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TED2005

Arthur Benjamin: A performance of "Mathemagic"

February 2, 2005

In a lively show, mathemagician Arthur Benjamin races a team of calculators to figure out 3-digit squares, solves another massive mental equation and guesses a few birthdays. How does he do it? He’ll tell you.

Arthur Benjamin - Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio

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Double-click the English subtitles below to play the video.
Well, good morning ladies and gentlemen.
00:13
My name is Art Benjamin, and I am a "mathemagician."
00:14
What that means is, I combine my loves of math and magic
00:18
to do something I call "mathemagics."
00:21
But before I get started, I have a quick question for the audience.
00:24
By any chance, did anyone happen
00:27
to bring with them this morning a calculator?
00:29
Seriously, if you have a calculator with you, raise your hand, raise your hand.
00:33
I -- was -- your hand go up?
00:37
Now bring it out, bring it out. Anybody else?
00:39
I see, I see one way in the back.
00:41
You sir, that's three,
00:43
And anybody on this side here?
00:45
OK, you over there on the aisle. Would the four of you with calculators
00:48
please bring out your calculators, then join me up on stage.
00:50
And let's give these volunteers a nice round of applause.
00:53
(Applause)
00:55
That's right. Now, since I haven't had the chance
00:58
to work with these calculators, I need to make sure
01:01
that they are all working properly.
01:04
Would somebody get us started by giving us a
01:06
two-digit number please?
01:08
How about a two-digit number?
01:11
Audience: 22.
01:13
Arthur Benjamin: 22. And another two-digit number, Sir?
01:14
Audience: 47.
01:16
AB: Multiply 22 times 47, make sure you get 1,034,
01:17
or the calculators are not working. Do all of you get 1,034? 1,034?
01:21
Woman: No.
01:26
AB: 594. Let's give three of them a nice round of applause there.
01:27
(Applause)
01:31
Would you like to try a more standard calculator, just in case?
01:33
OK, great.
01:35
What I'm going to try and do then --
01:37
I notice that took some of you a little bit of time to get your answer.
01:39
That's OK. I'll give you a shortcut for multiplying
01:42
even faster on the calculator.
01:45
There is something called the square of a number,
01:47
which most of you know is taking a number
01:49
and multiplying it by itself.
01:51
For instance, five squared would be?
01:53
Audience: 25.
01:55
AB: 25. Now, the way we can square on most calculators --
01:56
let me demonstrate with this one --
01:58
is by taking the number, such as five,
02:00
hitting "times" and then "equals,"
02:02
and on most calculators that will give you the square.
02:05
On some of these ancient RPN calculators,
02:08
you've got an "x squared" button on it,
02:10
will allow you to do the calculation even faster.
02:12
What I'm going to try and do now is to square, in my head,
02:15
four two-digit numbers
02:18
faster than they can do on their calculators, even using the shortcut method.
02:20
What I'll use is the second row this time, and I'll get four of you --
02:24
one, two, three, four -- to each yell out a two-digit number,
02:26
and if you would square the first number,
02:29
and if you would square the second, the third and the fourth,
02:33
I will try and race you to the answer. OK?
02:35
So quickly, a two-digit number please.
02:38
Audience: 37.
02:41
AB: 37 squared, OK.
02:42
Audience: 23.
02:44
AB: 23 squared, OK.
02:45
Audience: 59.
02:47
AB: 59 squared, OK, and finally?
02:48
Audience: 93.
02:49
AB: 93 squared. Would you call out your answers, please?
02:51
Woman: 1369. AB: 1369.
02:54
Woman: 529.
02:56
AB: 529.
02:57
Man: 3481.
02:58
AB: 3481.
02:59
Man: 8649.
03:00
AB: Thank you very much.
03:01
(Applause)
03:02
Let me try to take this one step further.
03:07
I'm going to try to square some three-digit numbers this time.
03:10
I won't even write these down --
03:13
I'll just call them out as they're called out to me.
03:15
Anyone I point to, call out a three-digit number.
03:17
Anyone on our panel, verify the answer.
03:20
Just give some indication if it's right.
