# 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. What. Does it have left. Math. And, the. Beauty of it and. Collegial. Conversation. Never, competitive. People. A very. Small group of people always, ten. 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 ten 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. We have self. 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. Let's. Guys. Here's. The question. I'm. Going to draw a. A grid, of, 5x5. Grid. By. Grid I mean kind of a chest. You know a board. Of. Five. Lines so I have five spaces. Four squares. No, only for this. Is weird, I need. Six. Lines to get five. Horizontal. Spaces. So, I guess I'm gonna need six. Vertical. Lines to get. How. Many squares do I have. Are. You sure. Okay. No, I'll take your word for it. One. Two, three, four. I. Was. Counting, I didn't, say one two. You. Have a very orderly mind okay, one, two three four five six seven, eight, nine ten. Is that okay 11, 12 13 14 15, 16, 17 18, 19 20, oh. I. Forgot. Sorry. I'll, take your word for it there 25. Wonderful. Idea okay. I, know. Their tool awesome you guys. If. You were to start in. The upper. Left-hand. Square do I have that right okay. Thank you if. You were to start there and draw. A line that's. Gonna 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. Don't want to go up and down and left and right okay, is, it possible, to do that starting from here or you know probably, not possible, yes.

You. Think it's possible I. Don't. Think so. Fantastic. Okay. I'm. Convinced, thank. You that's that's great idea thank you do. You agree, yeah, that's. The only way you can do that that's terrific, Samuel, would. You. She'll. Be okay. All yours Samuel oh. Oh. That's. Neat you nine forgiving against them oh you've got to the middle one terrific terrific. Thank you very much so there are two ways of doing it yes Petra. Come. On really. You're. Saying there's a third way. Yes. Then you can also go. Oh. You're. Doing it kind of in the. Rotated. Way it's. A Diaz thing kind of rotated. 90 degrees beautiful. There. Are three. Over. 100. Nathan. Did you calculate this or. No. I'm, sorry I should have said can't, go diagonally. Only. Up and down and left and right it. Was a brilliant. Idea, I like. Diagonal, lines as a lot of you know but. Not here. Right. Could. You do that without using, a diagonal but with an up-and-down motion, is. That possible. Try try it. Yeah. Down. All the way. Right. One. Up. All. The way. Right. One. Oh right. -. Is. This gonna work wait. One, not now I. Hear. Oh. Just. No I, erased, the line you want you want to connect that back up there oh that's, nice, you've got a part of a. Renovation. Of a castle. One. Two three four five. I. Think, one two three four five one. Two three, four, five, you were right this is an extra. Box. Of mind. No. It's not okay then go, ahead go ahead. It's. Gonna work I. Think. Go. Ahead. You. Want it, so. Let me. Know. Go. Ahead. But. You see you leave because they all happen anyway. They. Wait. Wait you had a good question Leia do they all have to connect they, all have to connect destiny one continuous, line yeah I think. This is really nice Daniel, does, his work. There, you. Can do it you can go on forever just explaining, how many ways there wait I think Nathan says 100 you say forever. Which. Is more or less forever. Fantastic. Hold on to it okay Daniel this is great thanks a lot, I will remember well you remember Daniels, how. About do, you uh sure. Petra. OH. The. Question, is how, many times how many ways can you draw, a line through all the boxes that's. A question yeah. Yeah. I'm, not sure it's the question, it's it's one of the questions that that, I thought of and you've come up with fantastic answers. One two no three no four no, 100, no an infinite number. Sure. And then Leia. Well. I'm doing this think to yourself. Besides. The question, of, how, many ways are there we know there are at least, three. Or four there. Might be a hundred there might be. Because. Like eventually. You're gonna actually, you. You. Have to have an infinite number of boxes, to have an infinite number of what. A great idea okay. One. Two three four five. One. Two three. Four. Five. Okay. Oh. This. Is like Daniels. Yep. The. Trunk is telling me something oh. That's. Great. What. I sure, did was to combine. Previous. Ideas, that's, doing, math that's, taking previous I promise I'll come to your Leia immediately. Taking. Building, on these