Dr. Robert Kaplan: “Mathematics: Learning to Speak our Lost Native Language” | Talks at Google

Welcome to all of you. Welcome to “The
Math Circle,” which is extremely peculiar because
we never teach anyone anything. It’s a school without
lectures, without textbooks, without grades, without
tests, without homework. And what does it have left? Math and the beauty of it. And collegial conversation,
never competitive, people– a very small group
of people, always. 10 is about our
maximum, with a leader. All the leader does
is put an accessible mystery in front of them. An accessible mystery–
something which really leads behind appearances to
the underlying structure. Our book, “The Art
of the Infinite,” has 10 of those mysteries in it. Hidden structure, which it takes
a lot of imagination, goodwill, conversation with one
another to uncover, to come to grips with, and
still have doubts at the end, but informed doubts. Our motto– well,
one of them is, “Learning to speak our
lost native language,” but another is, “Tell
me and I forget. Ask me and I discover.” And not only do the students
in “The Math Circle” teach one another and me what
that hidden structure is in their discoveries, but they
come to admire one another and to have their
own self-esteem increased because they’re
not being told things. They’re discovering them. But look, what am I doing? I’m telling you things. This is wrong. You should be
discovering for yourself what “The Math Circle” is, so
let’s give you a demonstration. Guys, here’s the question. I’m going to draw a grid– a five-by-five grid. By grid, I mean kind of a board. Of five lines. Do I have five
spaces for squares? No, only four. And this is weird. I need six lines to get
five horizontal spaces, so I guess I’m going to need
six vertical lines to get– how many squares do I have? AUDIENCE: 25. ROBERT KAPLAN: Are you sure? AUDIENCE: Yes. ROBERT KAPLAN: OK. I’ll take your word for it. AUDIENCE: Count them. Yeah. ROBERT KAPLAN: One,
two, three, four, five– AUDIENCE: [INAUDIBLE] count. ROBERT KAPLAN: I was counting. Did you hear me say one, two– AUDIENCE: Just do
it line by line. Just do it line by line. That would be easier. Like, you could even lose
count, so you’re like uhhh. You should actually mark– ROBERT KAPLAN: You have
a very orderly mind. OK. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Is that OK? 11, 12, 13, 14, 15,
16, 17, 18, 19, 20. AUDIENCE: You’re doing it again. Hey, wait! ROBERT KAPLAN: Oh, I forgot. Sorry. I’ll take your word for it. There are 25. AUDIENCE: Mark the boxes
as you do it to see what you already counted. ROBERT KAPLAN: Wonderful idea. OK. OK. I’m going to learn
a lot from you guys. If you were to start in the
upper left-hand square– do I have that right? AUDIENCE: Yes. ROBERT KAPLAN: OK, thank you. If you were to start there
and draw a line that’s going to go through
all the squares, each square once and
only once, and never take your chalk off the board
or your pen off the paper– you can only go up and down
and left and right, OK? Is it possible to do
that, starting from here, or you know, probably
not possible? Yes? What do you think? AUDIENCE: I think it’s possible. ROBERT KAPLAN: You
think it’s possible? I don’t think so. AUDIENCE: Can you take turns? ROBERT KAPLAN: Yeah,
come up and show me. AUDIENCE: So can you go
down and then take a turn? ROBERT KAPLAN: Oh. Fantastic. OK, I’m convinced. Thank you. That’s a great idea, thank you. Do you agree? AUDIENCE: Yes. ROBERT KAPLAN: That’s the
only way you can do that. That’s terrific. Samuel, what? AUDIENCE: You can go– ROBERT KAPLAN: Come
up and show me. OK, all yours, Samuel. Oh. Oh, that’s neat. You’re not going to be able
to get to the middle one– oh, you’ve got to the middle one. Terrific. Terrific. Thank you very much. So there are two
ways of doing it. Yes, Petra? AUDIENCE: There are
three ways of doing it. There’s more! ROBERT KAPLAN: Come on, really? AUDIENCE: There’s more than
three, but I’ll show you one. ROBERT KAPLAN: You’re
saying there’s a third way? AUDIENCE: Yeah. ROBERT KAPLAN: Yes, go ahead. AUDIENCE: If she goes
this way, then you can also go horizontally. ROBERT KAPLAN: Oh. You’re doing it kind
of in the rotated way. It’s a [INAUDIBLE] thing
kind of rotated 90 degrees. Beautiful. There are three– AUDIENCE: There’s more. ROBERT KAPLAN: Wait a minute. Wait. Daniel, Daniel. How many ways are
there going to be? AUDIENCE: There’s
probably over 100. ROBERT KAPLAN: Over 100. Over 100? AUDIENCE: There was a lot. ROBERT KAPLAN: Nathan, did
you calculate this, or? AUDIENCE: No. You could also go– ROBERT KAPLAN: No, I’m sorry. I should have said
can’t go diagonally. Only up and down
and left and right. AUDIENCE: Ah. ROBERT KAPLAN: It
was a brilliant idea. I like diagonal lines, as a
lot of you know, but not here. AUDIENCE: I was going
to leave out this one and then cover that end and
do it like that and yeah. ROBERT KAPLAN: Wait. Wait, wait, show me, Daniel. Without the diagonal. AUDIENCE: OK, so you
go diagonally down and you stop here. ROBERT KAPLAN: Right. AUDIENCE: And then
you go up and then you go like that and then you– ROBERT KAPLAN: Could you do
that without using a diagonal, but with an up and down motion? AUDIENCE: Yes. ROBERT KAPLAN: Is that possible? Try it. Yeah. Down all the way, right one,
up all the way, right one– oh, right two. AUDIENCE: There’s also
tons of ways of doing it. ROBERT KAPLAN: Is
this going to work? AUDIENCE: No. No. Wait. ROBERT KAPLAN: Why not? AUDIENCE: No, I should have– not that. ROBERT KAPLAN: Here. Oh, oh. AUDIENCE: OK, I got it. So there, I go like this. ROBERT KAPLAN: Just a minute. I erased the line you want. You want to connect
that back up there. AUDIENCE: Yeah. ROBERT KAPLAN: Oh, that’s nice. You’ve got a kind of a
crenellation of a castle. AUDIENCE: And then– no,
instead like that, yeah. Sorry. ROBERT KAPLAN: 1, 2, 3, 4, 5. AUDIENCE: Sorry, like that. And then you– ROBERT KAPLAN: I
think 1, 2, 3, 4, 5– 1, 2, 3, 4, 5. You were right. This is an extra box of mine. AUDIENCE: No, it’s just– ROBERT KAPLAN: No it’s not? OK, then go ahead. AUDIENCE: It’s not? ROBERT KAPLAN: Go ahead. It’s going to work, I think. AUDIENCE: Did I do this one? It’s an E! He did it right. It’s an E! AUDIENCE: It’s a double F. AUDIENCE: Oh, it’s– ROBERT KAPLAN: You want an E? AUDIENCE: E. ROBERT KAPLAN: The letter E. NATHAN: Oh, it’s actually– PETRA: Go up. Turn. ROBERT KAPLAN: Now what? DANIEL: What about this one? AUDIENCE: Is that a box? ROBERT KAPLAN: Go ahead. DANIEL: Up? LEAH: Do they all
have to connect? DANIEL: But you see,
you leave with this one. ROBERT KAPLAN: Do they
all have to be what? AUDIENCE: Go left! Go left! LEAH: Do they all
have to connect? ROBERT KAPLAN: Wait, you
had a good question, Leah. LEAH: Do they all
have to connect? ROBERT KAPLAN: They
all have to connect. It has to be one
continuous line, yeah. I think this is
really nice, Daniel. Now, does his work? DANIEL: I got to start from
the beginning by doing that. So you can do it. You can go on forever
just explaining how many ways there are. ROBERT KAPLAN: Wait a second. Nathan says 100. You say forever. NATHAN: Well, it’s about 100. ROBERT KAPLAN: About 100,
which is more or less forever. Dia? NATHAN: Well, a lot. DIA: I have another
way of doing it. ROBERT KAPLAN: Fantastic.
