Conway Checkers - Numberphile

[@williamnathanael412]

Rest in peace, John Conway.

[@Triantalex]

*RIP in peace

[@mitchdg5303]

@@Triantalex?

[@lox7182]

Rest in peace

[@yourlocalclosetedgaybestie3165]

I love Prof Zvezdelina's videos. I really missed her. Thank goodness, there is an extended version as well. Really happy about that :)

[@johnnygroebs4407]

What is that gorgeous accent?

[@frankbrody239]

Don't ya know? Its foreign, of course!!! : )

[@stevethecatcouch6532]

Bulgarian.

[@jasertio]

She's a 10/10 GILF

[@dabbydevito5930]

Genius I'd Like to Friend?

[@PaulPaulPaulson]

On an infinite board, "put it somewhere in the middle" is the same as "put it somewhere".

[@ragnkja]

Yup. "Somewhere in the middle" seems to mean "somewhere in the middle of your field of view".

[@josephlombardo1246]

William White every number is near the origin on an infinite plane.

[@PaulPaulPaulson]

"Near the origin" and "in the middle" is not the same!

[@SCM240298]

The act of numbering an infinite board is just as arbitrary as "choosing the middle" of an infinite board to put a line on, sure numerically you have a reference point now, but op's point is that if you choose any point in the lattice it will have as many numbers ranging for all cardinal directions as any other point. You can't find the PHYSICAL middle of such board, creating coordinates means little because the origin point could be anywhere, it's just an arbitrary pick

[@JorgetePanete]

William White No, it isn't*

[@elay127]

You always get me with you cliffhangers for the proofs. Every time I watch a complicated numberphile video, I have to sink another 30+ minutes into watching the proof (sometimes more than once)

[@3ckitani]

That parker square tho

[@codynatof5901]

R.I.P. John Conway

[@mathlegendno12]

I’ll avenge you John Conway!!!

[@Triantalex]

??

[@Felix-ve9hs]

a checker board = boring some checkers = boring a checker board and some checkers with some rules = *M I N D B L O W*

[@ApiolJoe]

TIL checkers not only play on 10x10 boards but also on 8x8 in some official rules it appears. Glad I checked (lol) before replying.

[@felixroux]

And a line.

[@Triantalex]

false.

[@BobStein]

The moral: if the only way to get ahead is by destroying others, then society's progress is severely limited.

[@BobStein]

Not sure competition is doomed. Would like to think predation could be.

[@triinkett3995]

or conway's checkers is pretty cool i guess

[@maxbaugh9372]

And what do you know, there are only 4 or 5 trophic levels in Earth ecology! :P

[@timsmith3346]

some competition and some cooperation is the only way. Life isn't zero-sum.

[@BobStein]

Max Baugh, that and Maslow's Hierarchy of Needs need more levels... Next Wednesday let's make some Eagle Pie!

[@bakmanthetitan]

Big fan of this professor, I hope she's on more

[@DanielSMatthews]

Is the fastest way to solve the game to play it in reverse?

[@silkwesir1444]

Interesting question! I haven't yet watched the proof video, but thinking about your question makes me suspect, playing the game in reverse might actually be the key to finding out why reaching row 5 is impossible...

[@Triantalex]

??

[@unknown360ful]

John Conway is a LEGEND!!! Love it! Edit: Awesome Video as always Mr. Brady!

[@sleepyfrog3640]

Her accent is very nice : ]

[@rogerhudson9732]

What a star! ( a zvezda joke).

[@vapenation7061]

Zeno Phu eastern european accent lol

[@wronski11]

@@vapenation7061 Bulgarian, former student of our schools for mathematically gifted children.

[@al3ph35]

"Chyeckurs" very nice

[@cormaniac13]

That brief Parker Square was a nice throw back

[@someguyusingyt9091]

i want more of her!!!

[@Insanelas]

Mendel Chow 😉

[@pakkiufung883]

I basically watched Parker’s book “Humble pi: A comedy of maths errors” and I found that square at 5:32 is in that book.

[@MichaelWarman]

I first read about this in The Curious Incident of the Dog in the Night Time, but referred to as Conway's Soldiers. That was all I knew him for for quite a long time until the Game of Life featured on Dara O'brien's School of Hard Sums.

