Conway Checkers (proof) - Numberphile

Here's the proof - the main video is at: • Conway Checkers - Numb...
Featuring Zvezdelina Stankova - more videos with her at: bit.ly/zvezda_v...
Pebbling a chess board: • Pebbling a Chessboard ...
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[@ClydeHobart]

Rest in peace, John Conway. Your math will inspire many for years to come

[@btfx]

Best handwriting on Numberphile so far.

[@nathanielgomez8350]

Vasiliy Sharapov You're looking for the real deal there. Stay Strong. *Insert appropiate emoji*

[@GordonHugenay]

Holly Krieger also has magnificent handwriting

[@tiavor]

I think this was one of the most complete and best explained proofs on your channels.

[@BunniBuu]

Yeah, there wasn't any "Okay let's just assume you know this formula" or "this would take too long to explain so I'm going to gloss over this one section". Very nicely done!

[@dante_rf]

Yes, they should do more videos like this

[@OMGclueless]

It's a long proof but it's totally clear the whole way. She's amazing at explaining this stuff without oversimplifying. One of my favorite proof videos you've done.

[@zh84]

This proof is given in Berlekamp, Conway and Guy's "Winning Ways", but the proof here is much clearer.

[@GijsvanDam]

This has to be one of the best Numberphile videos made untill now.

[@ely_mine]

How is it clear that moves are like powers of a number ? I understood the whole explanation, but not why remplacing cells with powers of X is relevant.

[@antonimaciag1259]

It's just because it lets you show what you want to show. There is no other explanation.

[@NinjarioPicmin]

@@antonimaciag1259 sure it lets us show exactly what we want to show, but there was no indication on how you would come up with exactly that formula

[@IchBinKeinBaum]

9:20 "It's a free country, I can put whatever I like in those cells." There's a political joke hiding in that sentence.

[@geekfanboy]

IchBinKeinBaum ,

[@peppybocan]

Communistic joke AF :D ... Communistic countries were "proforma" also free.

[@Doomsmoker]

Nope, that's a joke about US, where they tell you how free you are, but, actually, you are bound by strict laws.

[@KALSAFilms]

IchBinKeinBaum #lockherup

[@PhilBagels]

There was an old joke in the Soviet Union: In the Soviet Union, you have freedom of speech. In America, you have freedom after speech.

[@please.dont.]

Watched the whole thing. Did not regret a minute.

[@NoriMori1992]

I'm amazed at how organized Zvezda is with all the equations and notes she writes! She never seems to lose track of her previous notes, or run out of space by accident!

[@yourlocalclosetedgaybestie3165]

WOW!!!! I THINK THIS IS THE LONGEST VIDEO ON NUMBERPHILE2!!! I am really happy that it's by Prof Zvezdelina I really missed her. I guess I just watched a couple of videos by her but I really like her videos. Thanks Numberphile for uploading such a long video by her. You made my day a lot better :)

[@Eurley66]

The longest is actually the interview of James Simons, I think!

[@Ledabot]

Yea and i found it at 1.30am on a day before work. I'm sad I'm goto have to skip it and sleep

[@AndersJackson]

Agree. Wow!!!

[@AaronHollander314]

You can never be too rich or too thin ;) or have enough Prof Z!

[@Patrickhh69]

The longest on numberphile2 is an hour of coloring the collatz conjecture

[@nomekop777]

"I will define what I like if it does what I want"

[@VagueHandWaving]

now THAT should be a shirt

[@cameronboyle2746]

Im quoting this given the first opportunity haha

[@Henrix1998]

I didn't plan to watch this all but something happened between 20:00 and 41:54 that made me lose my sense of time

[@Nathouuuutheone]

I had no clue the video was this long!!!!

[@TheNinjutsustudent]

This is my new favorite movie . . .

[@lezhilo772]

I never realised how many subtle relationships there are for the golden ratio :D Also what a nice proof! I don’t get to see these things as often as I like in physics.