but hold onto it. OK, Daniel, this is great. Thanks a lot. Will you remember Daniel’s? How about Dia, Asher, Petra? PETRA: No, I– ROBERT KAPLAN: Oh, Petra. PETRA: So the question is how
many ways can you draw a line through all of the boxes? ROBERT KAPLAN: That’s
our question, yeah. PETRA: OK, I just wanted to
make sure I knew the question. ROBERT KAPLAN: Yeah, I’m
not sure it’s the question. It’s one of the questions
that I thought of. And you’ve come up
with fantastic answers. 1, 2? No. 3? No. 4? No. 100? No. An infinite number. Asher and then Leah. While I’m doing this,
think to yourself, besides the question of
how many ways are there– we know there are
at least 3 or 4. There might be 100. There might be an
infinite number. ASHER: No, there’s not
an infinite number. ROBERT KAPLAN: Why not? ASHER: Because
eventually, you’re going to actually have to have
an infinite number of boxes to have an infinite number of– ROBERT KAPLAN: Ah,
what a great idea. OK. 1, 2, 3, 4, 5. 1, 2, 3, 4, 5. Oh! This is like Daniel’s. ASHER: Oh. ROBERT KAPLAN: Yeah. [CHUCKLING] The chalk is
telling you something. ASHER: It happened. I don’t think the chalk
likes my equation. [CHUCKLING] ROBERT KAPLAN: Whoa! That’s great! Well, ah! What Asher did was to
combine previous ideas. That’s doing math. That’s taking previous– I
promise I’ll come to you, Leah, immediately. Building on these
ideas together, you’re working collegially. It’s just super. With the kind of
top of a castle– you see those tops
of the castle? And then some straight
line right down the middle, could it have been elsewhere? And then zip over and
do the zigzags backward. It’s really nice. By the way, your letter
E is hiding in here. Leah? LEAH: I have another way. ROBERT KAPLAN: Different
from all these? LEAH: Yeah. ROBERT KAPLAN: OK, come on up. Maybe there really are
an infinite number. Think about another question
besides how many ways. LEAH: One day,
you’re accidentally going to copy one other one. ROBERT KAPLAN: Sorry? LEAH: One day,
you’re accidentally going to copy one of the
ones that you already did. ROBERT KAPLAN: That’s
such a good question. Is anyone keeping
track of whether we’re going to be doing something
again that we did before? Or your memory’s so
good– mine isn’t– that you’ll remember
all the previous ones? Very good point. Are you worried that
yours might be an old one? LEAH: No, I’m pretty
sure it’s not. ROBERT KAPLAN: OK. If you’re pretty
sure not, that means it must be a
different basic idea. LEAH: I’m actually kind of
surprised nobody did it. ROBERT KAPLAN: OK, huh. It shows that, although
we all have minds, and they’re all terrific,
our minds work differently. Uh huh. Oh! [CHUCKLING] Stop for a second. Is anybody worried? NATHAN: No. ROBERT KAPLAN: You’re worried? Yes, you’re worried. ASHER: I can’t see. ROBERT KAPLAN: Step
aside for a second, Leah. Don’t say why you’re worried. But you’re both worried? Anybody else worried? You’re worried. Go ahead. LEAH: I’m not. I know it’s right. ROBERT KAPLAN: Oh! AUDIENCE: I actually think– ROBERT KAPLAN: Oh! [CHUCKLING] You’re
really seeing things. You’re seeing things
that don’t work, which kind of tells
you what does work. Go ahead. Oh, excellent. You avoided the problem
of having two directions you had to go in. And you could only go in one. Beautiful. I do think that’s new. Isn’t it? That is super. You guys are amazing. Yeah, Dia? DIA: I have another way. ROBERT KAPLAN: OK, come on up. Can I leave this and
give you pink chalk? OK. [CHUCKLING] That’s nice. That’s really nice. Looking at Dia’s pink
one, can you automatically see another one that
hasn’t been done? Automatically? Leah? LEAH: I think I do. ROBERT KAPLAN: Don’t
come up, but tell me. Looking at the pink line,
do you automatically– let me let me recite
the pink line. Start here. Go right all the way. Go down all the way. Go across all the way. Go up part of the way,
up to the fourth square, and then start wiggling down. And stop there. Yes, Nathan? NATHAN: It reminds
me of one way. But I think you’ll
have to erase the pink, if you don’t have
any other colors. ROBERT KAPLAN: OK, yes. So you took part of what
you did and changed it. Nice idea. NATHAN: Wait, actually, no. It shouldn’t be there. ROBERT KAPLAN: I’m
now losing track of whether we’re repeating
any things we had before. That’s terrific. You guys have an enormous
amount of imagination, because math is done
with one’s imagination. Let me ask you a question
before you pick up. You’re basically running
this class, not me. But I’ve got a question. I’m going to take
one of the paths I know was done– in
fact, the very first one. And I want you to
tell me something. Here’s that first path. I think it went like that. Am I right? AUDIENCE: Yeah. ROBERT KAPLAN: Without
doing anything– without asking me
to do anything more, tell me a second
path that’s there. Yes, Dia? DIA: It’s the same thing,
except if you flip it. ROBERT KAPLAN: Start. Finish. Is it a start, finish? Every time you draw one path,
you get one more for free. That means whether the number
is 2, 100, or infinite, it’s going to have to be even. Because for every path you draw,
there’s the path with the end and the beginning reversed. [CHUCKLING] So weird. [CHUCKLING] There are at least 2 paths. There are at least 4. I mean, there are at least 6. I mean, what do I mean? There are at least 100. There might be 99. Yes? ASHER: We have done it 5,
and then because of the point you made, 10. PETRA: We’ve done 7. ROBERT KAPLAN: 10, yeah. ASHER: No, we’ve done 5. I don’t know. PETRA: It was 14. ASHER: [INAUDIBLE] 12? Or 6. ROBERT KAPLAN: I’ve got a
different sort of a question. This has worked so well. And we haven’t even
begun to finally settle on how many times you can do it. Starting here– this
is a weird question– is there any other
place you could have started that would have