[@skoockum]

Love the interlude music. Nothing like a little Bossa Nova to help the time pass.

[@Kane_prajit]

that's one amazing accent

[@UrakovAnton]

I from Ukraine, and have the same)))

[@hypnolobster]

Bulgarian!

[@vapenation7061]

Peter Rabitt eastern european :)

[@rhyboy1]

I love that without knowing I see him getting smarter.... makes me feel like maybe that’s happening to us?

[@halamadrid7512]

Logically, we can visualize and simplify the explanation to this: any given checker can only move in 4 directions and with each move it removes (to be more accurate, it must remove) the adjacent checker in that direction (without this removal, it cannot move, following the rules). After 4 moves, it had removed all 4 checkers around it, thus making it impossible to make any more step in any direction.

[@jeremiwieczorek6628]

Thank you very much. It feels like Christmas today.

[@mighty8357]

3:39 made me laugh out loud Brady :D

[@Jivvi]

This is the configuration I found for row 3: 🔵 🔵 🔵 🔵 🔵 🔵 🔵 🔵 The bottom two on the left jump the two above them, then the four that remain work the same as the previous solution for row 2, and then the leftmost one on row 1 jumps across so it's directly under the one on row 2, and then up to row 3.

[@JackSassyPants]

The Parker Square as the failure is hilarious

[@danielg1910]

John Horton Conway, a legendary mathematician who stood out for his love of games and for bringing mathematics to the masses, died on Saturday, April 11, in New Brunswick, New Jersey, from complications related to COVID-19. He was 82.

[@MloveHb]

I always had the easiest time processing what was going on when Zvezdelina was instructing in the videos.

[@5gonza541]

Trying to solve this puzzles is so much more fun than studying for my chemistry test for tomorrow :p

[@natashacougoule951]

I love the Campanile playing in the background!

[@sebastiansimon7557]

4:34 - I came up with a similar configuration: same number of checkers, a row of five checkers right beneath the line, then a row of three checkers at the left or right below the upper row.

[@NoriMori1992]

So nice to see Zvezda again!

[@metalmenrock1]

Cool Zvezda, Sending regards to from UK

[@ubererhs2898]

I started playing this game in reverse. I put a checker on the 4th row and then set as the goal to get every checker back behind the red line. The allowed move is to put a checker next to another checker and then jump over it. Got a different solution for the 4th row and ran out of pieces trying to figure out the 5th. I dont really have a system for playing, kinda just messed around, but getting all the pieces under the red line while still making the pieces below stay connected felt impossible to me starting on row 5. Turns out that intuition was correct. Time to watch the proof.

[@zunlise2341]

That intuition is nearly correct. You need infinite checkers for row 5

[@dypteseu]

I would watch this video multiple times just to hear the accent again.

[@XWurstbrotX]

Chuck Norris can reach the 5th row of Conway Checkers.

[@toferg.8264]

You mean, without skipping over two adjacent checkers; don't you? I ask because, Chuck Norris can skip over two checkers at a time in Conway Checkers.

[@Jivvi]

@@toferg.8264 I think that means he can reach row 10.

[@toferg.8264]

@@Jivvi well played.

[@respiir]

I feel like because every time you jump (which can only be done horizontally or vertically) and the checker that you jump over disappears, you eventually get rid of all possible adjacent checkers that can help you advance.

[@CR0SBO]

I've been waiting for this since I read The Mysterious Incident of the Dog in the Night Time years ago, but I never knew this is what it was called! Woo, what a pay off after zero investment.

[@TheJamesM]

I have nothing intelligent to contribute regarding the maths, but I do love the design on her top. Very cool.

[@cheaterman49]

I think I'm in love with Prof Stankova :-) she makes everything interesting and she's very motivating as a teacher!

[@Tytoalba777]

It’s like building a rocket, to go a larger distance, you need a exponential amount more of material.

[@fee8422]

except no, because exponential things don't reach infinity in finitely many itterations, they just get large very fast. this one became litterally impossible.

[@Tytoalba777]

Marijn Zwitserloot fair enough

[@gcewing]

This suggests another question -- what is the maximum height you can reach in an n-dimensional version of the game?