[@BigDBrian]

there's all sorts of neat little numerical properties of phi. because of the relation x² = x + 1 , it is plain to see that the decimal expansion of phi and phi² are the exact same. The same is true for 1/phi through slight algebraic manipulation (divide everything by x) Personally I find it extremely cool how you can take the inverse or the square of a number with infinite decimals and have the exact same digits!

[@hpekristiansen]

Yes also more simple - consider which of these numbers are the largest: 1/sqrt(2), sqrt(1/2), sqrt(2)/2 or (1/2)^(1/2)

[@BigDBrian]

all of them are the largest ;)

[@skilz8098]

The golden ratio, e, pi, sqrt(2), ln(2), they show up everywhere!

[@GRBtutorials]

@@skilz8098 Yeah, and π especially has the habit of appearing when you least expect it! Who would have said that the infinite sums of the reciprocals of the powers ≥ 2 of natural numbers (the zeta function ζ(x) for x ≥ 2) would contain π in it?

[@kala_asi]

However, if one was to make the first move in 1 second, the next one in 0.5 seconds, the third in 0.25 seconds and so on, you could get to the fifth row in 2 seconds

[@Maharani1991]

Hahaha :D

[@DerekLeClair]

Only true numberphile viewers understand

[@Krekkertje]

Well actually, if the first move took 1 second, then 2 seconds for the next move, then 3 etc. which makes sense because we have to keep reaching farther for each move. We get 1+2+3+4... And we'd win 1/12 seconds ago.

[@Xandawesome]

+Krekkertje Clever, very clever

[@jacksonmcclintock4230]

I was thinking of supertasks too

[@laurel5432]

Now that's a good proof And when your first row is below the fifth, the sum is bigger than one so you don't need infinitely many moves anymore, so you can get there Was also quite nice to have learned something practical about the golden ratio, not like those abstract things it's supposedly doing (the things you hear in movies etc.)

[@AlanKey86]

At about 21:48 you suddenly get a sense of where this is going... It's really exciting!

[@mymic208]

The eureka moment if you will.

[@ultearmilkojohn1145]

Happened at around 0:53 for me :/

[@Henrix1998]

About that same time I lost my sense of time and it felt like 5 minutes

[@sybilbirling8813]

ultear milkojohn it just didn't happen at all in my case..

[@rlicinio1]

Exactly!

[@stoirtap12]

4:02 Brady's so sick of the Golden Ratio constantly popping up.

[@ludomine7746]

came back years later, and can finally appreciate how cool and elegant this proof is.

[@stevethecatcouch6532]

I love the crazed look at 41:30 as she delivers the fatal blow. Great video.

[@LIES666]

Very satisfying once everything starts to come together.

[@RagingPanic]

LOVED the long proof, would really like to see more long videos like this and obviously more of Zvezdelina would be great as well.

[@99jdave99]

wow, this is insane. Not a mathematician, but I was able to readily follow this the entire way. That proof of phi^cubed blew my mind, even having to rewind a few times to figure out what was going on. Awesome stuff, and very inspiring!

[@tgwnn]

The fact that the entire sum of every checker below the line is *exactly 1* is why we couldn't have chosen any other number. Pick something smaller (say, 0.55), and suddenly x^2+x -> 1 is increasing our sum, so "uphill moves" would no longer be monovariant. Pick something larger (say, 0.7), and suddenly the sum of the checkers below the line is larger than 1. This is breathtaking.

[@johndinner4418]

Could you elaborate?

[@xCorvus7x]

"There can be only one!" A mathematical version of Highlander.

[@dolfinity7858]

Now im curious. if you add another infinte dimension to the checkers board, same rules apply, then what is the maximum distance you can travel from any starting point on your 3d board?

[@alexandersanchez9138]

The main computational result from this proof is that the sum of the grid beneath the line is x^(n-5), for a center point on the nth line above the middle-line. So, just add copies of the grid for 3 dimensions. Based on the same calcuation as they did for the row, we conclude that the sum for a 3 dimensional lattice is x^(n-8). And, in general for an n-dimentional lattice, you get the sum below the line to be x^(n-3d+1). So, the methods of the proof lead to the following conjecture: Given a grid of dimension d, you can reach up to, via analogous rules, the (3d-2)th hyperrow above some dividing hyperplane. This proof suffices to show that you can't do any better than 3d-2. However, to show that you can actually get to the (3d-2)th hyperrow, you'd probably have to demonstrate an algorithm.