worked at all or as well? Leah? LEAH: You can start from
the top right corner. ROBERT KAPLAN: Start here. How do you know
that’s going to work? LEAH: Well, you can just
go the opposite way, and go all the way down. ROBERT KAPLAN: Terrific. Possible start. Possible start. Petra? PETRA: Start from the center,
because you can spiral. And if you– ROBERT KAPLAN: Hey,
that’s wonderful. We had a path that
ended in the center. Therefore, there has
to be a path that starts from the center. Asher? ASHER: So adding off what
she said, that you could start in the top right corner– all the ones we’ve
currently done, you could just turn them right. ROBERT KAPLAN: Rotate
the board instead of rotating our drawing. Dia, what were you going to say? DIA: That any part on the board
would be a possible solution, but you just have
to alter the line. ROBERT KAPLAN: Fantastic. You could start in any square. So far, we know you can start
in this one, in this one, in this one, in this one, and
in the center, or in this one, in this one, in this one,
in this one, or there. I’m just taking the endings and
going back to the beginnings. I seem to have gotten
the same thing. There are at least five
places you can start from. And your suggestion is we can
start absolutely anywhere, and just be careful. I want to take what
you suggested, Dia. Do you feel pretty much in
charge of this question, right now? Yeah, Nathan? NATHAN: So I was
wondering, could I show? ROBERT KAPLAN: Yeah, come on up. NATHAN: Actually, you
don’t have to re-write it. Just do it 2 by 2. ROBERT KAPLAN: What
an interesting idea. NATHAN: For just a second. So could you go like this. ROBERT KAPLAN: You
lifted your chalk. NATHAN: What? ROBERT KAPLAN: You
lifted your chalk. NATHAN: Oh, you can’t
lift your chalk? ROBERT KAPLAN: You
can’t lift your chalk. Nevertheless, you can
do what you wanted, I think, without
lifting your chalk. Here– NATHAN: True. ROBERT KAPLAN: Let me just– NATHAN: I was just wondering if
you could use the start point and push it off two ways? ROBERT KAPLAN: Your
question is can you start– NATHAN: Actually, I could do
it without lifting my chalk. ROBERT KAPLAN: Go ahead. Oh! [LAUGHTER] Watch out for this guy. OK, I’ll be much more precise. You’re asking me to be more
precise in my language. Without lifting your
chalk and without going over the same line
a second time. NATHAN: OK. ASHER: He didn’t go over it. He just went right
next to the line. [LAUGHTER] PETRA: Only once, though. ROBERT KAPLAN: Only once? Wait a minute. Wait a minute. Wait a minute. AUDIENCE: No, wait, you can
go to the edge of the square. ROBERT KAPLAN: I’ve just
drawn that line 7,981 times. NATHAN: You’re
outside of the square. ROBERT KAPLAN: Haven’t I? They’re all next to one another. [CHUCKLING] I think, Nathan,
you could still get your pattern without– NATHAN: You can still
get your pattern if you just go like this. ROBERT KAPLAN: Do it. Oh, yeah. Right. NATHAN: It’s just a question if
you can go two ways, not one. ROBERT KAPLAN: Got you. That’s a really
nice other question, which we should explore. Nathan’s brought up a
completely new question. Take a 5 by 5 grid,
or a 2 by 2 grid. Are there two-fold double
paths that’ll work? And if so, how many? In other words, instead
of going like that, could you go like that? And then like that? Or something like that. Interesting question. Yes, Daniel? DANIEL: You actually can if
you have 2 pieces of chalk. [CHUCKLING] Which we have more than 2. ROBERT KAPLAN: Or
if you have 2 arms, because you have to have 2
hands for the pieces of chalk. If you’re an octopus,
you can do 8. [CHUCKLING] Yes, Asher? ASHER: You can also
program a robot to hold like 3 or
9 little pieces. ROBERT KAPLAN: Very good. I want to get back to
something Dia raised. Going back to our original– whoops– 5 by 5, Dia said– tell me if I’m quoting
you correctly, Dia– that really, you can
start from any square, as long as you’re careful. Right? DIA: I rethought what
I said, and any place that’s next to the corner,
you can’t start there. NATHAN: Actually,
I think you could. ROBERT KAPLAN: What
A fascinating idea. Let’s see. Let’s see. 1, 2, 3, 4, 5. I’ll try to be really
careful in drawing this. You’re saying, starting in
a place next to a corner, where would you like
me to start from? By the way, somebody suggested
at the very beginning that we do something which would
make talking about this work easy. Do you remember what it was? When I was counting
the squares– when I was counting
all 25 of them, and I was counting
this way and that. SPEAKER 1: Oh, you
should go in rows. ROBERT KAPLAN: What? Say it again. SPEAKER 1: You
should go in rows. ROBERT KAPLAN: Yes,
what should go in rows? DIA: The way you’re counting? AUDIENCE: Your fingers? ROBERT KAPLAN: There was
something besides my fingers that somebody suggested I do. AUDIENCE: Dots? ASHER: You could mark every
square you’ve already counted, so you wouldn’t double count. But you could still do the– ROBERT KAPLAN: What do
you want to mark it with? ASHER: What? ROBERT KAPLAN: What do
you want to mark it with? PETRA: Chalk. ASHER: Chalk. Anything! ROBERT KAPLAN: I was thinking,
Asher, of your question about is this all going
to be geometry. It could be something else. ASHER: Well, it’s
not really geometry. But I meant like equation math. ROBERT KAPLAN: What do
you mean equation math? Go on. What do equations involve? ASHER: Numbers? ROBERT KAPLAN: What if I marked
these squares with numbers? AUDIENCE: If you did that– ROBERT KAPLAN: Somebody
said at the beginning, why don’t you just
number the squares? ASHER: It takes
longer, but I was thinking when I said that more
about speed, and not really– ROBERT KAPLAN: I know. But I think– DIA: You can count by 5’s. ROBERT KAPLAN: What? DIA: You can count by 5’s. ROBERT KAPLAN: I’m good at that. But last time, what I did