[@tomsheppard5145]

I managed to get to the first row with 26 by forming a mechanism (4 counters arranged in a 3,1) on the second and first rows above the line, and then using that to put 1 counter on the 4th row

[@Zeppeli84]

I honestly found a way with the given pattern in my first try. Just move all 5 stones from the second line over the middle line, then the 2 from the 4th line on the second, then the left one on the 3rd to the right and the right one on the 3rd to the left and then these two on the first line. Then the left one on the first to the right. After that all 3 left on the first line over the middle line to the second line over the middline. Theb the right one on the first line over the middline to the left and then on the 3rd line , the left one on the second line to tge right and then to the 4th line.

[@NoriMori1992]

How was this only a year ago? It feels like it's been ages!

[@aquawoelfly]

As you hlve nd halve gain the number of checkers on the board you eventually reach a point you cant bring ny more into the middle.

[@thomassynths]

She's back! She even tops Grime!

[@kanye9233]

At first thought she was wearing that v sauce t-shirt

[@kaitudhope9122]

v-sellout :D

[@andybird3956]

This is the first numberphile video that I even vaguely understood

[@magicalpencil]

Brady showing us why he is the doctor!

[@Gastel]

Once the coin arrangement is larger than 4 coins, you can't create a 'side' arrangement to replace enough of the coins to continue. This is because you need a 4 coin dimension arrangement to move a coin up 4 levels.

[@TykoBrian7]

Gastel yep

[@Henrix1998]

Yeah, I used side arrangements to feed the middle line but also before seeing 5 is impossible, I thought I could make the same configuration that I used to go to 4th line, to feed the line from the sides

[@OlafDoschke]

Excellent idea, for reaching higher, we just need to add dimensions to feed in coins from further directions. Is there a 16 coin solution for row or level 4 in 3 dimensions?

[@suicidalbanananana]

Not in 3, but there is in 4. (3rd would go wrong way, like up/down instead of forward/back across the board) a 4th has no defined 'plane' or direction, so we as the theorist can decide how 4th dimension works, and it could be on the same plane/direction as one of the first 2 dimensions.

[@OlafDoschke]

Haven't imagined it yet, but the third dimension would be forth/back and you can feed layer z=0 from layers z=1 or z=-1. I'd still give it a try, there might be other reasons, but you might also shift the goal to reach level y=4 not just on any x, but also on any z coordinate. (If you can do so, you can shift the starting configuration to reach z=0 anyway).

[@KipIngram]

I really like this lady'a personality. 🙂

[@happypiano4810]

RIP John Conway.

[@TooManyEditsProductions]

The return of the Parker square.

[@RealClassixX]

I wonder if, on my deathbed, I will regret becoming an engineer instead of a mathematician.

[@Harrs2]

I FUCKING LOVE DR. STANKOVA'S VIDEOS.

[@seanm7445]

Damn. The red bar on the icon made me think I had watched this already.

[@bronktug2446]

I love that Bulgarian accent! 😍😍😍

[@vasilzhekov9245]

Здр бебце кп? :D

[@jyrgenruut]

Oh yay! I actually managed to solve the row 4 (well, with the checker pattern shown, though)... not in a bad time, either, actually.

[@Triumvirate888]

I haven't seen the proof video, but I would assume that it has something to do with the idea of a bell curve with upper and lower limits applied to a system with irreducible complexity. To get to the first line, you have to sacrifice a minimum of 50% of your checkers. Then you have to sacrifice a minimum of 75% of them to reach the second line. Then 87.5% of them for the 3rd, and 95% of them to the 4th. The pattern is like this: First Row: 50% Second Row: 50% + 25% Third Row: 50% + 25%+ 12.5% Fourth Row: 50% + 25%+ 12.5% +7.5% Fifth Row: 50% + 25% + 12.5% + 7.5% + ?? If the last number is 5% or higher, the fifth row is impossible. And since the difference between 12.5% and 7.5% is 5%, even if the number of checkers exactly doubles, you'd still not make it to the 5th row. You would have to use fewer than twice the number of checkers from the last row, which is not possible on such a curve due to the space needed to perform the actions. Now you might say "But there is infinite space and infinite checkers." But that doesn't matter, because of the efficiency vs. space problem. There are only a certain number of checkers that matter (the ones required to move a checker up the lines). Those are the only spaces that matter, and each of those spaces is exactly the same size as a checker. Everything else is just wasted moves, non-efficient checkers that don't do anything. So even though you have an infinite amount of space, you are actually limited by the size of each checker being the same as the size of one unit of space. Each time you make a move, you are shrinking the actual usable space that matters. When you shrink the usable space each time you move, there are only a certain number of moves you can make before you've divided the USEFUL space down to below the size of a checker square. The size of a checker square is irreducible, as is the number of moves each checker can make. So you end up sacrificing and subtracting over 100% of the checkers needed to get to the 5th row. And if you subtract 100% of infinity from infinity, you are left with 0 checkers remaining.