[@alexandersanchez9138]

We know that you can't get 8 or more, but we don't know for sure that you can get 7.

[@shamus030]

So then the question is for which dimensions is it impossible to get to the (3d-2)th hyperrow?

[@alexandersanchez9138]

Right. I'm inclined to think that you can always get to the (3d-2)th row, but that's just a conjecture: it requires a separate proof.

[@luciano.a]

I need this answer!

[@taingbunhong4085]

I have not seen very many proofs like this, but still want to say: this is the most beautiful use of the concept of infinity in a proof. Worth the length. Thank you for investing the time.

[@StefanoMersi]

Absolutely perfect demonstration. One of the very best videos on this channel!

[@h0axyboi486]

One of my favourite proofs on numberphile ever. It never gets boring throughout. I was surprised that forty minutes went by so fast.

[@StreuB1]

Professor Stankova has just this amazing ability to explain complex proof and make it intuitive to the viewer. Her explanation of geometric proof in the past really shows just how absolutely solid and fundamental euclidean geometry was in schools her native Bulgaria.

[@amaarquadri]

Its incredibly beautiful that this proof works despite being so close to not working. If you replace 1/phi with a number just a tiny bit (epsilon) smaller, then x + x^2 will be less than 1. Then capturing a piece and moving towards the center will result in the sum increasing which breaks the proof. On the other hand, if you replace 1/phi with a number just a tiny bit (epsilon) larger, then the sum of all the squares below the line will be greater than 1 which also breaks the proof. 1/phi is the perfect sweet spot, and the only number that could be used for this proof.

[@nikolayhidalgodiaz9463]

Also, the problem (conway checkers) was clearly not specifially engeneered to produce such a miracle, it all happened in purely a natural way. Mind bending.

[@dinofirechief40]

This my third watch of this video. The first time brought tears to my eyes. It's amazing how much beauty is locked up in our ability to understand as much as in our ability to sense. Thank you numberphile and Mrs Stankova for showing us incredible worlds.

[@Czeckie]

prof Stankova is a treasure

[@lobrundell4264]

11:47 "brilliance" describes Professor Stankova's brain but also their smile :D

[@Bibbedibob]

This is one of the most beautiful proofs I've ever seen

[@Allanfallan]

This is a really good proof. As a high school graduate, I was still able to follow what was being explained. As soon as I realized only one piece could remain in row 5 for the puzzle to be solved, I knew that it was impossible. It all boils down to the fact that you can't start with an infinite number of something and end with a finite number. The journey was finding out that you had to end with 1. Very interesting!

[@Vhite]

I usually don't have the interest/capacity for proofs, but this really got me hooked.

[@HarperGamble]

This video was exactly as long as it needed to be. I had the ah-ha moment right as the video ended. Brilliant.

[@TreuloseTomate]

This is one of the best Numberphile videos.

[@Quintinohthree]

Now I'm wondering what the inverse algorithm looks like. If you start with a single checker in row 5, and every time you jump it two squares further a new checker appears in the square that was jumped over, how can you populate the entire board on the other side of the line.

[@ahmedouerfelli4709]

That would be a game with an end but no beginning. Maybe we can call such thing a reverse supertask.

[@ely_mine]

That's how I tried to find the arrangements in the main video, but it guess it would require an algorithm which can establish a new configuration to pop a new cell, given a configuration. That surely would involve recursivity which makes it even harder to compute, let's find a better way. (I will not show you the way)

[@jarredallen3228]

I'm sure there exists a path you can take that would do that. If you wanted to fill the entire board (not just below the line), you could follow a Hilbert Curve (en.wikipedia.org/wiki/Hilbert_curve ), which would cover the entire board in checkers. There probably exists some path which does the same as the Hilbert Curve but staying below the line, if you wanted to only cover that segment.