was I counted in this way. But I could really be very
regular and count like this. If I put these numbers
in just to make it easier to talk about,
when Dia suggested I start in a square next
to a square in the corner, I guess she meant
square number 2. Thanks for filling this in. 21, 22, 23, 24, 25. OK, Dia. DIA: See, it
couldn’t start there, because if you went
this way and this way– ROBERT KAPLAN: Is
this going to work? [INTERPOSING VOICES] NATHAN: Can I say
something about that? ROBERT KAPLAN: Yes. NATHAN: I can show
you how you can do it. If you start right here– ROBERT KAPLAN:
That’s great, Dia. Thank you. NATHAN: –pretty much
exactly what Dia did, except when you go
here, go like this. Well, actually, no. Not that. This– she’s right. [CHUCKLING] AUDIENCE: Nope. ROBERT KAPLAN: She’s wrong? Nathan thinks Dia’s
right that you can’t start from square number 2. NATHAN: Well, I guess
it possibly could– ROBERT KAPLAN: Asher
and then Daniel. Oh, you’ve all got
your hands up at once! Asher first, then
Daniel, then Samuel. NATHAN: I guess you could. ROBERT KAPLAN: I think Leah. Did you have your hand up? NATHAN: It’s not in the way. I tried. ASHER: Watch. [CHUCKLING] ROBERT KAPLAN: Oh! ASHER: That worked better
in my head than actually on the board. ROBERT KAPLAN: It’s
looking as if Dia– ASHER: I just did one
thing wrong, I think. ROBERT KAPLAN: What is
going on with the problem that Dia has raised? PETRA: If he takes it over
here, he can make it easier. Because he doesn’t
have to go like that. He can do it here, instead. Sorry, it’s hard to keep track. ROBERT KAPLAN: Let’s see. So are you saying 2, 1, 6, 7,
8, 3, 4, 5, 10, 9– so far, so good– 14, 15, 20. PETRA: And then you go to 25. ROBERT KAPLAN: Down to 25. Don’t draw that in blue. Go ahead. PETRA: I’m just going to go
over it in blue, because it’s confusing me right now. ROBERT KAPLAN: Is
this going to work? AUDIENCE: Yes. ROBERT KAPLAN: She’s
retracing the previous path. But when she gets to 20, she
goes down to 25, across to 24. After 20, 19, 18,
23, 22, 17, 16. AUDIENCE: I think I did it. ROBERT KAPLAN: And skipped 13. PETRA: Yeah, I skipped 13. That was a mistake. ROBERT KAPLAN: Yes, Leah. LEAH: Well, I have another
way of doing it from the– ROBERT KAPLAN: Another
way that’s going to work? LEAH: I think. ROBERT KAPLAN: Starting with 2? LEAH: Yes. ROBERT KAPLAN: Great. And Samuel, you’ve had your
hand up for a long time. Starting in 2– SAMUEL: Maybe you could go– ROBERT KAPLAN: Now what? You’ve left out 15. SAMUEL: Oh, right. ROBERT KAPLAN: Hmm. [CHUCKLING] NATHAN: I have a question. ROBERT KAPLAN: Yes. NATHAN: Can you put the
same line into squares? ROBERT KAPLAN: What do you mean? NATHAN: Like if
there was one square, could you put two lines in it? ROBERT KAPLAN: Oh, no. No, absolutely not. One, the line is a
straight steady line. And it never goes back
in a square it’s been in. NATHAN: It’s not
exactly moving back. ROBERT KAPLAN:
What is it moving? Helicoptering down. SAMUEL: I have another idea. ROBERT KAPLAN: Dia? DIA: I think I know
a way it would work. You could just spiral it. ROBERT KAPLAN:
Let’s try a spiral. SAMUEL: Hold on, I have
another idea, though. ROBERT KAPLAN: Go on. Go ahead. SAMUEL: It’s similar
to what I did. Oh, hold on. AUDIENCE: What about 19? ROBERT KAPLAN: Yeah,
so we ended 19. AUDIENCE: It’s
exactly what you did. ROBERT KAPLAN:
Yeah, stuck again. Is this something wrong
with our imaginations or with the problem? Leah? LEAH: I think I’ve got
one that would work. ROBERT KAPLAN: Terrific. Dia, you think you have one? How about Dia then Leah? Ask yourself, what is going on? It was so successful
starting from a corner with lots and lots of
an even number of paths. And we can’t even get one
starting from square 2 started. Well, we can get it started. But we can’t go on. 1, 2, 3, 4, 5. 1, 2, 3, 4, 5. And we’re starting in 2. This is going to be your spiral. Ah! [CHUCKLING] It was
beautiful up to there. What is wrong? Why can’t we do this? You guys are so good. Why– DIA: I think we can’t do it,
because the corner piece– ROBERT KAPLAN: Yes? DIA: It’s just kind of
wedged in between two pieces. But this piece has– ROBERT KAPLAN: The
corner piece is wedged in between two pieces. Is that what you mean? DIA: Well, except one
piece has an edge. And you can’t really go past it. ROBERT KAPLAN: Got you. Leah, your idea? Come on up. This is great, Dia. This is a really fascinating
trouble we’re having. I don’t quite
understand why we’re having so much trouble with it. You guys now all know that
you’re really good at this. You’re good at
imagining, counting– you’re actually better, I
think, at counting than I am. And yet just saying, can we
start in the second square from the corner– LEAH: Mine may not
work, but it’s OK. ROBERT KAPLAN: Well, try it. Try it. So far, so good. [CHUCKLING] Oh, thank you. Oh, wait a second. 1, 2, 3, 4, 5. It stops here. LEAH: Oh. ROBERT KAPLAN: Yeah. Stop for a second. Is it OK so far? AUDIENCE: Yes. ROBERT KAPLAN: Do you see
any trouble in the future? SAMUEL: Yes. ROBERT KAPLAN: You
see trouble, Samuel? Where do you see trouble? SAMUEL: When she
gets to the top, she’s going to get the end
of the line and the box right below 2. Aha! OK, go ahead. So far she’s avoided that. Now, look what’s happened. Look what’s happened. You have a choice. If Nathan’s idea that we could
go in two directions at once, we’d be done. But we don’t have that. We can only go in one
direction at once. And you can either go up– you
certainly are welcome to go up, but then you leave that out. You can go left, and
then you leave that one. LEAH: Why is math so hard? ROBERT KAPLAN: Why
is it so difficult? Why doesn’t it work? What’s the trouble with us? Yes? SPEAKER 1: I got a way. ROBERT KAPLAN:
