Pebbling a Chessboard - Numberphile

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Will the clones escape "prison"?
More links & stuff in full description below ↓↓↓
MAIN VIDEO: • Pebbling a Chessboard ...
CONTINUATION: • More about Pebbling a ...
AND EVEN MORE: • Pebbling a Chessboard ...
PLUS CUT FROM PART ONE: • Pebbling a Chessboard ...
Featuring Zvezdelina Stankova - Professor of Mathematics at Mills College,
Director of Berkeley Math Circle, UC Berkeley
Spreading clones across a chessboard and escaping "prison" - commonly known as pebbling a chessboard.
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Videos by Brady Haran
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Опубликовано:

18 дек 2013

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Комментарии : 1 тыс.

@motmahp 10 лет назад

I know she was saying "clones," but I must mention that infinitely many clowns is a terrifying prospect.

@gedstrom 6 лет назад

Send in the Clones!

@StraightEdgeHippie 5 лет назад

0:18 "Every clown clones itself..."

@1truefreedomfighter 5 лет назад

Can a finite limit on clowns be defined? To be serious, this is the first time I have seen and understood the value of attrition. Thank you Math, also thank you Zvezdelina Stankova for showing this to me.

@keithwilson6060 4 года назад

Kramer agrees.

@lawrencedoliveiro9104 3 года назад

I can think of something worse: infinitely many mimes.

@gfetco 9 лет назад

Why is there only about 4 videos of Zvezdelina Stankova? We want more! she's has remarkable skills in telling stories, making mathematics more interesting.

@amblyo2706 8 лет назад

+Enlightenment Agreed

@zorm_ 8 лет назад

+Enlightenment and a nice accent too lol

@enclave2k1 7 лет назад

@danamoldovan9135 5 лет назад

Absolutely right!

@juanpedrolardet2088 Год назад

after 7 years, i'm afraid to think that there will never be enough videos of Zvezdilina

@oafkad 10 лет назад

You could easily fit two or more clones in one square. You'll just need a clone car. ...this joke only makes sense if she says it :/.

@rogerab1792 6 лет назад

Hey, Folks! genius

@joegillian314 5 лет назад

Lol, I thought the problem was about clowns at first, too.

@jonaskoelker 4 года назад

My dad fought in the clown wars?

@HighKingTurgon 10 лет назад

When she introduced that infinite sum I half-expected James Grimes to appear in a jump cut clapping his hands together.

@varthadivyeshyashvantb15me26 4 года назад

@CogitoErgoCogitoSum you just replied to a 5 year old comment

@alexismiller2349 4 года назад

@CogitoErgoCogitoSum Wait a second I recognise your username from some comment a while ago anyway.... News flash, did you know that RU-vid was not made only for you? Personally, I think if these videos don't assume that the viewer has worked on the Riemann hypothesis, taken a PHD in physics and been working on the cutting edge of solving the Navier-Stokes equations then smh...

@electromorphous9567 4 года назад

Bruh 😂😂😂😂🤣🤣🤣🤣

@rokronroff 3 года назад

@CogitoErgoCogitoSum it's almost like these videos are for casuals or something.

@huh968 3 года назад

@CogitoErgoCogitoSum lol that's like saying teachers should never teach their class anything that they've already taught to any of their previous classes. of course numberphile has explained the geometric series multiple times, but this will be the first time for any new viewers. every single one of those new ppl also only need it explained once, just like you. don't pretend you're special.

@Treblaine 10 лет назад

Game difficulty modes: -Easy -Medium -Hard -Extreme -Infinite...

@TruthNerds 5 лет назад

In fact, this was the original design for Bioshock Infinite… ;-)

@numberphile 10 лет назад

I've ended up producing four videos on this topic. They won't all appear in your subscription boxes (phew!) but links are in the video description.

@MrOperettalover 8 лет назад

There was no proof in this video. To prove the game is impossible one must show it is impossible at some stage not to leave a hole. As far as I remember it has been proven that this game is possible to play without leaving holes. Thus, we can easily integrate that it indeed is possible to free the clones. This is just as obvious as 1 == 1.999... or that achilles reach the goal.