[@cesarantonioenriqueramirez]

@@jarredallen3228 the real question would be: could you find a path that leaves the top of the board empty?

[@hansonbai8808]

It would be impossible to end up with no checkers on the starting side with finitely many moves

[@yordandar]

That's so interesting and satisfying - the steps to discovery, the 'scientific spirit', the intuitive explanations...! Also, the really beautiful handwriting :). Prof. Stankova, thanks a lot and hello from Bulgaria :).

[@Bluhbear]

Came here from Vsauce. RIP, John Conway. :'(

[@NoNTr1v1aL]

Close your eyes and listen 34:20 dirty talk maths edition!

[@wesselbindt]

All math talk is dirty talk to me

[@skoockum]

That accent does to me what it does to Jamie Lee Curtis in A Fish Called Wanda.

[@dontfeelcold]

This video has me feeling giddy with excitement and hanging on the edge of my seat. The simple pleasures of life.

[@nathanielgomez8350]

Let me get this straight. You can win the game by making an infinite number of moves, but if you do win the game it means that you've made a finite number of moves and thus you can't have won? Wow.

[@Macieks300]

No, if you did win the game it doesn't mean that you've made a finite number of moves

[@martimlobao]

You also don’t show that it’s possible to win just by showing the sum at the beginning is equal to the sum at the end, all you show is that you can’t exclude that possibility using this method. But you can obviously come up with several initial and final configurations in which the sums are the same and yet there is no way to move from one configuration to the other (for example, if no two pieces are adjacent).

[@SirFloIII]

make your first move at time t_0 = 0 make the nth move at time t_n = t_(n-1) + 1/2^n you are done with infinite moves at t = 1

[@aleksandarvlasev4030]

It's a little more subtle. What's it's saying is that in order to have enough "energy" to get to row number 5, you need to have infinitely many checkers underneath the line. In fact, the numbers work out such that you need ALL of the checkers underneath the line. If you don't use some of them, you don't have enough energy to get to row number 5. Therefore, it's impossible to do in finitely many moves.

[@stevethecatcouch6532]

SirFloIII. The moves are countable, so each move has a positive integer associated with it. The first move is associated with 1, the second with 2, etc. Which positive integer is associated with the move which first reaches row 5? Hint: it's a rhetorical question.

[@twwc960]

Definitely worth watching all the way through. The proof is simple to understand, but I never would have come up with it on my own.

[@earthbjornnahkaimurrao9542]

very beautiful maths. absolutely love it. need more of these long version numberphiles!

[@someguyusingyt9091]

i need more conways game

[@alexpotts6520]

Incidentally, there is at least one other mathematical game where the golden ratio appears. It's called Wythoff's game. In Wythoff's game, two players have two piles of (not necessarily equally large) coins in front of them. A turn consists of removing any number of coins from either pile, or removing the same number of coins from both piles. The winner is the person who takes the last coin. It turns out that the winning strategy is to keep the ratio of the sizes of the two piles as close to the golden ratio as possible. Phi is one of those numbers that has a habit of popping up when you least expect it.

[@masked0warrior]

Mindblowing. It's interesting how seeing the proof made me see this game in a completly different way!

[@Omikron35]

That's one really pretty proof right there :) really interesting maths emerging from a simple problem, Conway magic

[@lowercaserho]

What a wonderful video. Professor Stankova is always a joy.

[@stephendavies5968]

Whoa, longest numberphile video ever(?) but engrossed whole way through and ultra-satisfying conclusion.

[@orestisfraSPDR]

ok my brain is fried but i understood everything

[@hitoshiyamauchi]

This is just amazing. I literary cried at the last. Thank you so much.

[@antonlushankin9568]

You know what? I don't know why, but the proofes like these always give me a huge smile at the end, as if all in sudden after all these 40 minutes has turned out to be so ridiculously smart and yet so simple. It gives just an explainable burst of joy, that comes a solving of some secret or mystery. Absolutely wonderful!