ideas together you're, working illegally it's just super, with, the, kind. Of top, of a castle you see those tops. Of a castle and then, some, straight line right. Down the middle. Could. It have been Ellsworth and, then, zip. Over and do the. Zigzags. Backward. It's. Really nice by the way your letter E is hiding in here okay. Leia. Different. From all these. Okay. Come on up maybe. There really are an infinite number. Think. About another question. Besides how, many ways. Sorry. That's. Such, a good question, is. Anyone keeping track of whether we're gonna, be doing, something again that we did before or, you remember he'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. Okay. If. You're pretty sure not that. Means it must be a, different. A different, basic. Idea. Okay. It. Shows that although we all have minds, and they're, all terrific. Our. Minds, work differently, oh. Stop. For a second is, anybody worried. You're. Worried yes. You're worried, okay. Well. 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 Oh. What. Oh. You're. Really seeing things. You've. 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 then you could want to go in one it's. Beautiful, I do. Think that's new isn't it, okay. That isn't super, you. Guys are amazing. Okay. Come on up can, I leave this and give you a pink chalk okay. Haha. That's. A nice that's, really. Nice, looking. At this looking at the pink one looking. At Diaz pink one can, you automatically. See another, one that hasn't been done, automatically. Lea, I think, I do, don't. Come up and 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. Okay. Yes, so you took part, of what you did and. Changed. It, nice. Idea actually. I'm. Now losing track of whether we're repeating anything's we had many of the things we had before, that's. 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 your. You're basically running this class not me but I've got a question. I'm. Gonna take one. Of the paths I know. Was. Done in fact the very first one. And. I'm I want, you to tell me something, here's. That first path. I figured. When. Like, that am i right, okay. Without. Doing. Anything. Without. Asking me to do anything more, tell. Me a second, path that's, there. Yes. Do you. Start. Finish. Is. That a start. Finish. Every. Time you draw a one path you get one more for free, that. Means whether. The number is to, a. Hundred. Or, infinite. It's. Gonna have to be even. Because. For every path you draw there's, the path with the end of the beginning reversed. So. Weird. Well. That's, there. Are at least two paths there at least four I mean there at least six I mean what, do I mean there. Are at least a hundred there by be 99, yes. We have got five, and, then. You. Made ten. Yeah. Myself. I've. Got a different sort of a question this. Has worked so, well and, we, haven't even began to finally. Settle on how many times you can do it starting. Here is there. This. Is a weird question is, there, any other place you could have started, that would have worked at. All or as well Leigha. Start. Here how. Do you know that's going to work. Terrific. Possible. Start possible. Start, Petra. In, the sector. Spiral. And 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. Passion. So uh adding, on foot she said alright, that you, could start on the top right corner uh you. Could actually just all blends be of current item but you could just turn them right. Rotate. The, board. Instead. Of rotating, our drawing idea, what were you going to say. Fantastic. You could start in any square. So. Far we know you, can start. 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 doing the. Taking. The endings and going back to the beginnings I see, we've gone 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 okay I want to take what. You. Suggested, dear. Do. You feel pretty much in charge of this question right now yeah Nathan. Okay. All right I do, bite you. What. An interesting idea. Okay. You. Lifted your truck. You. Lifted your truck. You. Can't lift your truck nevertheless, you, can do what you wanted I think without. Lifting your truck here. Let me let me just I, mean. You could I was, just wondering. If you could use, the start point and put. Push it off two ways. Your. Question, is can. You. Go. Ahead. Watch. Out for this guy. Okay. I'll be much more precise you're asking, me to are more precise in my language without. Lifting your chalk and without going over the same line a second, time.