@jamma246 8 лет назад

MrOperettalover There is no hope for you.

@RichardHoman9009 8 лет назад

I'm sure you hate me at this point, but can you see previous comments? I'm asking you specifically because I know for sure you have previous comments here like I do, but I can only see mine. When I try to view them from another account (checked on my friend's computer), I can't see any. I can only see them if I use my phone... Very strange issue...

@MrOperettalover 8 лет назад

jamma246 Try valid arguments.

@jamma246 8 лет назад

MrOperettalover It wasn't an argument, it was an asserted statement of fact without supporting evidence supplied. That doesn't make it untrue. I'm constantly amazed that people are unable to follow a basic mathematical proof. It shows what an awful state mathematics education is in. And, even worse, people who are unable to understand mathematical proof somehow have an overinflated sense of their own ability in these things, and to apparently see what mathematicians, who do this for a job, are not able to see. I guess it's just an example of the Dunning-Kruger effect.

@SteamsNightcore 9 лет назад

This is basically just a game to tell your friends to play and then watch as they spend 15 minutes trying to figure it out.

@MistaAcorn 9 лет назад

The clowns must escape the prison! To Greenland it is!

@ValkyRiver 4 года назад

Answer: Move your clones to coordinates 64.182032, -51.695202

@TheLetterJ10 4 года назад

@@ValkyRiver *clowns

@rcb3921 3 года назад

But... they'll all be unemployed! In Greenland!

@masterlangtau 2 года назад

@@rcb3921 Underrated comment

@MitchBurns 8 лет назад

What a twisted game. It is almost possible to win, but still impossible. A game where you always think you have almost won, but can never quite do it, so cruel.

@ImASDFx2 7 лет назад

Reminds me of the tale of Sisyphus.

@Rayhuntter 7 лет назад

sounds like being in the friendzone: so close yet so far.

@ZoctorDhivago13 6 лет назад

Just as life

@vincentc978 4 года назад

If you think about it, it's actually impossible not because of the infinite stuff. The bottom and left most rows will never be filled because the previous clones are needed to fill in their positions now.

@hassanakhtar7874 3 года назад

@@vincentc978 that is correct. It is common to prove things using extreme cases and overestimates though because it makes the solution easier.

@centraldoxadrez 10 лет назад

This channel is the best finding in youtube I have made this year, and maybe even in my youtube's life. Congratulations for everyone that help with this project. You people are helping the world to be a better place.

@AidenOcelot 7 лет назад

"Let's use the very first number you learned in school" Seven! "One" Darn it!

@s4ad0wpi 10 лет назад

We must free the clowns!!!

@T-Fyre 5 лет назад

Must we though?

@5gonza541 4 года назад

Xhinope Keep them in please... I don’t like clowns

@Opaqu.e 9 лет назад

Poor Greenland! Cold enough as it is, now they've gotta put up with the sum of 2 clones pouring out of an infinite prison!

@mrglibb 9 лет назад

I just can't take this seriously, I keep hearing 'clowns' and imagining an infinite number of clowns cloning themselves and attempting to escape a prison xD

@jevicci 8 лет назад

+Mr_Glibb ...and fleeing to Greenland!

@U014B 8 лет назад

I CAN'T STOP LAUGHING HOW COULD YOU DO THIS TO ME

@yangshi2388 7 лет назад

they are really CLOWNING

@panda4247 6 лет назад

infinitely many clowns are not escaping a prison, but a VW Beetle

@ValkyRiver 2 года назад

@@jevicci The Inuit Zeno found a way. If you halve the time needed on every clone, then the time is also a geometric series, which means you completed an infinite task in a finite time.

@Khronosian 10 лет назад

FREE THOSE CLOWNS, WOMAN. YOU CAN DO IT.