[@OriginalPiMan]

On a tangential note, I think the mile needs to be redefined as φ kilometres. It is already pretty close.

[@toferg.8264]

OriginalPiMan , oh cool!

[@angelmendez-rivera351]

That would make the unit more impractical than it already is. We do not need that really. It is a mathematically neat concept, but units of measurement should stay on the practical side of things. After all, they're made for measurements, not abstract thinking.

[@tgwnn]

That would be gold.

[@UnlimitedRadioButNoSoap]

Well boys, we did it. After an infinite number of moves, we've finally made it to the 5th row

[@ragad3]

Umm… who are you talking to?

[@PenguinMaths]

Every time Brady talks I get so happy, he's so willing to participate and try his best even when it's an area of math that he may not be familiar with.

[@wierdalien1]

John Conways games lead to some real interesting maths

[@bluekeybo]

Alistair Shaw i think a lot of people have come up with interesting ideas that lead to cool maths, but not many others will pursue them and make them popular. Conway is such a big name that people will devote careers to solving his ideas

[@wierdalien1]

bluekeybo my point wasnt that other people havent done created lots of important and cool maths. Far from. Nor was it an overt celebration of conway himself, although he does in many ways deserve it. No its more that cool maths arises from both interesting and trivial places.

[@zh84]

You might like the book on mathematical analysis of games which he wrote with Elwyn Berlekamp and Richard K Guy, "Winning Ways". I have had it for twenty years or so and read bits of it many times, but never come close to mastering it. In one game I used to play at school, Fox and Geese (though we called it Fox and Hounds), it turns out that the geese have an advantage of one plus the reciprocal of the largest possible infinite number!

[@cendyywarlos]

His study of games actually led to the development of a new set of numbers! Look up "surreal numbers;" they're incredibly interesting!

[@wierdalien1]

Ethan Smoller thats exactly my point

[@OhDannyBoy512]

THIS WAS SUCH A WILD RIDE! WHAT A CONCLUSION! So glad to have another video by Professor Zvezdelina :D

[@AlanKey86]

This is excellent! Thank you Zvezdelina and Brady!

[@prydin]

A beautiful proof by a great educator. We want more Zvezdelina!

[@obsidianHelix]

Massive respect to both of you, but especially to this wonderful educator. What a great video.

[@shubhamgune1168]

Lets get to solving Reimann's Hypothesis now!

[@GaMatecal]

Just for giggles, I actually drew up a 25x25 board, with the 1st row being the 5th row, just like the video has done. I used buttons as pieces, and filled the line below for the solution for the 4th row. I stopped their, because I realized that I couldn't add enough pieces to get a piece close to the 1 piece in the 4th row. It would always be >2 spaces away, making it impossible to reach that 1 piece. After having it stare you right in the face, you realize that there isn't enough room. The solution for row 4 takes up waaaay too much space.

[@AlbertJoeElectronicKid]

The model and numbers used in this proof is particularly beautiful!

[@Juggler1097]

"I will define what I like if it does what I want." I wish that answer would have been accepted in my math classes.

[@AlaskaSkidood]

I made a checker board out of quartered sticky notes, and used paper clips as pieces. Great fun!

[@be_clay6785]

This is far better than the first part. I’m glad I watched the whole thing.

[@pcfilho425]

Professor Zvezdelina, I love your math, your eyes and your accent. ❤️

[@leo17921]

when you do an insanely hard calculation and you get 1 out the other side its just really satisfying

[@izakj5094]

Amazing, do more proofs whenever possible

[@JaredNeil]

I'm not sure why, but this was my favorite proof I've seen from Numberphile.

[@rikicraft9476]

the video is one of the best i've ever seen and even these comments are incredible

[@eduaft1]

Pay off at the end was worth the whole explanation!

[@christophersavarese7356]

Wow, that was a great proof. It reminded me of Simpson's paradox at the end.

[@kayleighlehrman9566]

(cracks knuckles and pulls out my pencil, slide rule, and checkerboard) let's get to work!