Fantastic, come on up. OK. The old top of the castle. Ah! [CHUCKLING] Wait, wait! Just stay around. I’ll just get rid
of that much, OK? NATHAN: There’s no way
you can get through. ROBERT KAPLAN: You mean
there’s already trouble? It’s too late, already? Leah, what do you think? Too late already? LEAH: Yeah, but I
have another way. NATHAN: Actually, no. It isn’t. ROBERT KAPLAN: No, it’s not. [CHUCKLING] SPEAKER 1: Actually,
can I check my paper? I think I have– ROBERT KAPLAN: I think
your idea was great. NATHAN: If you go
down from here– ROBERT KAPLAN: Yes? NATHAN: Done. ASHER: But there’s a problem. ROBERT KAPLAN: Let’s see. What’s happened here? NATHAN: I’m not using this. ROBERT KAPLAN: OK,
in other words, you’ve missed that square. How did I know that there
was going to be a problem? NATHAN: Because there’s been
a problem every other way. ROBERT KAPLAN: You’re right. [CHUCKLING] That’s
called induction. NATHAN: I saw Dia
got a way to do it. ROBERT KAPLAN: Oh, Dia. Why don’t you all say the minute
you see a problem coming up? We start here. Anybody see a problem coming up? AUDIENCE: No. ROBERT KAPLAN: OK, go ahead. AUDIENCE: 6. ROBERT KAPLAN: OK, so far? OK, so far? AUDIENCE: Yeah. Yes. ROBERT KAPLAN: Why
is it OK, so far? AUDIENCE: Because it is. ROBERT KAPLAN: Yeah, say it Dia. DIA: You can go straight back. ROBERT KAPLAN: We
know she’s only got one choice when she’s here. She has to go this way. And you choose to go
all the way across. Everything OK, so far? NATHAN: Is the grid 6 by 5? ROBERT KAPLAN: It’s
meant to be 5 by 5. It is 5 by 5. NATHAN: No, but I’m asking her. DIA: No, it’s a 5 by 5. ROBERT KAPLAN: OK. AUDIENCE: 6 by 5. ROBERT KAPLAN: Keep that
idea in mind Yes, Asher? Wait, go ahead. Go ahead with what you’ve got. AUDIENCE: Someone
else did that, too. ROBERT KAPLAN: Oops, you’ve
gone on the square twice. AUDIENCE: Someone
else did that, too. ROBERT KAPLAN: You went in
the square twice, again. AUDIENCE: May I help her? ROBERT KAPLAN: You
want to, because you want to get to that square. And that’s the only
way you can do it. And you’re stuck. AUDIENCE: May I try? ROBERT KAPLAN: You’re just
one square off finishing it. DIA: 1, 2, 3, 4, 5. 1, 2, 3, 4, 5. I think I got it. ROBERT KAPLAN:
That’s interesting. 1, 2, 3, 4, 5. 1, 2, 3– yours is 5 by 5. And you’ve done that. AUDIENCE: But are there
intersecting squares? DIA: 1, 2, 3, 4, 5. ROBERT KAPLAN: Should
we number her steps? This is 1, 2, 3, 4, 5. DIA: Oh, it’s a 6 by 5. ROBERT KAPLAN: 6. I’m sorry, that’s because
I started with the number 2 in the corner. It still is 5 squares. I’m numbering, now, the
steps instead of the rows. 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, and 17, 18, 19– DIA: Then I went down again. ROBERT KAPLAN: –20, 21, 22, 23. I’m not sure we’re ever
going to get to that. Wait a second. 1, 2, 3, 4, 5. 1, 2, 3, 4. Ah, sorry. My board seems to
have 1, 2, 3, 4, 5. Yes, it’s 6 wide. That’s interesting. When it was 6– NATHAN: No, it’s not 6 wide. It’s broken by one of
the lines that we drew. ROBERT KAPLAN:
Oh, right you are. Right you are. 1, 2, 3, 4, 5. That line going
through, it’s just going through that one square. Has this worked? DIA: It worked on my paper. NATHAN: I might have– ROBERT KAPLAN: Take a
look at Dia’s paper. So far, so good. By the way, 1, 2, 3,
4, 5 by 1, 2, 3, 4. 1, 2, 3, 4. DIA: There’s 1, 2, 3, 4, 5. ROBERT KAPLAN: Wait a second. Oh, I see. Those are 2. DIA: There’s a tiny box there. ROBERT KAPLAN: 1, 2, 3, 4, 5. 1, 2, 3, 4, 5. OK. So you go down the left hand
side and across the bottom. Up the right– ah, you’ve
gone into this square twice with this line
and with that line. DIA: Oh. ROBERT KAPLAN: Oh, it was
so beautiful up to there! 10 minutes left. How can this be? Can we do this for
another 2 hours, please? [CHUCKLING] DANIEL: Yes. ROBERT KAPLAN:
Let me ask you a– yes, Daniel says
yes, 10 or 2 hours. [CHUCKLING] Is it
going to work when you start from the second square? NATHAN: I– ROBERT KAPLAN: Nathan, what? NATHAN: Can I show you
a way that I did it? I might be able to
do it on the board. ROBERT KAPLAN: Yeah, please. Shall I erase this, or– NATHAN: It’s kind
of the same as this. But I think our problem is
that we’ve been going routes. Oh, this doesn’t work. AUDIENCE: Yeah. NATHAN: I didn’t realize. So I was just estimating that
might be the problem, because I like to estimate some things. ROBERT KAPLAN: That’s good. Well, I liked your estimate
of 100 a while ago. By the way, Asher, you
asked a long time ago, is this going to be geometry? This might be arithmetic
instead of geometry. I don’t know. Asher and then Petra. ASHER: I don’t think this
really counts, though. ROBERT KAPLAN: Oh. Ah! So far, that is the backward
of something that, I think, Dia did. Now, zigzagging. Stop. You’ll never get into the left– yeah, exactly. [LAUGHTER] ASHER: [INAUDIBLE] You didn’t
say you couldn’t go outside! See? ROBERT KAPLAN: Great, yes. I got a funny question to
ask before Petra and Leah. How big is this board? NATHAN: 5 squares by 5 squares. ROBERT KAPLAN: Namely,
how many squares? NATHAN: 5 squared. ROBERT KAPLAN: 5 squared, true. What is 5 squared? NATHAN: It’s 25. ROBERT KAPLAN: It’s 25. What kind of a number is 25? DIA: A square number. ROBERT KAPLAN: It’s a square
number and it’s an odd number. It’s an odd number squared. It’s an odd number. Can I go back to
Nathan’s 2 by 2 board? 2 by 2 is what? What’s 2 times 2? This is very difficult, I know. AUDIENCE: It’s 4. ROBERT KAPLAN: More or less, 4. I mean, you estimate. So it’s kind of 4. And on a 2 by 2 board,
a board with 4 squares, can we start from here? Will it work? AUDIENCE: Yes. ROBERT KAPLAN: Did
we start from here? AUDIENCE: Yes. ROBERT KAPLAN: Here? AUDIENCE: Yes. ROBERT KAPLAN: Here? AUDIENCE: Yes. Dia? DIA: The reason
the 2 by 2 works is that there’s no square in the
middle of either two corner pieces or a corner piece
