Beautiful Card Trick - Numberphile Numberphile English (auto-generated) English

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okay so I'm gonna do a card trick based

MATT: OK, so I'm going to do a card trick based

on the number 27 and this is my all-time

on the number 27.

favorite math card trick I'm going to

And this is my all-time favorite maths card trick.

show it for you today and I'm going to

And I'm going to show it for you today, and I'm going to

explain it and I found this trick in an

old 1950s math book written by Martin

I found this trick in an old 1950's math book written by

Gardner and for me it is the mass card

Martin Gardner. And for me it is the maths card trick with the most

trick with the most beautiful maths

beautiful maths behind it out of all of them.

behind it out of all them and because it is a math card trick it does involve a

And because it is a math card trick, it does involve a lot

lot of long tedious counting but bear

of long tedious counting.

with us here so did this involve 27 cards so I'm going to take 27 off and

But bear with us here. So this involves 27 cards, so I'm going to take 27 off.

this is a genuine count 1 2 3 4 5 6 7 8

And this is a genuine count. One, two, three, four, five, six, seven, eight, nine, 10.

9 10 27 is actually one of my favorite

27 is actually one of my favorite numbers--

numbers 110 because it's a cube number 1

One, two, three, four, five, six, seven, eight, nine, 10--

2 3 4 5 6 7 I guess 27 cards and this

because it's a cubed number. One, two, three, four, five, six, seven.

works with any 27 cards and and none of

OK, that's 27 cards. And this works with any 27 cards, and none of this trick

this trick is sleight of hand none of it

is slight-of-hand.

is a YouTube magic where I'm using

None of it is YouTube magic where I'm using something

something sneaky or a sneaky edit and

sneaky, or a sneaky edit.

I'll explain the trick afterwards so

And I'll explain the trick afterwards, so it's OK.

it's ok but this is how it works you get

But this is how works--

27 cards you shuffle them up I'm

You get 27 cards, you shuffle them up.

actually going to get brady to both film

I'm actually going to get Brady to both

and be the volunteer so I'll flick

film and be the volunteer.

through which do you want to tap which

So I'll flick through, do you want to tap which one you

one you want which one of these just

want, which one of these?

just like that one there you want to

OK that one there.

show the camera that color let me see it

Do you want to show you the camera that card?

obviously and you wanna put it back in

Don't let me see it, obviously.

wherever you want I can always and do is

And do you want to put it back in wherever you want?

just remember what that card was and

Thank you. Now all he needs to do is just remember what that card was,

believe me people in the comments will

and believe me, people in the comments will mention

mention upwards if you don't Oh Brady

afterwards if you don't.

what's your favorite number from 1 to 27

Brady what's your favorite number from 1 to 27, if you

if you had to pick bigger number 10 10

had to pick a number?

any particular reason why 10 I just like

BRADY: 10. MATT: 10, any particular reason why 10?

how it looks you're like ok are you

BRADY: I just like how it looks.

looking for your card by the way what I

MATT: You like it? OK. Are you looking for your card by the way?

want you to do is I have a look and see

What I want you to do is have a look, and see if you can

if you can spot which pile your card goes into and people may have seen this

spot which pile your card goes into. And people may have seen this trick done before.

trick done before it's a variation in fact it's a generalization on a 21 card

It's a variation, in fact, it's a generalization on a 21

trick which pile is it in it's in middle

card trick. Which pile is it in? BRADY: It's in that pile.

part there okay I'm going to pick them

MATT: In the middle pile there?

up from from well from the viewers right

OK, I'm going to pick them up from the

to left and what people tend to do is

viewer's right to left.

they do this tedious counting out each

And what people tend to do is they do this tedious counting

time what I'm actually doing is last

out each time. And what I'm actually doing is last time I

time I memorized all the cards and so I

memorized all the cards.

knew when you taught me which pile I've

And so I when you told me which pile, I had narrowed it

narrowed it down to nine possible cards

down to nine possible cards it could be.

it could be if I do it again because of

If I do it again, because of the way I'm dealing it out, if

the way I'm dealing it out if you tell me which pilots in this time I will

you tell me which pile it's in this time, I will narrow it

narrow it down to one of three possible cards which pile is it

down to one of three possible cards.

in the stock this time it is in it's in

Which pile is it in this time? BRADY: This time it is in the middle pile.

the middle one again there we are

MATT: The middle one again, there we are.