03:22
A three-digit number, sir, yes?
03:24
Audience: 987.
03:27
AB: 987 squared is 974,169.
03:28
(Laughter)
03:31
Yes? Good.
03:33
Another, another three-digit --
03:35
(Applause)
03:37
-- another three-digit number, sir?
03:38
Audience: 457.
03:41
AB: 457 squared is 205,849.
03:42
205,849?
03:45
Yes?
03:47
OK, another, another three-digit number, sir?
03:49
Audience: 321.
03:52
AB: 321 is 103,041. 103,041.
03:53
Yes? One more three-digit number please.
03:56
Audience: Oh, 722.
03:59
AB: 722 is 500 -- ooh, that's a harder one.
04:01
Is that 513,284?
04:04
Woman: Yes.
04:07
AB: Yes? Oh, one more, one more three-digit number please.
04:08
Audience: 162.
04:11
162 squared is 26,244.
04:12
Thank you very much.
04:16
(Applause)
04:18
Let me try to take this one step further.
04:25
(Laughter)
04:28
I'm going to try to square a four-digit number this time.
04:30
Now you can all take your time on this; I will not beat you to the answer on this one,
04:33
but I will try to get the answer right.
04:36
To make this a little bit more random, let's take the fourth row this time,
04:38
let's say, one, two, three, four.
04:41
If each of you would call out a single digit between zero and nine,
04:44
that will be the four-digit number that I'll square.
04:46
Audience: Nine.
04:49
AB: Nine.
04:50
Audience: Seven. AB: Seven.
04:51
Audience: Five. AB: Five.
04:52
Audience: Eight. AB: Eight.
04:53
9,758, this will take me a little bit of time, so bear with me.
04:55
95,218,564?
04:58
Woman: Yes.
05:07
AB: Thank you very much.
05:08
(Applause)
05:10
Now, I would attempt to square a five-digit number --
05:18
and I can --
05:21
but unfortunately most calculators cannot.
05:23
(Laughter)
05:25
Eight-digit capacity -- don't you hate that?
05:27
So, since we've reached the limits of our calculators --
05:30
what's that? Does yours go --
05:33
Woman: I don't know.
05:36
AB: Does yours go higher?
05:37
Oh -- yours does?
05:39
Man: I can probably do it.
05:41
AB: I'll talk to you later.
05:42
In the meanwhile, let me conclude
05:44
the first part of my show by doing something a little trickier.
05:46
Let's take the largest number on the board here, 8649.
05:49
Would you each enter that on your calculator?
05:52
And instead of squaring it this time,
05:55
I want you to take that number and multiply it
05:57
by any three-digit number that you want,
05:59
but don't tell me what you're multiplying by --
06:02
just multiply it by any random three-digit number.
06:04
So you should have as an answer either
06:07
a six-digit or probably a seven-digit number.
06:09
How many digits do you have, six or seven?
06:13
Seven, and yours? Woman: Seven.
06:14
AB: Seven? Seven?
06:16
And, uncertain.
06:19
Man: Yeah.
06:21
AB: Seven. Is there any possible way that I could know
06:22
what seven digit numbers you have? Say "No."
06:24
(Laughter)
06:27
Good. Then I shall attempt the impossible --
06:29
or at least the improbable.
06:31
What I'd like each of you to do is to call out for me
06:33
any six of your seven digits, any six of them,
06:35
in any order you'd like.
06:38
(Laughter)
06:40
One digit at a time, I shall try and determine the digit you've left out.
06:41
So, starting with your seven-digit number,
06:46
call out any six of them please.
06:48
Woman: One, OK, 197042.
06:50
AB: Did you leave out the number 6?
06:56
Woman: Yes, AB: Good, OK, that's one.
06:57
You have a seven-digit number, call out any six of them please.
06:59
Woman: 44875.
07:01
AB: I think I only heard five numbers. I -- wait -- 44875 --
07:04
did you leave out the number 6?
07:09
Woman: Yes. AB: Same as she did, OK. You've got a seven-digit number --
07:11
call out any six of them loud and clear.
07:13
Man: 079044.
07:16
AB: I think you left out the number 3?
07:20
That's three. The odds of me getting all four of these right by random guessing
07:22
would be one in 10,000: 10 to the fourth power.
07:26
OK, any six of them.
07:29
Really scramble them up this time, please.
07:32
Man: 263972.
07:34
AB: Did you leave out the number 7?
07:38
And let's give all four of these people a nice round of applause.
07:40
Thank you very much. (Applause)
07:42
For my next number --
07:52