@Auravoo 10 лет назад

It took me a while to realize it was clones, not clowns. I kept thinking about why we want to free the clowns. Shouldn't they be kept up in the circus?

@matthewseelow 8 лет назад

LEAVE THE CLOWNS ALONE!

@Joeobrown1 10 лет назад

that animation was ridiculously useful and well done, good work whoever did it

@that-guy-pearce 7 лет назад

There sure will be a lot of clowns in Greenland if those prisoners escape

@Reionder 10 лет назад

I know its "clones" but I can only hear "clowns"! Heheh

@DragonAurora 9 лет назад

"Is it possible to prove that something is impossible?" Mind officially blown...

@evetheeevee2977 4 года назад

No, it is impossible to prove that something is impossible Paradox

@omikronweapon 4 года назад

nice soundbite, but if that actually blew your mind, it didnt take much.

@mintsjams8862 4 года назад

@@evetheeevee2977 thats not true at all you can just use proof by contradiction.

@hassanakhtar7874 3 года назад

That blows your mind? Even number + even number = odd number. Easy to disprove. Is there an equation that solves fifth degree polynomials? Hard but can be disproven. Many things in between those 2 results in terms of difficulty can be proven. Math is about proofs and logic and it is sad people don't get taught this no more.

@huh968 3 года назад

they actually do this a lot in maths

@JoeMama-sy8ty 9 лет назад

oh noes the clones have been captured by the seperatists

@tabularasa0606 10 лет назад

I love the proof, brilliant.

@panteleimonnielsen225 3 года назад

Love these vids! Such a great feeling to them - thanks so much! I used to wish there were things like this on TV. Your teachers are tops, and Zvezda's becoming my guru.

@Agnostic080 10 лет назад

very interesting video, I finally understand the concept of 1 + 1/2 + 1/4 + 1/4 + ... adding up to 2. Thanks for sharing.

@gauravsingh3007 5 лет назад

You have written 1/4 twice. It should be 1 + 1/2 + 1/4 + 1/8 + ...

@Intrafacial86 9 лет назад

I got lost half way through. I thought the point of the game was to empty the prison, not fill the "greenland".

@LeoMRogers 8 лет назад

+Intrafacial86 the point of the invariant is to show that thats the same. The total weight of occupied squares stays the same, the total weight of squares outside the prison is the same as the total weight of squares inside prison, therefore to go from all the squares in the prison occupied to none of the squares in the prison occupied, you need to occupy all the squares outside the prison.

@ragnarthepotato3523 8 лет назад

+Intrafacial86 I know a way to win get a bunch of clones like millions! Then stage a revolt XD

@DevonBernard 10 лет назад

This game is amazing, thanks for uploading this awesome video! Looking forward to your next video, keep up the great work!

@bjzaba 10 лет назад

This lady is awesome! More from her please! This series of videos was so neat because it demonstrates what the study of mathematics _really_ is about. I could not get my head around the last video, but that's perfectly ok. I'm glad you included it for those who might be interested.

@KleinLieu 10 лет назад

Hey it's my math professor from Berkeley. Very Cool video!

@greysquirrel404 5 лет назад

It would still not be possible to get all the clones out, even with infinite moves. This is because when you make a move you cannot change the number of clones on the lowest row, or the leftmost column. So a maximum of two squares each on the lowest row and the left most column are occupied, with the others being empty. Therefore you have to leave a non-zero amount of space in greenland empty, so the space the clones can occupy in greenland is less than 2.

@DemoniteBL 4 года назад

This should've been the answer in the video.

@huh968 3 года назад

everyone knows that, but just pointing that out isn't a mathematical proof

@voveve 10 лет назад

2:20 Is there a flash game of this?

@lakejizzio7777 4 года назад

Yes

@CSBMathewKJ 2 года назад

The Gaussian integral is an example of how adding up a grid is sometimes easier than adding up a row of that very same grid

@MichaelFergusonVideos 9 лет назад

Thanks for sharing this. I love Prof. Zvezdelina Stankova's enthusiasm!