getting pure coincidence I'll pick them up again and then we'll do it one last

OK purely coincidence, I'll pick them up again.

time again dividing by three and this is

And then we'll do it one last time again dividing by 3, and

why 27 is 3 cubed if you say which one

this is why 27 is 3 cubed.

it's in I will know having memorized all

If you say which one it's in I will know, having memorized

the cards exactly 1 in 1 I will know

all the cards, exactly one in one, or I will know precisely

precisely which card it is and that's

which card it is.

just the pure information of this trick

And that's just the pure information of this trick.

which ones do all over that cook ok so

Which one's it in? BRADY: That one. MATT: That one over there. Cool, OK.

now to be fair all of that wasn't true

So now to be fair all of that wasn't true.

were the numbers were true and the

Well the numbers were true, and the number of cards it

number of cards it could have been going from 27 to 9 to 3 to 1 that's completely

could've been going from 27 to nine, to three, to one, that

accurate I wasn't bothering to memorize

is completely accurate.

and I was doing something else slightly

I wasn't bothering to memorize them though, I was doing

different um what was your card you can

something else slightly different.

tell me now it was the king of her King

What was your card? You can tell me now. BRADY: It was the king of hearts.

of Hearts present what was your favorite

MATT: King of hearts, and what was your favorite number?

number 10 okay watch this here we go

ready 1 2 3 4 5 6 7 8 9 10 King of

MATT: OK. Watch this. Here we go. Ready?

One, two, three, four, five, six, seven, eight, nine, 10.

Hearts so this trick you can put the

King of hearts. So this trick, you can put the card-- even though you don't

card if you know you don't know what it is as long as they tell you which pilots

know what it is-- as long as they tell you which pile it's

in you can put it anywhere in that deck

in, you can put it anywhere in that deck.

so if you say any number after 3 lots of

So if you say any number, after three lots of dealing it

dealing it out I can put the card into

out, I can put the card into that position.

that position and that is my all-time

And that is my all-time favorite

favorite maths paste card trick do you

maths based card trick.

wonder how it works yes this is this is

Do you want to know how it works?

BRADY: Yes please.

ok so can I have some of your famous brown paper alright ok excellent now

MATT: This is brilliant. OK, so can I have some of your famous brown paper?

let's look at why this trick works now

OK, excellent. Now let's look at why this trick works.

ok you're going to bear with me here I'm

Now you're going to have to bear with me here.

going to set up a slightly unusual way

I'm going to set up a slightly unusual way

to look at the cards because when you

to look at the cards.

get the 27 cards the very last step if

Because when you get the 27 cards, the very last step-- if

we go from the end of the trick I pick

we go from the end of the trick--

them up into there were three piles of nine cards let's say from now I'm going

I pick them up into three piles of nine cards.

to call the top one the zeroth pile and

From now on I'm going to call the top one the 0th pile, and

then the first pile on the second pile

then the first pile, and the second pile.

and there's a reason for that in a

And there's a reason for that in a moment, but just bear

moment but just bear with me while I set

with me while I set up some notation.

up some notation so when the cards go back together there are nine cards in

So when the cards go back together there are nine cards

the top pile 1 2 3 4 5 6 7 8 9 so that

in the top pile one, two, three, four, five, six, seven,

was why I called the 0th pile on top

eight, nine. So that was why I called the 0th pile on top.

then there was 1 2 3 4 5 6 7 8 9 in the

Then there was one, two, three, four, five, six, seven,

first pile and the bottom one 1 2 3 4 5

eight, nine in the first pile. And the bottom one-- one, two, three, four, five, six, seven,

eight, nine, that was the second pile.

that was the second part and as it turns out your one was the King of Hearts that

And as it turns out your one was the king of hearts.

ended up being the tenth card down that

That ended up being the 10th card down.

was because you said the very beginning you said if ever number is ten and

Because you said at the very beginning your favorite number

you're King of Hearts ended up there and

is 10, and your king of hearts ended up there.

so now when you think about it these top

And so now when you think about it these top three from

three from the final pass because this

the final pile-- because this is the very last top, middle,

is the very last top middle and bottom

and bottom pile--

pile that top one came from the previous

that top one came from the previous top pile.

top piles that was the previous 0th pile

That was the previous 0th pile, that was the previous

that was the previous middle pile that was the previous bottom tile that was

middle pile, that was the previous bottom pile.

the previous top middle bottom top

That was the previous top, middle, bottom.

middle bottom and so actually if you

Top, middle, bottom.

watch it you can see how that happens

And so actually if you watch it you can

because I've picked them up from the

see how that happens.

second time I've got the top the middle

Because I've picked them up from the second time.

in the bottom packets each of nine cards

I've got the top, the middle, and the bottom packets.

