Beautiful Card Trick – Numberphile


MATT: OK, so I’m going to
do a card trick based on the number 27. And this is my all-time favorite
maths card trick. And I’m going to show it for
you today, and I’m going to explain it. I found this trick in an old
1950’s math book written by Martin Gardner. And for me it is the maths
card trick with the most beautiful maths behind it
out of all of them. And because it is a math card
trick, it does involve a lot of long tedious counting. But bear with us here. So this involves 27 cards, so
I’m going to take 27 off. And this is a genuine count. One, two, three, four, five,
six, seven, eight, nine, 10. 27 is actually one of my
favorite numbers– One, two, three, four, five,
six, seven, eight, nine, 10– because it’s a cubed number. One, two, three, four,
five, six, seven. OK, that’s 27 cards. And this works with any 27
cards, and none of this trick is slight-of-hand. None of it is YouTube magic
where I’m using something sneaky, or a sneaky edit. And I’ll explain the trick
afterwards, so it’s OK. But this is how works– You get 27 cards, you
shuffle them up. I’m actually going to
get Brady to both film and be the volunteer. So I’ll flick through, do you
want to tap which one you want, which one of these? OK that one there. Do you want to show you
the camera that card? Don’t let me see
it, obviously. And do you want to put it back
in wherever you want? Thank you. Now all he needs to do is just
remember what that card was, and believe me, people in the
comments will mention afterwards if you don’t. Brady what’s your favorite
number from 1 to 27, if you had to pick a number? BRADY: 10. MATT: 10, any particular
reason why 10? BRADY: I just like
how it looks. MATT: You like it? OK. Are you looking for your
card by the way? What I want you to do is have
a look, and see if you can spot which pile your
card goes into. And people may have seen
this trick done before. It’s a variation, in fact, it’s
a generalization on a 21 card trick. Which pile is it in? BRADY: It’s in that pile. MATT: In the middle
pile there? OK, I’m going to pick
them up from the viewer’s right to left. And what people tend to do is
they do this tedious counting out each time. And what I’m actually
doing is last time I memorized all the cards. And so I when you told me which
pile, I had narrowed it down to nine possible
cards it could be. If I do it again, because of the
way I’m dealing it out, if you tell me which pile it’s in
this time, I will narrow it down to one of three
possible cards. Which pile is it in this time? BRADY: This time it is
in the middle pile. MATT: The middle one again,
there we are. OK purely coincidence, I’ll
pick them up again. And then we’ll do it one last
time again dividing by 3, and this is why 27 is 3 cubed. If you say which one it’s in I
will know, having memorized all the cards, exactly one in
one, or I will know precisely which card it is. And that’s just the pure
information of this trick. Which one’s it in? BRADY: That one. MATT: That one over there. Cool, OK. So now to be fair all
of that wasn’t true. Well the numbers were true,
and the number of cards it could’ve been going from 27 to
nine, to three, to one, that is completely accurate. I wasn’t bothering to memorize
them though, I was doing something else slightly
different. What was your card? You can tell me now. BRADY: It was the
king of hearts. MATT: King of hearts, and what
was your favorite number? BRADY: 10. MATT: OK. Watch this. Here we go. Ready? One, two, three, four, five,
six, seven, eight, nine, 10. King of hearts. So this trick, you can put the
card– even though you don’t know what it is– as long as
they tell you which pile it’s in, you can put it anywhere
in that deck. So if you say any number, after
three lots of dealing it out, I can put the card
into that position. And that is my all-time
favorite maths based card trick. Do you want to know
how it works? BRADY: Yes please. MATT: This is brilliant. OK, so can I have some of
your famous brown paper? OK, excellent. Now let’s look at why
this trick works. Now you’re going to have
to bear with me here. I’m going to set up a
slightly unusual way to look at the cards. Because when you get the 27
cards, the very last step– if we go from the end
of the trick– I pick them up into three
piles of nine cards. From now on I’m going to call
the top one the 0th pile, and then the first pile, and
the second pile. And there’s a reason for that
in a moment, but just bear with me while I set
up some notation. So when the cards go back
together there are nine cards in the top pile one, two, three,
four, five, six, seven, eight, nine. So that was why I called
the 0th pile on top. Then there was one, two, three,
four, five, six, seven, eight, nine in the first pile. And the bottom one– one, two,
three, four, five, six, seven, eight, nine, that was
the second pile. And as it turns out your one
was the king of hearts. That ended up being the
10th card down. Because you said at the very
beginning your favorite number is 10, and your king of
hearts ended up there. And so now when you think about
it these top three from the final pile– because this is
the very last top, middle, and bottom pile– that top one came from the
previous top pile. That was the previous 0th pile,
that was the previous middle pile, that was the
previous bottom pile. That was the previous
top, middle, bottom. Top, middle, bottom. And so actually if you
watch it you can see how that happens. Because I’ve picked them up
from the second time. I’ve got the top, the middle,
and the bottom packets. Each are nine cards, I’ve
put them together. I deal out the next three piles,
and the first three come from that top
pack of nine. And then the next three come
from that top pack of nine, and then the next three from
the same top pack of nine. So that’s why over here the
top three come from the previous top 0th pile. The next three of each one come
from the middle pile. So that’s the first three off
the middle, next three off the middle, next three
off the middle. And I’ve got nine left, that was
the previous bottom pile. That’s why now I get three from
the bottom, three from the bottom, three
from the bottom. So they end up going
down like that. And if you get some cards and
you start playing around with this, within the final ordering
it turns out from the very, very first time you put
them together this is the top, the middle, the bottom. The top, the middle,
the bottom. The top, the middle,
the bottom. And don’t lose too much sleep
over exactly why this happens. If you get a pack of cards
and deal it, you’ll start to see why. And what you end up here is this
is the ordering from the first time we dealt
the cards out. That’s the ordering from the
second time we dealt the cards out, and that’s the ordering