@parismars7866 10 лет назад

Those pesky clowns, damn them.

@pmacg 10 лет назад

Great video! Thanks for this, Brady. My favourite numberphile video for in a while :)

@bhanupratapsingh7842 7 лет назад

one of the best curious and amazing video i have ever seen , thank you @numberphile

@RedXiongmao 10 лет назад

I must've watched this video a dozen times just because I like the sound of Dr Stankova's voice.

@PRASHANTKUMAR-sn2gs 3 года назад

it's funny.

@RedXiongmao 3 года назад

@@PRASHANTKUMAR-sn2gs :(

@PRASHANTKUMAR-sn2gs 3 года назад

@@RedXiongmao nice to see you'r alive

@szymongorczynski7621 8 лет назад

Clowns, clowns and clowns

@jimmy000 10 лет назад

One of the best videos by far on numberphile!

@TheDarkerPath 10 лет назад

Brilliant! Thanks for posting extra videos.

@Ziellos 8 лет назад

Id like to hear more about this terrifyingly prolific and fascinating clown prison.

@counterpicky 10 лет назад

Similarly, could you say the problem is unsolvable even with an infinite amount of time, since the last square (the one in the very top right) is unfillable since it would require a move that would cause a clone to leave the board? I suppose this coincides with the other solution since the concept of "last" doesn't really exist in infinity I guess

@eyesofphysics97 10 лет назад

Was going to answer your question before you answered it yourself :) But yes, that is an interesting way to also disprove the possibility of solving it in a finite space. Something tells me you tested this and found it out?

@counterpicky 10 лет назад

Well, thought experiment but yeah I did :P

@Drachenbauer Год назад

and there can never be more than 2 clones in the first row and column. No matter, what you try, moves in this L-shaped region always remove one clone in this region and place one there. So you never can fill this region full.

@lake4ishikawa 10 лет назад

This was my favourite Numberphile^^ Astonishingly elegant!

@nom654 10 лет назад

Excellent. One of the best in a while.

@naverilllang 9 лет назад

sounds like the 10th level of hell "play this game for eternity until you beat it"

@AdrianLikesFlags 10 лет назад

This reminds me of that Joke that goes like "An infinite number of matemathicians walk into a bar..."

@Devilyaki 10 лет назад

Found out how to free them in one move. .. *flips board* done freed :)

@raykent3211 9 лет назад

I have the great privelege of having fairies living at the bottom of my garden. At the last full moon I asked them "where is the bottom left hand corner of infinity?" They said there isn't one. I asked them if they knew of anything which divided in two, disappeared, and it's offspring never grew bigger, and their offspring were half that size, and so on to infinity. They said no. If anyone's interested in relating this to physics, note that in this game space is strictly quantised a priori, but the size of entities in that space is not. The entities are allowed to be infinitely small, while still occupying the same space. OK, it'a game. I think it has little consequence because of the peculiar rules. Something like zeno's paradox in 2d.

@NoriMori1992 8 лет назад

...What?

@raykent3211 8 лет назад

+NoriMori I was probably drunk, sorry.

@NoriMori1992 8 лет назад

+Ray Kent Haha! And I assume you don't remember what you were getting at?

@shambosaha9727 4 года назад

@@raykent3211 Wow, the videos are so great that people try to watch them even when drunk.

@huh968 3 года назад

@@raykent3211 haha, you're right tho

@pilifx 10 лет назад

This was absolutely beautiful. And very easy to follow. Thank you so much for sharing this with us. If only math at school would have been done like this.

@nerf1van1nerfingen 10 лет назад

so if i make a move in 1 minute and the next one in 1/2 minute and the next in 1/4 minute ... , would i be able to free the clones in 2 minutes?

@KartonRealista2 7 лет назад

nerf van nerfingen That's called a supertask, look it up

@tomwhipp3245 6 лет назад

I was seeing if someone suggested they could solve it in a super-task!