[@pedrohenriquecontente332]

My mind was blown infinite times during this video

[@daanroelofs119]

"I will define what I like if it does what I want" DAMN

[@ManuelRuiz-xi7bt]

Little error on 25:21: you don't always decrease by 1 (only in the middle) , but you do decrease by someting strictly positive.

[@rebase]

That hole in the board really bothers me

[@U014B]

Right? All of my infinite checkers keep leaking out through it!

[@dragoncurveenthusiast]

That was so worth my time! Great explanation using nothing but high school math. Amazing!

[@FirstnameLastname-hg5gt]

Great video, I am amazed. The value that actually worked is [sqrt(5)-1]/2, the opposite of the evil twin. But φ (the golden ratio) was found as the positive solution of the equation x^n+x^{n+1}=x^{n+2}. On the other hand the desired number is the positive solution of the equation x^n=x^{n+1}+x^{n+2}. This is in my opintion one easier description of the numbers that works rather than describing it as the opposite of the evil twin.

[@AaronHollander314]

Wow! Brilliant. Just following along. I can't even imagine truly understanding how to reach this conclusion.

[@AaronHollander314]

Watched again. Mind blown!

[@ianbarton1990]

Todd Rogers got to the fifth row in the 80s, he has the tape somewhere he just needs to find it....

[@toferg.8264]

Ian Barton , Ha ha ha.

[@Tracequaza]

well obviously he did it by already starting in the second row

[@madlad255]

@Ian Barton@@Tracequaza Underrated comment + underrated reply

[@wouterlahousse9637]

10:01 -What do powers of negative numbers do? -They alternate? .... -They FLIP-FLOP !!!

[@elderhickory0776]

Who’s here from vsuace 😂

[@AminGhomati]

You came here from Vsauce2 right, WRONG

[@frozen_antifreeze]

@@AminGhomati maybe?

[@anindyabiswas1551]

I saw this before, but this time, from vsauce 2

[@dilemmacubing]

Me.Right?WROOONG!!!!!!

[@skrrskrrrr99]

or is it?

[@Rivenbourn]

First time watching this blew my mind. Rewatching it is the most beautiful thing i've done

[@WildAnimalChannel]

That's blown my mind. I would never have known how to prove that in a billion years.

[@yeetzera]

I cannot admit i understand all of it but the way it wrapped up in the end was beautiful.. When you realize if you need infinite moves to reach to a point, then you will be always reaching for it :)

[@toferg.8264]

That jump cut at 9:16 is hilarious!

[@MathLady42]

What I imagined at first was a game of checkers played with normal rules between two people, but the pieces follow the rules of Conway's game of life.

[@angelindenile]

I feel like if I had known the quadratic formula was related to Phi in high school I would have actually understood my algebra classes a bit more... Two thoughts: 1: a gut feeling I have is, the reason you can't get to row five is because it's stuck at two dimensions and might work at higher dimensions. 2: this feels like a stepping stone to the understanding of the proof as to why there's never enough energy in the universe to get anything with mass up to the speed of light....

[@aliasmask]

You're going to need 2 more checker boards stacked on top of the first.

[@antonimaciag1259]

A very clear and understandable proof of a truly badass thesis.

[@fatpie2.0]

I enjoy these more detailed explanations

[@Toprak135]

GOLDEN RATIO 20:48 “let’s start summarizing” THERE’S STILL 21 MINUTES LEFT

[@JAzzWoods-ik4vv]

This is such a great video, i absolutely love explanations of complete proofs. I hope there are more to come! I do wonder, is there an algorithm to reach row 5? The same way you can never equal pi but we know of an algorythm that allows us to get there

[@davidknight247]

Fantastic accessible explanation, worth every minute.

[@epliroforiki]

Amazing explanation, and nice use of mathematical tools.

[@retepaskab]

That's what I like in maths: making up silly rules to prove useless things, and getting bonus facts like the sum below the line is 1.

[@evgenykuznetsov7759]

😦😧😮😮 Great Conway!!! Sad to don't have him anymore with us!!! Great presentation thank you.