I put them together

Each are nine cards, I've put them together.

I deal out the next three piles and the

I deal out the next three piles, and the first three

first three come from that top pack of

come from that top pack of nine.

nine and then the next three come from

And then the next three come from that top pack of nine,

that top pack of nine and then the next three from the same top pack a nine so

and then the next three from the same top pack of nine.

that's why over here the top three come from the previous top

So that's why over here the top three come from the

0th pile the next three of each one come

previous top 0th pile. The next three of each one come from the middle pile.

from the middle pile so that's the net first three off the middle next three

So that's the first three off the middle, next three off the

off the middle next three off the middle

middle, next three off the middle.

and I've got nine left that was the previous bottom pile and that's why now

And I've got nine left, that was the previous bottom pile.

I get three from the bottom three for

That's why now I get three from the bottom, three from

the bottom three from the bottom so they

the bottom, three from the bottom.

end up going down like that and then

So they end up going down like that.

actually within those and if you get some cards and you start playing around

And if you get some cards and you start playing around with

with this within the final ordering it

this, within the final ordering it turns out from the

turns out from the very very first time you put them together this is the top

very, very first time you put them together this is the top,

the middle the bottom the top the middle

the middle, the bottom.

the bottom the top the middle the bottom

The top, the middle, the bottom.

the top then don't don't lose too much sleep over exactly why this happens if

The top, the middle, the bottom. And don't lose too much sleep over exactly why this happens.

you get a pack of cards and deal it

If you get a pack of cards and deal it, you'll

you'll start to see why and what you end

start to see why.

up here is this is the ordering from the

And what you end up here is this is the ordering from the

first time we dealt the cards out that's

first time we dealt the cards out.

the ordering from the second time we

That's the ordering from the second time we dealt the cards

dealt the cards out and that's the ordering from the third time we got the

out, and that's the ordering from the third time we dealt

cards out and to get it here at tenth I can see that to get this position here

the cards out. And to get it here at 10th, I can see that to get this

it's the zero zero first or top top

position here it's the 0th 0 first.

middle and so when each time Brady point to where his card was the first time I

Or top, top, middle. And so each time Brady pointed to where his card was the

put that pile back on top the second

first time I put that pile back on top, the second time I

time I put that pile and back on top the third tire put that pile in the

put that pile back on top, The third time I put that pile in

middle sashka to secondary in fact very

the middle. The first time I put that pile back on top. The second time I put that pile back on top. The third time I put that pile in the middle.

do you wanna pick a different number so

In fact, Brady, do you want to pick a different number?

say I told you my favorite number was 13

BRADY: So say I told you my favorite number was 13, what

what would you have done

would you have done?

okay so 13 I need to put 12 cards on top

MATT: OK so 13, I need to put 12 cards on top of that, and

of that and 12 is 1 9 1 3 and no units

12 is one 9, one 3, and no units.

so I'm going to put that on the top the

So I'm going to put that on the top,

middle the middle 13 is 9 10 11 12 13

the middle, the middle. 13 is, nine, 10, 11, 12, 13.

Yesi 0 top middle middle that the way I

Yeah see? 0, top, middle, middle.

work it out is I'm actually working a

But the way I work it out is I'm actually

tower in base 3 because this whole trick

working it out in base-3.

uses base 3 ternary numbers which I

Because this whole trick uses base-3 ternary numbers, which

think are absolutely amazing and the

I think are absolutely amazing.

first time you put the piles back

And the first time you put the piles back together you're

together you're doing the unit's column of your base 3 number the next time you

doing the units column of your base-3 number.

put them back together you're doing

The next time you put them back together you doing 3's

threes column and then the last time

column, and then the last time you're doing the 9's column.

you're doing the ninth column and so when you give me your number I work out that number in base 3 and

And so when you give me your number I work out that number

then that tells me how to put the piles

in base-3, and then that tells me how to put

back together ok so now we're going to

the piles back together.

redo the very first trick I did in

OK so now we're going to redo the very first trick I did in

effective almost slow motion in annotated mode if you will and so you

almost slow motion, in annotated mode if you will.

had a look at one card and then I

And so you had a look at one card, and then I started

started dealing these and then I talked

to you about your favorite number you

And then I talked to you about your favorite number,

said 10 you're looking for the King of

and you said 10.