@greysquirrel404 5 лет назад

It would still not be possible, because it's not possible to have clones on every square in greenland, even with infinite moves. This is because when you make a move you cannot change the number of clones on the lowest row, or the leftmost column. So a maximum of two squares each on the lowest row and the left most column are occupied, with the others being empty. Therefore you have to leave a non-zero amount of space in greenland empty, so the space the clones can occupy in greenland is less than 2.

@U014B 4 года назад

🤔🤔🤔🤔🤔

@lawrencecalablaster568 3 года назад

I was JUST watching the Vsauce video about supertasks

@KipIngram 2 месяца назад

This was just beautiful. Best math video I've seen in ages.

@picknikbasket 10 лет назад

I think you hit the jackpot with this one, Brady. Great job and more from Prof. Stankova please, please!

@12gabriel3 10 лет назад

Chuck Norris can free the clones with 3 moves.

@BrandonStark0 10 лет назад

I really like this lady, Brady. Please do more videos with her!

@Qwerasd 7 лет назад

This was great! I really enjoyed it!

@goncalofreitas2094 4 года назад

Beautiful problem, this is why I love mathematics! Thank you Zvezdelina Stankova and Numberphile

@SPACAR-RESCUE 10 лет назад

The accent with the red hair and all the smart things coming out of her, is really doing it for me.

@notanimposter 10 лет назад

Her accent reminds me of Dexter from Dexter's Lab.

@tunamayo21 10 лет назад

This is the BEST Numberphile video

@habibaghasafari2237 10 лет назад

an amazing video. a really good point at end, thanks Brady.

@hacatu 10 лет назад

I thought I could solve it easily, but I realized that the same pattern I was trying to solve kept recurring and it really was impossible. Interesting. If anyone still thinks it's possible or would just like to play with a bigger board, I've made a javascript version. Note that it's not infinite, but is faster than real checkers or paper. jsfiddle.net/hacatu/LcDsC/embedded/result/

@MrBobmarcellino 9 лет назад

I am da boss. I decide da order. I DECIDE WHICH CLOWN TO CLOWN

@max-king 10 лет назад

This was very interesting! I enjoyed watching, thanks Brady :)

@avadakedavra696 10 лет назад

Love her teaching/speech/explanation.

@badatphilosophy 7 лет назад

What if you perform the first move after one second, then the second move after half a second, then the third move after a quarter of a second, and so on and so forth. So like, a supertask. Wouldn't the game be complete after 2 seconds?

@TVFILMBUFF 10 лет назад

I love Zvezdelina's accent. Clowns!

@vF3ARv1 10 лет назад

What an awesome subject. She did a great job explaining as well.

@axelsolhall5830 10 лет назад

This have got to be one of the better episodes of Numberphile. It was really interesting :D

@knuckstrike 10 лет назад

I think I've found an easier way to solve it. The amount of clones on the first vertical column will always be two because by cloning you have the same amount of clones on the vertical column you clone it from. No clones can be added to the first vertical column because you can't clone to the left and there's nothing left of the first vertical column. So with only two clones on the first vertical column, the maximum value of the first vertical column is 1/4+1/8 which equals 3/8. The total sum of the first row is 1/2, so there's 1/8th that can't be filled, right?

@knuckstrike 10 лет назад

Oh wow. I just watched the second video and this is basically what it says. Nice work me...

@theniceneighbor 10 лет назад

Clowns? Where!?!

@Noonycurt 10 лет назад

Great episode!

@pratyushbose6758 4 года назад

I have never seen such a beautiful explanation than this

@rosssharma542 8 лет назад

This is my favourite Numberphile video

@Radii_DC 8 лет назад

+Ross Sharma Why?

@rosssharma542 8 лет назад

I like the way it's explained

@boyinapeatbog 10 лет назад

This was a fantastic video.

@OhDannyBoy512 7 лет назад

I love this explanation! So very clever!

@user-wm7nk7db6p 10 лет назад

Excellent!! Perfect explanation!!

@NoahTopper 10 лет назад

I really like her accent.