from the third time we dealt the cards out. And to get it here at 10th,
I can see that to get this position here it’s
the 0th 0 first. Or top, top, middle. And so each time Brady pointed
to where his card was 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. 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. In fact, Brady, do you want to
pick a different number? BRADY: So say I told you my
favorite number was 13, what would you have done? MATT: OK so 13, I need to put
12 cards on top of that, and 12 is one 9, one 3,
and no units. So I’m going to put
that on the top, the middle, the middle. 13 is, nine, 10, 11, 12, 13. Yeah see? 0, top, middle, middle. But the way I work it
out is I’m actually working it out in base-3. Because this whole trick uses
base-3 ternary numbers, which I think are absolutely
amazing. And the first time you put the
piles back together you’re doing the units column of
your base-3 number. The next time you put them back
together you doing 3’s column, and then the last time
you’re doing the 9’s column. And so when you give me your
number I work out that number in base-3, and then that
tells me how to put the piles back together. OK so now we’re going to redo
the very first trick I did in almost slow motion, in annotated
mode if you will. And so you had a look at one
card, and then I started dealing these. And then I talked to you about
your favorite number, and you said 10. You’re looking for the king of
hearts, and I’m thinking how am I going to get that
king of hearts? Well I don’t know
what card it is. How’m I going to get
whatever the card is to the 10th position? And 10, nine goes
into that once. And so I want to get
nine cards on top. So I actually have to put it in
the top, the top, and then the middle. So has the king of
hearts gone past? Where was it? BRADY: It was there. MATT: OK so I now know it has
to go top, top, middle. So when I pick them up from left
to right– these two I don’t care about– that can be
bottom, that can be middle. The king of hearts is in this
one, so it has to go on top. Which means it’s going to
be one of the first nine to get dealt out. And so it’s going to be either
the top card of the next piles, or the second card of
the next piles, which it happens to be, or
the third card. And then the rest we actually
don’t care about. Because those other two piles
I know it wasn’t in those. These are just padding to get it
into the correct position. So now which one was it in? The middle one? OK so again it’s top,
top, middle. So it has to go top again. And if you watch, when I pick
them up I still pick them up in the same order. But I put them together
in a different order. So that goes on top, and then
I’ll get this last one, and I’ll just shove it underneath. So now I know it’s on top. In fact I know it’s in the top
three of the top pile. So when I go down this time
it has to be the top card, there it is. And then the rest go on top, and
then the last time it has to go in the middle. And so you can see what’s
going to happen now. Because if it goes in the middle
it’s going to get nine cards put on top of it. It’s going to be the top card in
the middle pile, it’s going to be the 10th card. So it was in this one? Well how about that? Pick that one up first, pick
that one up and put it underneath. So it was the middle one, put
that one underneath like that, and so now it has to
be the 10th card. One, two, three, four, five,
six, seven, eight, nine, boom. So in fact one way you can
think about it is I like drawing a time versus
card height diagram. So the first time you do
it– this is the first time you deal out– you’ve got the bottom pack,
you’ve got the middle pack, and you’ve got the
top pack when you put them back together. And the reason I use 0, one, and
two is actual units column in ternary. The second time you’ve got the
bottom pack, you’ve got the middle pack, you’ve got the top
pack, and again that’s 0, one, and two. And that’s the second
time you deal. And then the third time you
deal, again you’ve got the bottom, the middle pack,
and the top, and that’s 0, one, and two. So there are the three
packs when you put them back together. And in fact this is your
units column, or that your 1’s column. That there’s your 3’s
column, and that there’s your 9’s column. 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. So it’s going to go top,
bottom, middle. To put 15 cards on top. And it’ll end up being
the 16th card. BRADY: If someone at home wants
to do this trick do they have to be pretty
good at maths? MATT: You have two options. You can either be pretty good
at maths, or you can spend a lot of your free time practicing
until your brain gets used to doing this. Which to be fair, are both
exactly the same thing. Maths is all about practicing
something, and developing a new way of thinking for your
brain to get used to it. So either option, learn maths,
of learn card tricks. You’re ending up with the same
skill set to be honest. BRADY: You said at the start
this was your favorite trick to some extent. MATT: It is. BRADY: There are
lots of tricks. What is it about that one
that resonates with you? MATT: People know the 21 card
trick, where you put it back in the middle each time, and
then it ends up being the middle card. And so people kind of
know that, but they don’t know why it works. Whereas this one you know why it
works, and then you can do so much more with it. And there’s a huge difference in
math– indeed in anything– between just memorizing the
steps so you know how to do it, versus knowing why
those steps get you where you want to be. And so this utilizes the
advantage of knowing why the steps are doing something,
and then you can tweak it as you go. So instead of always putting it
in the middle you can put it anywhere you want, because
you understand how it works. Because you’re putting three
piles back together three times there are 27 possible
arrangements of putting it back across the trick,
which correspond to all 27 possible positions. In fact you can do this trick
with a lot more cards if you really want to. It’s the number of piles to the
power of how many times you deal the cards out. If you get 10 billion cards,
which is a lot of cards, and you deal them out into 10 piles
10 times, you can put any of those 10 billion cards
into any position just through 10 deals. Although admittedly you are
dealing a million cards into each pile, so it does take
a very long time. In fact in Martin Gardner’s book
Magic, Maths, and Mystery he describes that if you want
to do the 10 billion card version his recommendation is
to be very, very careful as you’re doing the 10 piles
of 10 each time. Because if you make a mistake
very few audiences will sit through that trick for
a second time.