[@AshArtois]

I did it on my first try, but i think i got lucky. Thought i was stuck, looked at it for a second and finished the last few jumps.

[@keithwilson6060]

If the board is infinite, how can you find the ‘middle’ to draw a line?

[@keithroberts1401]

Checker has been disappeared

[@KamiKuzi]

Awesome. Fun to watch. And a surprise lightsaber at the end...

[@ricosuave666]

I can't get passed that bullet hole on the chess board.

[@BobStein]

I know right. Seems to hover like an animation. Had to replay because I wasn't listening.

[@NoriMori1992]

Yes, it's somehow really distracting.

[@РумянаСтанкова-с7ш]

Excellent!

[@vapenation7061]

Румяна Станкова you got the same family name!

[@mow184]

Brady, you are frickin awesome.

[@Nicoder6884]

I wonder how much it would change if you can just jump straight over the line without having to jump another checker, but only when it comes to jumping the line

[@juanpedrolardet2088]

Omg this is so addictive!

[@hansmuller1846]

I thought I'd watch the proof. Until I saw it's 40 mins long :D

[@MisterAppleEsq]

Watch it anyway.

[@JimmyLundberg]

I thought I'd watch until I saw the episodes are all 40 mins long. ... If you would have liked a 10 min video, wouldn't 40 mins be even better?

[@41-Haiku]

If you watch mathematical proofs with the same degree of engagement as a movie or TV show, there is something objectively wrong with you. ...Now If you don't mind, I have a proof to watch.

[@JimmyLundberg]

Haiku Metzger I guess so. I find this more relaxing than Netflix.

[@MisterAppleEsq]

+Haiku Metzger I agree, there's something wrong with you if you find as TV as entertaining as mathematical proofs.

[@Grassmpl]

What rules apply involving checkers already above the line? Why can't we use 2 checker vertically and keep jumping after being above the line forever and ever?

[@KeNi666]

You didn't fail. You ParkerSquared!

[@realpatrik]

This may be a philosophic question but if the board expands to infinity in all directions, is there something like the middle? I don't think so but I'm excited to read what you think or even better what the mathematical proof/argument for or against is.

[@knurbsi]

That manic star face at 3:39, 4:34 and 5:32 - hilarious. :)

[@Filip_Wessman]

I guessed on 4! HAd a gut feeling it just gets harder and harder till it dont go any more.

[@ArslanShahh]

How about we think the solution in reverse order? If we're gonna end up in nth row, then simply place it in that row first and then undo the process.

[@ArslanShahh]

Probably much easier than brute forcing possibilities of positions right off the bat.

[@fee8422]

it wasn't necessarily, you can spot paterns, for example the solution for row 2, the L-shape, ends up 2 spots in front of the small part of the L; a square of 2x2 ends up in either of the 4 corners diagonally from the 2x2 square etc. I tried it when I paused the video, I got to 1 in 2; 2 in 4; 3 in 8 and 4 in 22 quite easily, it wasn't optimal it seems, but it was very easy once I spotted some patterns.

[@MisterAppleEsq]

That's how I did the ones in the video.

[@arnavjoshi796]

This guy invented Conway's game of life, right??

[@iAmTheSquidThing]

Yes. I can see parallels with this.