Only. Once wait a minute wait, a minute wait, a minute. I've. Just drawn that line seven, thousand nine hundred eighty-one times. Haven't. I they're. All next to one another I. Think. Nathan you could still get, your putter. Oh. Yeah, right. This. Is a question if you could go to two ways. Got. You that's a really nice other question, which, we should explore, Nathan's. Brought up a completely new question, take. A five by five grid or a four 2x2 grid, aren't. 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. Or. If, you have two arms did. You have to have two hands for the pieces of draw if you're an octopus you can do eight. Very. Good I want, to get back to something dia raised going. Back to our original. Whoops. Our original 5x5. Deal. Said. Tell. Me if I'm if I'm quoting you correctly do, you they, really you can start from any square as long, as you're careful right, I need. That what I said. Any place, that's like next, to the corner. You can't start there. Fascinating. Idea let's see listen. 1. 2 3 4 5 oh really. Careful, in drawing this, you're. Saying, starting. In a place next, to a car. 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 board, easy. What. I was counting the squares when I was counting all 25 of them I was counting this way of that oh. Say. It again. Yes. What, should go in rows. There. Was something besides my finger that somebody suggested, I do. You. Played mark every, square you party counted, she, wouldn't double kill'. Do. That what do you want to mark it with what. Do you want to mark it with I. Was. Thinking ah sure of your question, about is this all gonna be geometry. It. Could. Be something else. Well. It's not geometry. But like. I mean, I meant like hey. Equation. That what. Do you mean an equation math go on what do our equations, involve, numbers. What. If I mark these squares with numbers. Somebody. Sells the beginning why'd you just number the squares. It. Takes, longer but you can actually, I'm. Thinking when. I said that more about speed and not really, I know I know but I think Mike you can count by fives. You. Can count by fives I'm. Good at. 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 Wendy. I suggested, I start in a square next, to a square, of a corner, I guess. She met square number two. No. Thanks for filling this in 21. 22. 23. 24. 25. If, you went this. Way. Is. This gonna work. Yes. I I. Can show you how you do. If. You start right here that's great to you thank you pretty much exactly, what, dia, did. Except. When, you go here. Go, like this well. Actually no not that. This. She's. Wrong, Nathan, thinks G is right that you can't start from square number two. Asher. And then Daniel, oh you're got your hands up on once after. First then Daniel then Samuel yes you could I think layer you under hand up. Watch. Then. Actually on the board. It's. Looking like. Do you. What. What is going on with the problem, that Diaz. Raised if he takes. It over here he. Can make it easier by cuz he doesn't have to go, like. That he, can do it here instead let's. See so, are you saying 2 1 6 7 8, 3. 4 5 10 9 so far so good, 14. 15. 20. Then. You go to 20. Down to 25. Don't draw that in blue go ahead. 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, and. Skipped. Yes. Another. Way that's gonna work. Starting. With two, great. And, Samuel. You've had your hand up for a long time. Starting. To maybe. You could go. Now. What, you've. Left out 15. Yes. Can. You put, the same line in the juice where's. What. Do you mean like. If. There was once. Where would, you live. Oh no no absolutely not. One the line is a straight steady. Line and it never goes back in the square in a square it's been in I mean, exactly. What. Is it moving. Helicoptering. Down I have another idea. Do. You. Could. Just. Let's. Try a spiral, go on, go. Ahead go ahead. Yeah. Yeah he's, that way into 19. Yeah. Yep. Stuck. Again, is. This something wrong, with our imaginations. Or with the problem. Laya. Terrific. At DIA you think you have won how, about dia, then Leia.