100 thoughts on “Beautiful Card Trick – Numberphile

  1. Could someone show the math behind why the 21 card trick works? I wrote out the numbers for the position of the card and found it ends up in the 10th 11th or 12th position, but I'm not seeing why mathematically.

  2. This is a brilliant enhancement to the famous 21 card trick. I have always been intrigued as to why the 21 card trick worked. Your insight was very helpful.

  3. Maths is not about practice.. By practice, you develop a habit which without regular practice, you may forget how it worked in the first place.

  4. I presented this card trick to my class when I was 11 and I immediately became a permanent nerd. Thanks Numberphile ๐Ÿ‘๐Ÿฝ

  5. Just wanted to point out (as was noted below by Sonja Quan) that this trick works equally as well with 64 cards. You have to convert the chosen number (number of cards from the top), minus 1, to base 4 using 1, 4, 16 instead of 1, 3, 9, and number the positions in the deck 0 to 3, top to bottom, and deal 4 piles each time (3 times), Otherwise it works the same way. This can be extended similarly to any deck of N-cubed different cards. Other generalizations are possible, if you don't mind dealing LOTS of cards.

  6. You can also add the digits together and what group of three the sum is in determines the position, so 1 2 and 3 are the top 3, 4 5 and 6 are the middle 3, and 7 8 and 9 are the bottom 3. Then you find the final position by which group of nine the chosen number itself is in: 1-9, 10-18, or 19-27. For instance, if 25 the chosen number, 2+5 is 7, and 7 is the top number of the bottom 3, and 25 is in the bottom 9, so in the order is top bottom bottom

  7. Even if you know the trick its still magical! It's not a slight of hands but a slight of brains trick. Very impressive. Hello from the future past.

  8. To make the trick even better, I make a premonition of the card on the final turn. I write downย the card and its position, namely the favourite number chosen by the amazed party. It's easy to do.ย  To locate the card on the turn simply count from either 0,9 or 18 until you reach the favourite number. Eg. to reach 25, count 7 turns of 3 from 18. To reach 14, count 5 turns from 9. Just memorize these 3 cards and when your guest choses the final pile, continue to place it as normal. You now know the card and can prognosticate it by writing it down before the big reveal. You can even do it without seeing the cards on the first two turns. Have pulled it off, to much bemusement.ย Many thanks for the inspiration for it Matt.

  9. Oh yea!!! The 21 Card trick. Don't know how that works though. lolol. But this 27 card trick is pretty cool.

  10. "Very few audiences would sit through that trick a second time."

    That was obviously a tongue-in-cheek joke by Gardner, perhaps to end the article. Nobody could sit through it one time. Assume you could deal two cards per second. You deal 10 billion cards 10 times, so that's 100 billion deals. That would take 50 billion seconds, or 1584 years.

    Not only that, but each pile of a billion cards would be over 180 miles high.