and a regular piece. ROBERT KAPLAN: Wow. NATHAN: Well, a 3
by 3 would work. ROBERT KAPLAN: So a 3 by
3 would or wouldn’t work? NATHAN: Would. ROBERT KAPLAN: 3 by 3. There’s that middle square. [CHUCKLING] Trouble. 3 by 3 is trouble. 5 by 5 is trouble. What’s 729 by 729? Trouble or easy? NATHAN: Actually, I got it. AUDIENCE: Trouble. ROBERT KAPLAN: Trouble! NATHAN: Well, actually, no. I didn’t. I did it. Because I started from here. ROBERT KAPLAN: Oh, you
started from the first square? NATHAN: No, this square. ROBERT KAPLAN: Oh, you started
from the second square? NATHAN: I went like this,
and this piece of chalk got– oh, wait. Actually, no. ROBERT KAPLAN: Yeah, right. Starting from the second
square on an odd number board– NATHAN: Does not work. ROBERT KAPLAN: –is trouble. Why? Petra? PETRA: There probably
is a way to do it, but it’s hard to find. ROBERT KAPLAN: That’s great. There’s probably a way to do
it, but it’s hard to find. Asher, you were saying there’s
just no way to find it, no matter how you try. Five whole minutes,
say it, Asher. ASHER: So I’m just
going to draw a 3 by 3. ROBERT KAPLAN: OK, good. ASHER: So if you
start from here, you can’t really do
anything about it. Because there’s always going
to be this extra corner. And if you try to
eliminate that corner, one of the middle pieces
is not going to work, see? So here– ROBERT KAPLAN: Right. ASHER: –won’t work– ROBERT KAPLAN: Right. ASHER: –because once
you get to the point where it’s either a
corner or the center, you’re always
going to get there. You have to only go to one. ROBERT KAPLAN: That’s terrific. ASHER: So it’s going to
leave the center out or one of the corners. ROBERT KAPLAN: Is a 6 by 6
board going to work or not? AUDIENCE: Yes. ASHER: It will, because it’s– ROBERT KAPLAN: And a 7 by 7? AUDIENCE: No. ROBERT KAPLAN: Right. Dia? That’s terrific, Asher. DIA: If it’s an even
number by an even number, then it’s going to work. Because the reason– NATHAN: Because there’s
no middle square. DIA: There’s no middle square. ROBERT KAPLAN: There’s
no middle square. DIA: So whatever
you do, it won’t cross that one odd square. ROBERT KAPLAN: That
one what square? DIA: Odd square. ROBERT KAPLAN: That
one odd square. On a 25 square board, 5 by 5,
how many odd numbers squares are there? And how many even
number squares? Yes? LEAH: The odd number
square, since it starts with an odd
number, there’s going to be one
more [INAUDIBLE].. ROBERT KAPLAN: There’s always
one more odd number square than even on an
odd number board. And on an even number board? Yes, Leah? LEAH: 5 by 5, it’s
not going to work. But I think it’s going to. ROBERT KAPLAN: But I think
you’re onto the hidden deep structure of this problem. There’s always an
extra odd number square on an odd number board. And if you’re going
to cover every square, you better start
from an odd one, or you’re not going
to cover them all. Because there’s an
extra odd one somewhere in the center, or in a corner,
or something like that. NATHAN: You can’t
start on an even number on an odd number of squares. ROBERT KAPLAN: Fantastic, guys. Leah, what do you want to say? LEAH: At the end of
this, are we able to ask you a bunch of questions? ROBERT KAPLAN: Please do! [LAUGHTER] I’m not going
to have any answers, but go ahead and ask. LEAH: What are those books? ROBERT KAPLAN: Oh, those are the
books that Eleanor and I wrote. The first one, “Out
Of The Labyrinth” is about this approach,
the math circle approach, where we do math in the
real way with you guys doing the thinking. So you own it. It’s yours. This is a way we invented
with no textbooks, no tests, no quizzes, no being
told, but being asked, and people working together the
way you all did fantastically, to come to the deep,
hidden, underlying structure of the world. The other book, “The
Art Of The Infinite,” which is mathematics, because
mathematics is almost always about the infinite. Infinite number
of possibilities? It has 10 different
problems in it, like this one, which lead
you from confusion and chaos, on the surface, to deep
structure that makes sense. That gives meaning
to the whole thing. [APPLAUSE] These are the people to applaud. AUDIENCE: We have 20 minutes
for question and answers. ROBERT KAPLAN: How about
Petra, Asher, and then parents. AUDIENCE: We’ll give you
one kid and then parents. ROBERT KAPLAN: Petra? PETRA: If you end
on an odd square, you can do it backwards. And it will be the same lines. ROBERT KAPLAN: Absolutely. PETRA: I did it
on an odd square. ROBERT KAPLAN: Absolutely. That’s right. Start on 1, end on 25
in one of our patterns. Start on 25, end on 1
in the paired pattern. Asher? ASHER: Actually, with “The Art
Of The Infinite,” basically, why you named it that. Actually, I’m pretty sure
infinity is not a number. ROBERT KAPLAN: You’re right. It’s like big and small. Infinity is not a
number, but it is an idea we have about how many– well, actually, I’m sorry. How many counting
numbers are there? AUDIENCE: Infinite. ROBERT KAPLAN: Oh, I
thought there were 10,981. AUDIENCE: Wait, how many what? ROBERT KAPLAN: Counting numbers. I thought they end. I thought the last
one was 10,981. AUDIENCE: What about 10,982? ROBERT KAPLAN: Ah! [LAUGHTER] So any time I name a
counting number, you can say, well, there’s one more. And that must suggest there are
not an ended number of them. So the infinite is
hiding behind this. How many boards
does this problem work on for any
possible starting place? An infinite number. All the even number boards. And there are an infinite
number of even numbers. [CLICKING TONGUE] AUDIENCE: Any questions
amongst the audience here? ROBERT KAPLAN: Parents, yes. AUDIENCE: Are we going to do
something like this again? ROBERT KAPLAN: You mean
in the math circle? AUDIENCE: Yeah, like a group. ROBERT KAPLAN: Or maybe