Ask. Yourself, what, is going, on it was so successful. Starting. From the corner, with. Lots and lots of an even. Number of paths, and. We. Can't, even get one, starting. From square to started. But, we can get it started but they can't go. On one two three four, five. One. Two three. Four five. And, we're starting it. This. Is gonna be your spiral. It. Was beautiful. Up to there. What. Is wrong why, can't we do this you guys are so good but we can't do it because the corner, piece yes. It's just. It's. Kind of wedged in between two, pieces but. Quarter, pieces wedged, in between two. Pieces that, what. You mean, well except, one piece has an edge and, you can't go past gotcha. Come. On. 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 oh, you. Guys now all know that you're really good at this you're, good at imagining counting. You're. Better. I think it count even I am and yet. Just. Saying. Can we start in the second. Square, from the corner. Mind, may not work. Well. Trying. So. Far so good. Oh thank. You oh. Wait. It's like one two three four five that stops, here. Yep. Okay. Stop. For a second, is it okay so far. Any. Trouble do you see any trouble a future you, see trouble Samuel, where. Do you see trouble. Okay. Go, ahead so. Far she's avoiding that. Now. Look what's happened. Look. What's happened, you have. A choice if. Nathan. 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, yep, that would. Why. Is it so difficult. Why. Doesn't it work what's, the trouble, with us yes. Fantastic. Come, on up. Okay. That. The old top of the castle. It's. Just six around Syria I'll just I'll just get rid of that. Much okay. Give. Me there's already trouble. It's. Too late already, Leah, what do you think too late already I. Think. Your idea was great. Yes. Done. Let's. See. What's. Happened here well this is. Okay. In. Other words you've missed that square how. Did I know that there was going to be a problem. That's. Called induction oh. Dear. Why. Don't. You. All say, the, minute, you see a problem coming, up. We. Start here anybody. A problem coming up okay. Okay. So, far. Okay. Okay. So far. Why. Is it okay so far. Yes. A good idea. Okay. 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. Okay so far if, you grin since my high, it's. Meant to be five it, is five by five. Keep. That idea in mind yes, sure go. Ahead go ahead go ahead with what you've got. Oops, you've gone to the square twice. You. Can't be wet in the squares place again. 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. You're. Just one square, off finishing. It. For, the back. That's. Interesting, one two three four five one two three, yours is 5x5, and, you've done that and. You. Shall. We number her steps. This. Is one, two. Three, four. Five. Six. I'm. Sorry we don't we well, that's can I start with the number two in the corner this, still is five squares I'm numbering now the steps, instead of the rows. 7. 8. 9. 10. 11, 12. 13. 14. 15. 15. 16. 17. 19. 16. 17, 18. 19. 20. 21. 22. 23. I'm. Not sure we're ever. Going. To get to that wait wait a second. That's, long 2 3 4 5 1. 2 3 4, ah sorry. My board seems to have 1. 2 3 4 5 yeah it's, 6 wide. That's. Interesting, well. 6 oh. Right. You are. Right. You are 1. 2, 3 4 5, that line going through is that it's just one, square. Has. This worked. Take. A look at Diaz paper. So. Far so good. By. The way 1 2 3 4 5 by 1 2 3. 4. One. Two three four. Wait. A sec oh I, say those are two one. Two three four five one two three four five okay so, you go down. The. Left-hand, side, and across the bottom up. The. Right. Ah you've gone into the square twice. With. This line and with that line, oh it. Was so beautiful, up to there. Ten. Minutes left how can this be, did. We do this another two hours please. Let. Me ask you what yes, daniel says yes ten more two more hours okay. Is. It gonna work when you start from the second square, Nathan. What can, I show you something.

Yeah. Please sorry, races. But I think our problem is that we yeah we've, been going across. Yeah, well. I didn't. I didn't, realize so, I was just estimating, that that might be the problem because. I like to ask to me. Well. I like your estimate above a hundred a while ago by, the way after, you. Asked, a long time ago, is. This gonna be geometry. This might be. Arithmetic. Instead, of geometry, I don't know ah sure. And then Petra I don't. Think this really counts though so, oh. So. Far. That is backward. Of something that I think dia did. Now. Zigzagging. Stop. You'll you'll never get into the lift. Great. Yes. I've. Got a funny. Question to ask before Petra, and Leia. How. Big is this board. Namely, how many squares. 5-squared. True, what. Is 5 squared it's, 25. It's. 25. What. Kind of a number is 25, possibly. It's. A square number and some odd number it's an odd number squared. It's. An odd number. Can. I go back to Nathan's. 2x2. Board, 2x2. Is what. What's. 2 times 2 this is very difficult I know but more. Or less for I mean you estimate so it's the kind of four and on, two by two boards, a board. With four. Squares. Can. We start from here will it work can. We start from here here. Here. Dear. And. The reason the 2 by 2 works, is that there's, no square in the middle of, their, two corner pieces 4. Wow. So. A 3x3. Would. Or wouldn't. 3x3. There's. That little square. 3. By 3 is trouble 5. By 5 is trouble. What's. 729. By 729. Trouble, or easy. Oh. You. Start from the first square no oh you're, from the second squared. Oh, actually. No yeah right. Starting. From the second. Square. On. An odd number. Board. Is. Trouble. Why. Petra. There, probably is a way to do it but. It's. Hard to find that's. Great if there's probably a way to do it but it's hard to find after you're saying there's just no way to find it no matter how you try five. Whole minutes oh, say. It ah sure so I'm, just gonna okay. Good so, if you start from here. You. Can't really do. Anything about it because there's. There's. Always gonna be this extra. Corner. And if you try to eliminate that corner. Middle. Pieces is not gonna work see so. Here right won't work right because. You. Have to once, you get to like when, do to the point where it's, either a. Corner. Or the center. You're. Always gonna get there you, have to you have to only go to one that's, terrific. It's gonna leave the center. Out. Or, just, one. Of the corners is, a 6x6 board gonna work or not it, will because, it's, and a 7x7. Right. You, dear, that's terrific, it's. Getting. Even number by an even, number then, it's going to work because. Like. The. Reason. There's. No middle there's no middle square, so whatever you do it won't cross that one odd square. That. One what square I square, that. One odd square. On a 25, square. Board 555. How. Many odd number squares are there and how many even number squares.