  11. I love that you can do this trick with pretty much any number. As long it a multiplication of any other numbers it's is possible to do this one. My favourite is probably doing it with 25 cards, many are confused because I have to do the ordering only twice, whileas they kinda understand the 27 card trick. It's also fun to do it with 24 cards, but I haven't fully grasped that one yet, but it works ๐Ÿ˜‰

  12. guys, it' s simple. pick the number, subtract 1 to it, make modulo 3 number from 000 to 222. read from right to left :0 means top, 1 middle, 2 under.

    remember to subtract 1 to the starting number or you will get mad because the card appears always of 1 position later to what expected!!!

  13. I've tried this several times & keep ending up with the card 1 after the number the person selects. Confirmed that my conversion to base-3 is right… any ideas of what I'm doing wrong?

  14. You can make this trick even more impressive my using all 52 cards and the two jokers and asking the person to pick a number 1-54. To start with you split the pack into two piles of 27, and ask what pile their card is in. If they choose a number 27 or under, you can discard the pile they didnโ€™t pick and to the trick as normal. If they pick a number 28 or over, you have to put the other pile to one side, do the trick with the number they picked minus 27, then add the other 27 cards back to the pack at the end. In fact, this way, you can just ask the other person to think of a card rather than pick on, giving the illusion of reading minds.

  15. +Numberphile Could you provide the links to the recommended viewing shown at the end of this video in such a way that mobile viewers can follow them? Annotations don't show up when viewing from the mobile app, so if you could either add them to the description, put them in a pinned comment, or use some of those boxes built specifically for linking to other videos in place of the annotations in the video itself, that would be great, thanks.

  16. This can actually be done with any number to any power (so long as you have enough cards). The amount of cards you need would be A^B where 'A' is the number of equal piles and 'B' is the number of times you sort the piles. And also, the number chosen by the participant can be larger than the amount of cards used

  17. +Numberphile I feel like there must be some parallel here with a some kind of sorting algorithm, can we get a Computerphile crossover?

  18. I learnt a variation to this trick, but you know their card by memorising the bottom card in the deck and stacking them in such a way that their card is underneath the card you memorised. You then know their card and make a stack with how many letters it takes to spell their card. Then stack the rest of the cards however you want. Get them to tell you their card and then ask them to spell their card out whilst putting the stack you made down and then the bottom card should be the card they chose

  19. I tried this on my friends today, with the added bonus that I did actually tell them what their card was before turning it over at the end. The way I did it was I took their chosen number (17) mod 9 = 8, and from that figured out that their card will be the 8th one dealt to its pile on the last go. So I memorized the 8th card dealt to each of 3 piles, then told them what their card was when they pointed to their pile. So in the case of 10, his card will be the first one dealt to its pile on the last go. Go ahead, verify that it's true

  20. ๐Ÿ˜€ tried this with my friend ๐Ÿ˜€ he told me 666 ๐Ÿ˜€ had to stop for a little while to do more maths ๐Ÿ˜€ fk… , now im about n^n card trick

  21. We also played this card trick, but we didn't choose any particular number of cards like 27 in this case. We just selected some odd number of cards and same procedure was followed as you explained in this video with one exception. That is, every time we were keeping that pile of card in the middle of other two pile of cards. And finally at the end, guessed card exactly falls middle of the total number of cards.

  22. done as the same with the 21 trick/ 3 piles, 21 cards, and the chosen card always ends in the 11th spot after three passes

  23. i still don't get it, everytime that I try it, i dont know where to put the pile he picked, on the ist and 2nd shuffle, the chosen pile with his chosen card, I put it on the top pile until the last shuffle, i always put his pile on the middle and ended up being a shame

  24. Best part of the trick is when you explain it as you are doing it.
    Its actually really simple, all I do is memorizing all the cards so I can work out where it is, and your audience thinks you are rainman ๐Ÿ™‚

  25. Memorising the steps VS knowing why the steps get you where you want to be… that's a very well articulated way of saying what I've been saying all my life (math's got no need for memorisation)

  26. Great math! But it is also possible to use base 2 and 36 cards, base 3 and 27 cards, base 4 and 16 cards, base 5 and 25 cards, base 6 and 36 cards, base 7 and 49 cards (my favourite!). A great video by Matt! ๐Ÿ™‚

  27. A magician once showed me a deck of cards and told me to pick a card so I grabbed the whole deck of cards off him and put them in my pocket. They're my cards now!

  28. I was literally in awe for this was exactly the card trick we used to play when we were in elementary…can't even remember who taught us this trick lol..glad i stumbled upon it again for i totally forgot how to do it again haha

  29. For 15 as favourite card it should be bottom middle and middle. 2+3+9. BTW thanks for the knowledge share. It can be generalized for anything.

  30. I decided to repurpose this trick with 16 cards, four rounds of two piles, to teach children binary. I think you could make another video like this, it's a quicker trick and more curriculum friendly…

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