@astroturf9367 10 лет назад

I giggled everytime she said clones like clowns

@Tyngdlyftning1 10 лет назад

i was first.

@SrmthfgRockLee 10 лет назад

...I kept thinking of Star Wars: Return of the Clones...

@joebailey8294 8 лет назад

#freetheclowns

@danielepicone1480 10 лет назад

Wonderful video, I love elegant puzzle solutions.

@darkaquatus 10 лет назад

That's not the real reason why you cannot free the clones, but whatever, it's been fun to watch.

@IMortage 10 лет назад

Very good guest, bring her back!

@Wompylulz 10 лет назад

I love how people comment with comments like :" Oh nice video, thanks" on a 13 minutes video that was putted on like 8 minutes ago lol

@MrZhabadasher 10 лет назад

Videos are not always available to every country at the same time. It might be 8 minutes where you live, and 15 minutes on the other side of the world. Typically, the western United States gets videos faster because Google is from California, and many of their servers are there. If you live on another continent, the region of that continent the Google servers are located at often get videos sooner.

@NoshNosher 10 лет назад

Mr Haberdasher That's news to me.

@AlexanderEVtrainer 10 лет назад

Loved this video a lot! and she gave a nice crash course on the geometric series for those who are unfamiliar with it.

@mumiuz 10 лет назад

Fantastic video, really interesting, would love to see her in more videos!

@Webcentaur 10 лет назад

What if you were playing in a 3 dimentional "chessboard"?

@jwhalstrom75 10 лет назад

Why are we talking about clowns?

@JakJakku 10 лет назад

Listening to Zvezdelina is absolutely mesmerizing, love the accent

@darkercheese 10 лет назад

I like these kind of videos. Solving a puzzle like this I think is really cool

@logancapes 9 лет назад

Can I get her voice on my GPS? I love it.

@hamfranky 8 лет назад

Free the clowns, for too long have they been oppressed by the Greenland administration.

@farstar31 8 лет назад

+hamfranky I'd like to think that, with that profile picture, that you're saying "Only you can free the clowns, ..." XD

@victorvictor4619 10 лет назад

I love these videos

@KipIngram 2 месяца назад

Oh. Oh my goodness. That is SUCH a nice, clean, elegant proof. I love that - it's one of those "Well, DUH..." proofs. Beyond obvious.

@GelidGanef 10 лет назад

A slightly more concrete reason why you can never fill the infinite board is that there will never be more nor less clones in the first column and first row than there are at the start of your game. (Play around with a board, this will become apparent pretty quickly). This makes the actual sum of your available space in greenland less than 2 (between 1 and 1.5 I think)

@numberphile 10 лет назад

check part 2...

@Nulono 8 лет назад

+GelidGanef *fewer

@GelidGanef 8 лет назад

Nulono I always use less when I have to use more cuz fewer has no separate antonym of its own and I feel it's more euphonious and promotes better parallelism. But thank you for defending the english language and all.

@JordanMetroidManiac 7 лет назад

And also as soon as you clone the bottom-leftmost prisoner, that space becomes impossible to fill again via cloning.

@htmlguy88 7 лет назад

GelidGanef try greater.

@HyperSpify 10 лет назад

Play it online: www.newgrounds.com/portal/view/631209

@anamay01 10 лет назад

Loved it!! Thanks a lot for this..

@ximomiagendainfantil484 5 лет назад

What a beautiful solution. Bravo!

@akabab23 10 лет назад

What's up Brady? It's hot back in Adelaide... Will Matt Parker or James Grime be in any new videos? Not that they are bad without them, just that those two tend to talk about really interesting topics :D

@PandaBashable 10 лет назад

Why is the starting weight not 3? The three pieces are the original pieces that have not yet split...?

@NoriMori1992 8 лет назад

Go ahead and use 3 if you want. Let me know if it makes the solution easier.

@itsokthen 10 лет назад

This was one of, if not the best numberphile video! Her voice is outstanding first off and she is extremely informative, and the whole problem was cool too. Great video!