since you’ve already seen this, something beyond this. AUDIENCE: Yeah, maybe. ROBERT KAPLAN: Maybe. So the answer is maybe. [CHUCKLING] AUDIENCE: I have a question. ROBERT KAPLAN: Yes, go ahead. AUDIENCE: In this setting, the
beginning was just a drawing. So everybody started
on the same page. What happens when you need to
use more mathematical notation or machinery that
people in the class may have different
levels of comfort with, coming in to the circle? ROBERT KAPLAN: The kids
should invent the notation and the symbols themselves. Why use the inherited,
old-fashioned, clouded symbols? Make your own. I’ll adapt to them. The world will adapt to them. If you don’t like writing
fractions this way, write them that
way, or that way, or any way you choose, as long
as you explain what you mean. You mean that many of
those kind of things. That many of those kind
of things, 11 of the 17’s. Whereas this would mean 17
of the 11 kind of things. The numerator tells
you the number. The denominator tells you
the denomination, the name. AUDIENCE: So I have a question. One of the things
that eventually happened was we switched from– many of us in the
audience heard, “can you start from the second
square” as being, “are you clever
enough to come up with a way to start
from the second square?” And the process led to
finally having to ask, can you, being can anyone? Is it possible? I’m curious whether
that is, having done this many times, often a
crux that these things turn on? And I was wondering if you
could talk a little bit more about that. ROBERT KAPLAN: Absolutely. And it’s a crux on which
the math circle turns– a fulcrum on which it balances. Self-confidence and lack of it. One comes in as a
human into this world and, after just a little bit of
experience, loses one’s nerve. And says that if
it can’t be done, it’s because I can’t do it. Until you begin to say,
well, maybe nobody can do it. A great British biologist,
WBS Haldane, in the 1930s said, it may not be that
the universe is stranger than we think. It may be that it’s
stranger than we can think. Well, what if this problem– what if it turned out we
hadn’t come up with a solution. And we came to the
conclusion that, well, we’ve hit the boundaries
of human ability. We’ve hit the limit
of our ability. As you gain in self-confidence,
you say, listen. The world is my oyster. I control this. Mathematics is made
by humans for humans. If we can come up with a
problem, we can answer it. Now, for those of you who know
some history of mathematics, you know that what I’ve
just said is a complete lie. [LAUGHTER] Because in 1950, Godel,
an Austrian logician, proved that there are questions
you can state and state very clearly and very
well, which you will never be able to answer,
which is devastating. Now, the answer to
Godel is not, oh, yeah? But with mathematics, as
it’s presently understood– with set theory, as it’s
presently understood– that is a limitation. But maybe the trouble
is with set theory. Maybe the trouble is with the
way we’re posing the problem. Maybe we just don’t
have our understanding of those relations deep
enough, imagined enough, yet. Why shouldn’t anything
be our oyster? There was a mathematician in the
17th century, an Italian named Viete, who said,
[SPEAKING ITALIAN].. There is no problem
that can’t be solved. And so say I. Let me leave you
all with a problem. This is not the end
of the conversation, but let me just leave you
with this problem, all of you. And email me when
you have some ideas. How many corners does a
10 dimensional cube have? Think about that. Send me an answer. AUDIENCE: Three part question. ROBERT KAPLAN: Good. AUDIENCE: The first part is
are you thoroughly convinced that the problem of starting
the path on the second square is impossible for all odd by
odd squares and possible for all even by even squares? ROBERT KAPLAN: Are you
asking if I’m convinced? Me? AUDIENCE: I might be asking
the rest of the class. ROBERT KAPLAN: Oh, good. They shouldn’t be,
because we did in 40, 50 minutes, what’s a
deep, complex problem. And we haven’t talked
about induction, if that’s what’s on your mind. AUDIENCE: Yeah, pretty much. ROBERT KAPLAN: Exactly. Once, and this brings up an
answer to a question someone else asked– when I guess [INAUDIBLE],,
when the symbolism, when concepts become deeper– when we come to
grips with induction, which we could do in 20 minutes,
we can re-ask that question. Can you generalize even
dimensional boards? And how can you know that
a 10,984 by 10,984 board, you can start from anywhere on? It’d be really nice to
say, listen, piece of cake. Yum, yum. AUDIENCE: This is just a
pretty practical question, not a very interesting question. But I was curious as to why
you decided to start with a 5 by 5 grid instead
of a 3 by 3 grid. Because it seems like maybe
some of the conclusions the kids would have
come up to might have been a little bit faster,
given the time constraints. So I was wondering if it
was a creativity thing or like an exercise? Or I guess, what led
to that decision? ROBERT KAPLAN:
Really good question. I started with
something, which I knew was going to be beautiful,
easy, a piece of cake, and we’d all be
tremendously confident. Confident enough to look
at a hard aspect of it that hadn’t been thought of. Start with a one
by one, and they would have run out of the room
saying, what are you doing? Of course, you start in
the middle of a one by one and end in it. Yeah, OK. Or if I’d said 10 million by 10
million, they would have said, I can’t think in those terms. 5 by 5 sounds comfortable. It turns out that the springs
in that arm chair are broken. And you feel uneasy, and you
get more and more uneasy in it, until it turns out [SIGH] I know
how to repair those springs. I understand how it works. But you need an
accessible mystery. It’s got to be appealing. It’s got to be mysterious. We were asked to do a
