, a young lady.

I'll try something different here.

If you are sure of your birthday,
what year was it?

What day of the week was it?
Tell me in advance the day of the week.

Audience: Thursday.

AB: Thursday. Ok, and what year was it?

Audience: 2002.

AB: 2002. Did it happen to be on May 2nd?

Audience: Yes.

AB: Yes, but that is my daughter,
I knew that one.

(Laughter)

I was there for that, on that Thursday.

Anyway, I never tried that before.

Do we have anybody here who doesn't know
the day of the week they were born on

but would like to find out?

Ok, now let's see. I'll see yours.

Now, of course, if you don't know
what it is, I could just make up an answer

and you will probably believe me.

But I don't want you to have to do that,
so I come prepared for that situation.

There is an app for everything these days.

So, I'll ask one of you here just
to take this and...

So give us your year and then
type in the year in that blank box.

What year?

Audience: 1995.

AB: 1995. So type in 1995. Great.

And what month?

Audience: September.

AB: September.
Press the September button there.

And that should give you the calendar.
September what?

Audience: 21st.

AB: 21st. I believe, it was a Thursday.
Can we get confirmation?

It was Thursday. Good.

I'll tell you what,

you know since you have the app with you,

let's try something trickier.

The app actually goes as far into
the future as 3000,

as far back into the past as 1600.

Don't go below 1600.

Then we get off the Gregorian calendar
and that messes me up a little.

(Laughter)

So, what year would you like?

Choose a year between
1600 and 3000. Go ahead.

Audience: 2730.

AB: 2730. So enter 2730 into that.
And what month would you like?

Audience: June.

AB: June what?

Audience: 13th.

AB: 13, will that be a Friday?

Audience: Yes.

AB: Yes, and it'll be cloudy
on that day too If I am not mistaken.

Thank you very much.

(Applause)

In fact, anybody else who wants
to find out their birthday,

see me in the lobby, perhaps in the break,
I'll be more than happy to tell you.

Now, just a little bit of time left
so I'd like to do one last thing for you

that I alluded to earlier when we had
the other calculators on stage.

I am going to try to square
a five-digit number,

requiring if you have a ten-digit
calculator or higher.

Fell free to bring out your
calculator at this point.

But to make my job more interesting
for you as well as for me,

I going to do this last problem
thinking out loud,

so you can actually honestly hear

what is going on in my mind,

while I do a calculation of this size.

Let's create a five-digit number.

Why we just go up to this aisle,

the first five people along the aisle,
each gives me a single-digit,

that will be my five-digit number.

Three.

Seven.

Six.

Nine.

One.

37,691 squared. Yuck!

Let me explain to you how
I'm going to attempt this problem.

I'm going to break the problem down
into three parts.

I'll do 37,000 squared,

plus, 691 squared,

plus, 37,000 times 691 times 2.

Add all those numbers together
and with any luck arrive at the answer.

Now, let me explain one more thing.

While I do this calculation you might hear
certain words, as opposed to numbers,

creep into the calculation.

Let me explain what that is.

This is a phonetic code,
a mnemonic device that I use,

that allows me to convert
numbers into words.

I store them as words, and later on
retrieve them as numbers.

I know it sounds complicated, it's not.

I just don't want you to think,

you're seeing something
out of "Rain Man" here.

(Laughter)

There's definitely
a method to my madness -

definitely, definitely. Sorry.

(Laughter)

One last instruction
for my judges with calculators.

Now who has got an answer
in front, raise your hand.

Okay, enough of you.

There is a 50% chance that I will make
a mistake on this problem.

If I do, don't tell me
what the mistake is,

just say, "You're close or something,"

and I'll try and figure it out which
could be pretty entertaining in itself.

If, however, I am right, whatever you do,

don't keep it to yourselves.

Make sure everybody knows
that I got the answer right,

because this is my big finish, OK?

So, without any more stalling,

here we go.

I'll start the problem in the middle,
with 37 times 691.

Now, let's see - Oh my gosh -
so that is 700 minus 9

I will take advantage on that,
700 times 37 is 25,900.

37 times 9 is 333, subtract
the two to get 25,567.

25,000 - do I believe that, yeah -
25,567 double that to get 51,134.

- 51,134 OK - 51,000 becomes
late tomorrow, late tomorrow is 51,134.

That seems right, I'll go on.

Next, I do 37 squared which is 1,369,
so I can say 1 billion.

Take the 369 add that to light.
Is there going to be ???

369 add that to light to get 420 million.
Tomorrow, tomorrow, OK.

Next, we do 691 squared that 700 times
682, plus 9 squared that is 477,481.

Raft, if I need it, raft, take the 477,

add that to tomorrow to get 611,481?

Audience: Yeah!
(Applause).

AB: Yes, good.

Thank you all very much.

I hope you enjoyed mathemagics.
I am Arthur Benjamin. Thank you.

(Applause)