[@ThisIsAYoutubeAccountAsd]

Yes. I find it to be an extremely curious coincidence. He discovered something which happens to have his very surname! That's amazing! Same thing with Richard Feynman, he discovered the Feynman diagrams which also happen to have his very surname. Crazy how nature do that.

[@BARRUTG]

I don't know about Conway, but the diagrams that Feynman came up with were called Feynman BECAUSE he discovered it.

[@miot22]

---------->joke TwiTwiPlanet

[@stevethecatcouch6532]

It's weirder when it doesn't happen that way, the Simson line, for example, was discovered by William Wallace.

[@meeDamian]

At 5:33 I started laughing aloud like a crazy person 😂

[@tamaspolyak5564]

He so got Parkered...

[@hristo.bogdanov]

Страхотно видео, поздрави и всичко най-хубаво :)

[@pavphone2616]

CYKA BLYAT

[@hristo.bogdanov]

Sorry - I'm not a russian.

[@ofernandofilo]

neither am I. I'm not a russian.

[@vapenation7061]

Pav Phone that’s bulgarian, professor stankova is bulgarian :)

[@pavphone2616]

England is my city TBF I'm an idiot.

[@PrimaryBandit]

so you can't jump pieces once they're above the line I'm assuming. Like having two row 3 checkers jump horizontally, and line up under a row 4 checker to then jump and get a row 5?

[@SIStefanov]

Звезделина!!!!!! Поздрави от българи в Израел!!!!!

[@adamgonzales8316]

Please explain Eulers method as used in the movie “Hidden Figures”

[@ThePacificOc]

Always knew Brady has the spirit of a mathematician

[@JacobPlat]

Mooie Zvezdelina!

[@ChrisLeeW00]

How far can you get if it were a 3 dimensional checker board? How about a 4 dimensional board?

[@27122712ful]

I am in love with Zvezdelina

[@kalo_yanis]

Поздрави от България!

[@KarstenJohansson]

Instead of binary (powers of 2), can't you use the Fibonacci sequence to get a similar result with simpler math? The reason you can't get over 5 is because the next number would have to be 8, which is too far to jump as per the game rules.

[@scottlerman1805]

Saw Prof Stankova, clicked in .0000001 seconds.

[@dylanrambow2704]

The number 5 seems to ruin everything. It stops polynomials being solvable by radicals. It stops rings of integers from being a PID. And now it stops this game from always working.

[@kikotanto2980]

Actually, i did the 20 checker thing with the same configuration but a different order in which i moved

[@OlafDoschke]

The game strongly reminds me of Peg solitaire.

[@akshat9282]

Had numberphile2 video in sub box but not this one even after 10 minutes of upload of the other video I'm subbed to both for sure

[@kfftfuftur]

What if you tryed row 4 on a three dimensional grid? Wouldn't you be able to make it? Because you need a side to supply the coins on. On row one you do it with just one dimension. For row to you need to supply additional coins from the left so you need 2 dimensions. But for row three to follow the rule you could add a two by two square on the right. And you wouldn't need an additional dimension because you still have an unused direction. My guess is that you would have to add a third dimension to get to row 4 with 16 coins Row 5 should also be possible with 32 coins on three dimensions, but row six would need a fourth dimension to be possible with 64 coins.

[@DjVortex-w]

One would think that one could build an arbitrarily long "ladder" of checkers with one space between them (so that you can then use one checker to jump over all of them) by bringing checkers from the sides, no matter how long of a process it may be. But apparently not.

[@gabor6259]

What about extending the problem in 3 dimensions and instead of a line we'd have a plane?

[@GermansEagle]

wow she is awesome

[@CrepitusRex]

Great challenge.

[@nanamacapagal8342]

One more like for the Parker Square.

[@apburner1]

I would drink Prof Stankova's bath water.

[@vapenation7061]

apburner1 that’s an interesting fetish you got there.

[@visorkatossa]

Can someone explain to me why the max number of y=x^(10-x) is 4.1336605 and not 5?

[@zerid0]

What about a n-dimentional checker board? Is there always a maximum?

[@zubmit700]

Holy crap I've missed her.

[@ZylarGray]

Took me about 7 tries but I did it with 20. But my moves were different than theirs. This is fun