math circle in a prison. And we said, oh,
love to do that. We’ll do it next year. They said, no. Do it next week. We said, well, we’re
very busy next week. You’re doing it Wednesday. All right. [CHUCKLING] So Ellen and I went home
and thought, oh, God, what are we going to do? Well, we’ll make sure
our wills are in order. And we worked out a routine
to have in the prison. It was a high security prison. If you’ve ever been
inside or visiting, it’s very frightening. The guards are much more
frightening than the prisoners. All your belongings
are taken away from you and put in a drawer. And there’s barbed wire
on top of the walls. Anyway, we were put in a room. And our guide left. And the room was small,
and the people were big. They spend their
time body building. There were 17– I counted– of them,
and there were two of us up against the blackboard. They filled the floor. They were sitting on the floor. And we said, this is the
routine we’d worked out. I said, math. It’s awful, isn’t it? “Right on,” they said. Ellen said, “yeah, you know
they ask you stupid questions like what’s 1 plus 2.” We fell silent. Somebody says, “3, what
do you take us for? Of course, it’s 3.” And then that dies. So then they ask,
“what’s 1 plus 2 plus 3?” And a voice over there, “6.” And then came words
which I will not repeat in front of the audience. And while I’m saying
this, Ellen is writing on the board 1 plus 2
plus 3 plus 4, all the way up to 20. And I said, “and
then they ask you a question like, well,
what’s all that add up to?” And a guy got up and
said, “that’s what I hate about this math thing. They ask you a
dang fool question that nobody could
answer and get right. And even if they did, what
would they come up with? Some stupid number. And we said, “right on. Boy, is that ever right. Yeah.” And then Ellen
said, “if only there were a nice way of doing it–
a beautiful way of doing it.” They said, “yeah,
well, big fat chance.” And I said, “yeah, if there
were just some different way of counting it.” Voice over there, you
could do 10 bonds. I said, what are 10 bonds? Talk about new terminology. So you know, 1 plus 9 is 10. And 3 plus 8 is 10. “No, you idiot,” says this guy. 2 plus 8 is 10. So we get 10 bonds. Yeah, we’ve gotten
four or five 10 bonds. Then we’ve run out of 10 bonds. Someone says, “well,
you could make 20 bonds. 1 plus 19.” He said, “no, we’ve already
used the 1 and the 10 bond.” Oh, it’s hopeless. And they’re getting
physically quite restless. And we’re sweating. And then a little
guy at the back of the room, who hadn’t said
a word up to that point, said, “I know this is wrong,
but–” which is always the preface to the insight. He said, “I know this is
wrong, but 1 plus 20 is 21. And so is 2 plus 19.” And I said, “oh, sheer luck.” He said, “no!” Another voice says “no,
it’s not sheer luck. They’re all 21’s.” I said, “what do you
mean they’re all 21’s.” Yeah, all of them are 21’s. All what? Someone says, “all
those numbers. All those 20 numbers are 21.” “No,” says a voice over here. “All those thingies are 21.” What do you mean thingies? “The thingies.” Someone says, “pairs.” All the pairs are 21. So I said, “how many
pairs are there?” Someone says, “there are 10– 210.” [LAUGHTER] High fives all around the room. Can we do another? I said, yeah. What’s the sum from 1 to 100? “It’s 5,050. Give us another.” We did this for a while. At the end, they said,
“what do you call this?” And we said, “math.” [LAUGHTER] Start with something which is
beneath the audience’s dignity. And then hit them with
the mystery, which turns out to be accessible. And they are slaves
to math for life. [APPLAUSE]

25 thoughts on “Dr. Robert Kaplan: “Mathematics: Learning to Speak our Lost Native Language” | Talks at Google”

  1. A ten dimensional cube has 1024 corners right?
    A square, a 2d cube, has 4 or 2^2 corners. A 3d cube has 8 or 2^3 corners, so it follows that a 10d cube has 1024, or 2^10 corners.

  2. I am guessing the problem has its roots and meaning in the Koenisgberg Bridge Problem that Euler solved eventually, but can't seem to realise how starting on the 2nd square maps to Euler's theory.

  3. I guess the answer is 10,368 for 5×5 matrix . Cause, we have 4 squares with two paths(4 corners) X 12 squares with 3 paths X rest 9 squares with 4 paths
    = 4 x 2 x 12 x 3 x 9 x 4 = 10,368.

    You can check this is a right proof using proof by induction approach. in the simple term just take 2 x 2 matrix. Now, all squares have 2 paths so 4 x 2 = 8. it is correct for 2X2 matrix then we will assume that it will also work for the above solution…

  4. Very great teacher . I wish I'd one when I was in my childhood.
    We want more such videos, on learning maths a fun way …🙇🙏🙇🙏…👏👏👏

  5. It looks decades old video, as they r using chalks n blackboard n mic . It looks like been recorded in 90's.
    Very good though 😄👌.

  6. Ends with great words: "Start with something which is beneath the audience's dignity, and then hit them with the mystery which turns out to be accessible and they are slaves to math for life".

  7. The pairing solution at the end was allegedly worked out by another smart school kid – Gauss who didn't want to get bored when his teacher asked the class to sum the numbers from 1 to 100. Which maybe proves that once kids start having fun by exploring the shape of a problem, they are already half way to being mathematicians. Perhaps Pythagoras started having fun with the same problem when he was a kid by representing the numbers with rows of stones and arranging them into a triangle.

  8. He used cardinals instead of ordinals when pointing at each square. Why do MATHEMATITIANS make this mistake? 0_o And digits are not numbers. Maaaaan!!!!

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