(Laughter)
07:54
while I mentally recharge my batteries,
07:57
I have one more question for the audience.
07:59
By any chance, does anybody here happen to know
08:02
the day of the week that they were born on?
08:07
If you think you know your birth day, raise your hand.
08:10
Let's see, starting with -- let's start with a gentleman first,
08:14
OK sir, what year was it, first of all? That's why I start with a gentleman first.
08:17
What year?
08:20
Audience: 1953.
08:21
AB: 1953, and the month?
08:22
Audience: November. AB: November what?
08:24
Audience: 23rd.
08:25
AB: 23rd -- was that a Monday? Audience: Yes.
08:26
Yes, good. Somebody else? Who else would like --
08:28
see I don't -- haven't seen any women's hands up.
08:30
OK, it's -- how about you, what year?
08:32
Audience: 1949. AB: 1949, and the month?
08:34
Audience: October. AB: October what?
08:36
Audience: Fifth.
08:37
AB: Fifth -- was that a Wednesday?
08:38
Yes, my -- I'll go way to the back right now, how about you?
08:40
Yell it out, what year? Audience: 1959.
08:42
AB: 1959, OK -- and the month?
08:44
Audience: February.
08:46
AB: February what? Audience: Sixth.
08:48
AB: Sixth -- was that a Friday? Audience: Yes.
08:50
Good, how about the person behind her?
08:52
Call -- call -- what year was it?
08:54
Audience: 1947. AB: 1947, and the month?
08:56
Audience: May. AB: May what?
08:58
Audience: Seventh. AB: Seventh -- would that be a Wednesday?
09:00
Audience: Yes.
09:02
AB: Thank you very much.
09:03
(Applause)
09:05
Anybody here who'd like to know the day of the week they were born?
09:09
We can do it that way.
09:12
Of course, I could just make up an answer and you wouldn't know,
09:14
so I come prepared for that.
09:16
I brought with me a book of calendars.
09:18
It goes as far back into the past as 1800, 'cause you never know.
09:21
(Laughter)
09:25
I didn't mean to look at you, sir --
09:27
you were just sitting there.
09:29
Anyway, Chris, you can help me out here, if you wouldn't mind.
09:31
This is a book of calendars, and I'll ask --
09:34
who was it that wanted to know their birth day? You sir? OK.
09:36
What year was it, first of all?
09:38
Audience: 1966.
09:40
AB: '66 -- turn to the calendar with 1966 --
09:41
and what month?
09:43
Audience: April. AB: April what?
09:45
Audience: 17th. AB: 17th -- I believe that was a Sunday.
09:47
Can you confirm, Chris?
09:50
Chris Anderson: Yes. AB: Yeah, OK. I'll tell you what, Chris:
09:52
as long as you have that book in front of you,
09:54
do me a favor, turn to a year outside of the 1900s,
09:57
either into the 1800s or way into the 2000s --
10:01
that'll be a much greater challenge for me.
10:04
What year, Chris, would you like?
10:06
CA: 1824.
10:07
AB: 1824, OK.
10:08
And what month? CA: June.
10:11
AB: June what? CA: Sixth.
10:13
AB: Sixth -- was that a Sunday?
10:15
CA: It was. AB: And it was cloudy.
10:17
Good, thank you very much.
10:20
(Applause)
10:22
But I'd like to wrap things up now
10:28
by alluding to something
10:30
from earlier in the presentation.
10:33
There was a gentleman up here who had a 10-digit calculator.
10:35
Where is he, would you stand up,
10:38
10-digit guy?
10:42
OK, well stand up for me just for a second,
10:44
so I can see where you are.
10:47
OK, oh, OK -- you have a 10-digit calculator, sir, as well?
10:49
OK, what I'm going to try and do, is to square in my head
10:52
a five-digit number requiring a 10-digit calculator.
10:56
But to make my job more interesting for you, as well as for me,
10:59
I'm going to do this problem
11:04
thinking out loud.
11:06
So you can actually, honestly hear
11:08
what's going on in my mind
11:11
while I do a calculation of this size.
11:13
Now, I have to apologize to our magician friend Lennart Green.
11:15
I know as a magician we're not supposed to reveal our secrets,
11:18
but I'm not too afraid that people are going
11:22
to start doing my show next week, so --
11:24
I think we're OK.
11:26
So, let's see, let's take a --
11:30
let's take a different row of people, starting with you.
11:33
I'll get five digits: one, two, three, four --
11:35
oh, I did this row already. Let's do the row before you,
11:38
starting with you sir: one, two, three, four, five.