Hearts and I'm thinking how am I gonna

You're looking for the king of hearts, and I'm thinking how

get that King of Hearts or I don't know

am I going to get that king of hearts?

what card is how am I going to get

Well I don't know what card it is.

whatever the card is to the 10th

How'm I going to get whatever the card

position and 10 9 goes into that once

is to the 10th position? And 10, nine goes into that once.

and so I want to get 9 cards on top so I

And so I want to get nine cards on top.

actually have to put it in the top the

So I actually have to put it in the top, the top, and then

top and in the middle so has the King of

Hearts come past yeah where was it it

So has the king of hearts gone past?

was there ok so now I now know it has to

Where was it? BRADY: It was there.

go top top middle so when I pick them up

MATT: OK so I now know it has to go top, top, middle.

from left to right these two I don't

So when I pick them up from left to right-- these two I

care about that can be bottom that can

don't care about-- that can be bottom, that can be middle.

be middle the King of Hearts is in this one so it has to go on top which means

The king of hearts is in this one, so it has to go on top.

it's going to be one of the first nine

Which means it's going to be one of the first nine

to get dealt out and so it's going to be

to get dealt out.

either the top card or the next piles or

And so it's going to be either the top card of the next

the second card in the next part which

piles, or the second card of the next piles, which it

happens to be over the third card and then the rest

happens to be, or the third card.

we actually don't care about because

And then the rest we actually don't care about.

there's other two piles I know it wasn't

Because those other two piles I know it wasn't in those.

in those these are just padding to get it into the correct position so now

These are just padding to get it into the correct position.

which one was it

So now which one was it in?

middle one okay so again it's top top

The middle one? OK so again it's top, top, middle.

middle says to go top again if you watch

So it has to go top again.

when I picked them up I still pick them

And if you watch, when I pick them up I still pick them up

up in the same order but I put them

in the same order.

together in a different order so that

But I put them together in a different order.

goes on top and then I get this last one

So that goes on top, and then I'll get this last one, and

and I'll just shove it underneath so now

I'll just shove it underneath.

I know it's on top in fact I know it's

So now I know it's on top.

in the top three of the top pile so when

In fact I know it's in the top three of the top pile.

I go down this time it has to be the top

So when I go down this time it has to be the top

cart there it is

card, there it is.

and then the rest go on top and then the

And then the rest go on top, and then the last time it has

last time it has to go in the middle and

to go in the middle.

so you can kind of see what's going to

And so you can see what's going to happen now.

happen now because if it goes in the

Because if it goes in the middle it's going to get nine

middle it's going to get nine cards put

cards put on top of it.

on top of it it's going to be the top card in the middle pile it's going to be

It's going to be the top card in the middle pile, it's going

the tenth card so it was in this one

to be the 10th card.

well how about that pick that one up

So it was in this one?

first pick that one up and put it

Well how about that? Pick that one up first, pick that one up and put it

underneath so it's the middle one put that one underneath like that and so now

underneath. So it was the middle one, put that one underneath like that,

it has to be the tenth card one two

and so now it has to be the 10th card.

three four five six seven eight nine so

One, two, three, four, five, six, seven, eight, nine, boom.

in fact one way you can think about it

So in fact one way you can think about it is I like

is I drawing a time versus card height diagram and so the first time you do it

drawing a time versus card height diagram.

this is the first time you deal out

So the first time you do it-- this is the first

you've got the bottom pack you've got

time you deal out-- you've got the bottom pack, you've got the middle pack,

the middle pack and you've got the top

and you've got the top pack when you

pack when you put them back together and

put them back together.

the reason I use 0 1 & 2 is that your

And the reason I use 0, one, and two is actual units column

units column in ternary the second time

you've got the bottom pack you've got

The second time you've got the bottom pack, you've got the

the middle pack you've got the top pack

middle pack, you've got the top pack, and again that's 0,

and again at 0 1 & 2 and that's the

second time you deal and then the third

And that's the second time you deal.

