[MUSIC PLAYING] ROBERT KAPLAN:

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]

1st commenter and viewer.

Privilege!!!

I feel privileged to have watched and learned from such a great teacher. Thanks Google

625 ways i think by watching until 28 minute

It's over 9000!

one of the best comments on maths learning appears here 47:21

Grandpa

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.

That's how a mathematician ought to look like!

Smart kids. Wish I got involved with something like this when I was younger

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.

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…

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 …🙇🙏🙇🙏…👏👏👏

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 😄👌.

i am gonna ask my dad to dress like him or i won't love him anymore

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".

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.

You should have brought regular children.

6250 ways of doing

this is really beautiful. Mr. Kaplan is so humble, empathetic, brilliant and amazing teacher

Fantastic 🙏🏻❤️

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!!!!

Watch a 30 mins video in 3 mins. The BEST extension in google chrome store. https://chrome.google.com/webstore/detail/threelly-ai-for-youtube/dfohlnjmjiipcppekkbhbabjbnikkibo

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Good 🔥

What's up with the kid crying??