11:40
Call out a single digit -- that will be the five-digit number
11:43
that I will try to square. Go ahead.
11:46
Audience: Five. AB: Five.
11:48
Audience: Seven. AB: Seven.
11:50
Audience: Six. AB: Six.
11:51
Audience: Eight. AB: Eight.
11:52
Audience: Three. AB: Three.
11:54
57,683
11:56
squared. Yuck.
12:01
Let me explain to you how
12:03
I'm going to attempt this problem.
12:05
I'm going to break the problem down into three parts.
12:08
I'll do 57,000 squared,
12:11
plus 683 squared,
12:13
plus 57,000 times
12:17
683 times two.
12:20
Add all those numbers together,
12:22
and with any luck, arrive at the answer.
12:24
Now, let me recap.
12:28
Thank you.
12:31
While I explain something else --
12:34
-- I know, that you can use, right?
12:37
While I do these calculation[s],
12:40
you might hear certain words,
12:42
as opposed to numbers, creep into the calculation.
12:44
Let me explain what that is.
12:48
This is a phonetic code,
12:49
a mnemonic device that I use,
12:51
that allows me to convert numbers into words.
12:53
I store them as words, and later on retrieve them as numbers.
12:56
I know it sounds complicated; it's not --
12:59
I just don't want you to think you're seeing something out of "Rain Man" here.
13:01
(Laughter)
13:04
There's definitely a method to my madness --
13:06
definitely, definitely. Sorry.
13:08
(Laughter)
13:10
If you want to talk to me about ADHD afterwards,
13:14
you can talk to me then. All right --
13:16
by the way, one last instruction,
13:18
for my judges with the calculators -- OK, you know who you are --
13:20
there is at least a 50 percent chance
13:23
that I will make a mistake here.
13:26
If I do, don't tell me what the mistake is;
13:29
just say, "you're close," or something like that, and I'll try and figure out the answer --
13:32
which could be pretty entertaining in itself.
13:35
If, however, I am right,
13:39
whatever you do, don't keep it to yourself, OK?
13:41
(Laughter)
13:44
Make sure everybody knows that I got the answer right,
13:45
because this is my big finish, OK.
13:47
So, without any more stalling,
13:50
here we go.
13:53
I'll start the problem in the middle, with 57 times 683.
13:56
Now, 57 times 68 is 3,400, plus 476 is 3876,
13:58
that's 38,760 plus 171,
14:01
38,760 plus 171 is 38,931.
14:04
38,931; double that to get 77,862.
14:07
77,862 becomes cookie fission,
14:11
cookie fission is 77,822.
14:13
That seems right, I'll go on. Cookie fission, OK.
14:17
Next, I do 57 squared, which is 3,249, so I can say,
14:19
three billion. Take the 249, add that to cookie, 249,
14:23
oops, but I see a carry coming --
14:28
249 --
14:30
add that to cookie, 250 plus 77,
14:32
is 327 million --
14:34
fission, fission, OK, finally, we do 683 squared,
14:38
that's 700 times 666, plus 17 squared
14:41
is 466,489, rev up if I need it,
14:43
rev up, take the 466,
14:46
add that to fission, to get,
14:48
oh gee --
14:50
328,489.
14:54
Audience: Yeah!
14:58
AB: Good.
14:59
(Applause)
15:00
Thank you very much.
15:01
I hope you enjoyed mathemagics. Thank you.
15:03
(Applause)
15:05

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Arthur Benjamin - Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty.

Why you should listen

Arthur Benjamin makes numbers dance. In his day job, he's a professor of math at Harvey Mudd College; in his other day job, he's a "Mathemagician," taking the stage in his tuxedo to perform high-speed mental calculations, memorizations and other astounding math 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.

Benjamin is the co-author, with Michael Shermer, of Secrets of Mental Math (which shares his secrets for rapid mental calculation), as well as the co-author of the MAA award-winning Proofs That Really Count: The Art of Combinatorial Proof. For a glimpse of his broad approach to math, see the list of research talks on his website, which seesaws between high-level math (such as his "Vandermonde's Determinant and Fibonacci SAWs," presented at MIT in 2004) and engaging math talks for the rest of us ("An Amazing Mathematical Card Trick").

The original video is available on TED.com
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