time you deal again you've got the

And then the third time you deal, again you've got the

bottom the middle pack and the top

bottom, the middle pack, and the top, and

bandits are at one enters they're the 3

that's 0, one, and two.

packs when you put them back together

So there are the three packs when you

and in fact this is your units column or

put them back together. And in fact this is your units column, or

that that's your ones column that

that your 1's column.

there's your threes column and that

That there's your 3's column, and that

there's your 9s column so any numbers if

there's your 9's column.

you want to put let's say 15 on top to get 15 you're going to need two threes 1 9 and no units so it's going to go top

So if you want to put 15 on top, to get 15 you're going to need two 3's, one 9, and no units.

bottom middle to put 15 cards on top and

So it's going to go top, bottom, middle. To put 15 cards on top.

end up being the 16th card if someone at

And it'll end up being the 16th card.

BRADY: If someone at home wants to do this trick do they

do this trick do they have to be pretty good at maths there you have two options

have to be pretty good at maths?

you can either be pretty good at maths

MATT: You have two options.

or you can spend a lot of your free time

You can either be pretty good at maths, or you can spend a

practicing until your brain gets used to

lot of your free time practicing until your brain

doing this which to be fair both exactly

gets used to doing this. Which to be fair, are both exactly the same thing.

the same thing maths is all about practicing something

Maths is all about practicing something, and developing a

and developing a new way of thinking for

new way of thinking for your brain to get used to it.

your brain to get used to it so I mean either option learn maths or learn card

So either option, learn maths, of learn card tricks.

tricks you're ending up with the same

You're ending up with the same skill set to be honest.

skill set to be asked you said at the start this was your favorite trick to

BRADY: You said at the start this was your favorite trick

some extent it is what what is that that

to some extent. MATT: It is.

there are lots of tricks what is it

BRADY: There are lots of tricks.

about that one that resonates with you

What is it about that one that resonates with you?

well I like the on one hand it takes a

trick a lot superman' people know the 21

MATT: People know the 21 card trick, where you put it back

card trick where you put it back in the middle each time and then it ends up

in the middle each time, and then it ends up being the

being the middle card and so people kind

of know that they don't know why it

And so people kind of know that, but they

works whereas this one you know why it

don't know why it works. Whereas this one you know why it works, and then you can do

works and then you can do it so much

so much more with it.

more with it and there's a huge difference in math did anything between

And there's a huge difference in math-- indeed in anything--

just memorizing the steps so you know

between just memorizing the steps so you know how to do

how to do it versus knowing why those

it, versus knowing why those steps get you

steps get you where you want to be and

where you want to be.

so this utilizes the advantage of

And so this utilizes the advantage of knowing why the

knowing why the steps are doing

steps are doing something, and then you can

something and then you can tweak it as

tweak it as you go.

you go so instead of always putting it

So instead of always putting it in the middle you can put

in the middle you can put it anywhere you want because you understand how it works

it anywhere you want, because you understand how it works.

because you're putting three piles back

Because you're putting three piles back together three

together three times there are 27 possible arrangements of putting it back

times there are 27 possible arrangements of putting it

across the trick which correspond to all

back across the trick, which correspond to

27 possible positions in fact you can do

all 27 possible positions.

this trick with a lot more cards if you

In fact you can do this trick with a lot more cards if you

really want to you can do it with as

really want to.

many it's the number of piles to the

It's the number of piles to the power of how many times

power of how many times you deal the

you deal the cards out.

cards out if you get 10 billion cards which is a lot of cards and you deal

If you get 10 billion cards, which is a lot of cards, and

them out into 10 piles ten times you can

you deal them out into 10 piles 10 times, you can put

put any of those ten billion cards into any position just through ten deals

any of those 10 billion cards into any position

although middlee you aren't you are

just through 10 deals.

dealing and million cards into each

Although admittedly you are dealing a million cards into

parser to take a very long time in fact

each pile, so it does take a very long time.

in Martin Gardner's book magic maths and

In fact in Martin Gardner's book Magic, Maths, and Mystery

mystery he describes it if you want to do the 10 million card version his sorry

he describes that if you want to do the 10 billion card

10 billion card version his recommendation is to be very very

version his recommendation is to be very, very careful as

careful as you're doing the 10 piles of

you're doing the 10 piles of 10 each time.

10 each time because if you make a mistake very few audiences will sit through that

Because if you make a mistake very few audiences will sit

trick for a second time

through that trick for a second time.

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