Yes. There's. Always, one. More odd, number, square than. Even, on. An, odd number, board and already even number board. Yes. But. I think I think you're on to the hidden. Deep structure. Of this problem, there's. Always an extra. Odd number, square on an odd number, board and, if. You're gonna cover every square. You. Better start from an odd one well you're not gonna cover them all because it's an extra odd one somewhere. In the center, or. In a corner or something like that. Fantastic. Guys. Please. Do. I'm. Not gonna have any answers but go ahead and ask, oh. Those. Are books that Ellen, 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. Is. Almost always about the infinite, the infinite. Number of possibilities. It. Has ten 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 a whole thing. These. Are the people to afford. Okay. Absolutely. Absolutely. That's. Right start, on one end on 25 in one of our patterns start on 25 end on one in the paired. Pattern. Ashley. Like, what the art. Basically. My man back i. Is. Not enough you're. Right it's. Like big and small infinity, is not a number but it is an idea we have about. How many oh actually, I'm sorry how many counting, numbers are there I. Thought. There were 10,000 181. Counting. Numbers I thought they'd end I thought the last one was ten thousand nine hundred and eighty-one. So. Anytime I name a counting, number you can say well there's one more and that, must. Suggest there. Are not a finite, 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. Are. We gonna do something like this again you. Mean in the math circle yeah. Oh maybe, something, since, you've already seen this something beyond this. So. The answer is maybe I. Have. A question, yes. Yes. In. This setting the beginning, was just you. Know drawing so everybody start 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 the. Kids should invent the notation, and the symbols themselves, why. Use the. Inherited. Old-fashioned. Clouded. Symbols, make, your own I'll but after them the world will adapt to them if, you don't like writing fractions. This. Way. Right. Some. 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 Seventeen'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 I have. A question. One. Of the. Things. That. Eventually. Happened. Was. We. Switched from can. You. Being. A challenge, of. Many. Of us and 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, you. Know the process, led. To, finally. Having to ask can you being, can. Anyone, it is it possible. I'm. Curious. Whether, that is. Do. You having. Done this many times do you do you see that that is often. A crux, yes, things turn on. 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, if it can't be done this because I can't do it until. You begin to say well maybe maybe. Nobody, can do it a great. British, biologist. WB. S Haldane in the, 1930s said, it. May not be that the universe is stranger, than we think, it. May be that it's stranger that 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. Limit of our ability, as you gain and self-confidence. You, say listen, the. World's 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 as a complete, lie, because. In. 1950. Girdle, of Austria. And 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 girdle 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 in Italian, in viata who said no, longer on problem, Assol worry 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, three-part. Question good. 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, are. You asking if I'm convinced, me I math I might be asking the the. Rest. Of the class oh good. They. Shouldn't be because. We did in 40, 50. 50, minutes, what's, a deep, complex. Problem, and we haven't talked about induction if, that's, what's on your mind yeah for yes exactly, once. And this, brings up, an, answer to questions, on what else asked when, when, I guess it not at all when. The symbolism, when the concepts, become deeper. When. We come to grips with induction which we could do in 20 minutes we. Can react that question, can you generalize, even.

Dimensional, Boards and haven't, even know, that a ten thousand, nine hundred and eighty-four, by ten thousand, nine hundred eighty-four board, you, can start from anywhere on I mean. It'd be really nice to say listen. Piece of cake. Yom-yom. This, is just a pretty practical question, a very interesting question but I, was curious as to why, you. Decided to start with a five by five grid instead of a three by three 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 like there's a creativity thing or like an exercise, or I guess yeah. Led, to that decision really. Good question I started with something which I knew was gonna 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. Start. An invisible one-by-one and envel in, it yeah okay or, if I'd said ten million by ten million they. Were I, can't, think in those terms five. By five sounds. Comfortable. It turns out that, the springs in that armchair are broken, and you, have to you, feel uneasy and you get, more and more uneasy in it until. It turns out oh 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're, very busy next but you're doing it Wednesdays, all. Right. So. Ellen and I went home and thought oh god, what are we gonna do well 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 when put in a drawer and there's a barbed wire on top of the walls anyway we, were put in a room and our guides left, and the room was small and, the people were big they, spend. Their time bodybuilding. There, were Oh 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've worked out I said. Math. It's. Awful, isn't it, they. Said Ellen. Said yeah you know they ask you stupid questions like what's one plus two. We. Fell silent, somebody. Says, three. What do you take us for it so of course it's three and then, that I said and then they asked what's one plus two plus three and. Voiceover. They're six. And then, came words, which I will not repeat from 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 asked you a question like what's all that out of - and. A, guy got up and said, that's. What I hate, about this, bad, thing, they, asked you a damn. Fool question, that nobody, could ask her and get right and even, if they did what. Would they come out with some stupid, number, and.

We Said right, on voice, not 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 hopefully, a big. Fat, chance. And I just. Something you, know different, way of counting, it, voice. Over there, you. Could do ten bonds. That's. What what are ten bonds talk. About new terminology. So. You know one plus 9 is 10 + and + 3 plus 8 is 10 no you idiot says, this guy to, fuss 8 is 10 so, we got 10 bonds yeah we've gotten several well. Four. Or five 10. Bonds then we've run out of 10 bonds, so, is that we could make 20, bonds, you. Know 1 + 19's, and no we've already used the one in the 10 vote ah it's, hopeless, and, they're. Getting they're getting a physically, quite restless and we're sweating, and then. A little, guy of the back of the room who hadn't said a word after, that point said I know this, is wrong but which is always the preface, to the inside he. Said I know this is wrong, but. 1 plus, 20. Is 21, and so, is 2 plus 19, and I saw a sheer, luck, he, said no another. Voice I said no it's not sure Lots they're all 20 ones, what. Do you mean they're all 20 ones they're. All all, of them are 20 ones, Oh what, so, it says all those numbers, all those 20 numbers are 21, no. Says the voice over here all those thingies. Are. 21. What. Do you mean thingies the the figis. Someone. Says pairs oh, all. The pairs are 21. So. I said well how, many pairs are there someone. Says there are 10. 210. High-fives. All around the room can. We do another I see yeah what's this Ellen said what's the Sun from 1 to 100 it's. 5050. Give. Us another we. Did this for whole hell at, the end they said what. Do you call this we. Said math. Start. With. Something which is. Below. 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. You.

*2019-01-15 17:23*

__Comments:__

*2019-01-16 22:09*

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".

*2019-01-18 06:04*

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

*2019-01-19 19:04*

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

*2019-01-20 21:58*

It is not decades old - it was recorded in 2018.

*2019-01-21 11:43*

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 ...

*2019-01-22 00:50*

I guess the answer is 10,368 for 5x5 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...

*2019-01-23 23:48*

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.

*2019-01-25 18:56*

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

*2019-01-27 04:52*

That's how a mathematician ought to look like!

*2019-01-28 02:23*

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.

*2019-01-28 12:45*

Grandpa

*2019-01-30 09:02*

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

*2019-01-31 15:20*

It's over 9000!

*2019-02-01 06:27*

625 ways i think by watching until 28 minute

*2019-02-01 09:56*

+आदित्य Aditya मेहेंदळे Mehendale I watched till the end now. How much do you think is the answer

*2019-02-02 20:27*

So you missed the "has to be an even number" part?

*2019-02-04 08:00*

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

*2019-02-06 03:01*

Privilege!!!

*2019-02-06 07:55*

1st commenter and viewer.

*2019-02-07 02:33*

Mahendra kumar congrats

*2019-02-07 23:54*

You should have brought regular children.

*2019-02-08 12:15*

6250 ways of doing

*2019-02-09 20:17*

Fantastic

*2019-02-11 08:59*

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

*2019-02-11 18:57*

good job

*2019-02-13 09:20*

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!!!!

*2019-02-15 03:46*

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*2019-02-17 02:47*

@आदित्य Aditya मेहेंदळे Mehendale I watched till the end now. How much do you think is the answer

2019-01-16 00:02The 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.