Conway Checkers - Numberphile 

Rest in peace, John Conway.

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

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

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)

That parker square tho

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

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

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

i want more of her!!!

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.

That brief Parker Square was a nice throw back

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

Her accent is very nice : ]

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

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.

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

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

3:39 made me laugh out loud Brady :D

The Parker Square as the failure is hilarious

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.

that's one amazing accent

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

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.

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

I love the Campanile playing in the background!

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

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.

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.

Thank you very much. It feels like Christmas today.

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

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.

Brady showing us why he is the doctor!

So nice to see Zvezda again!

RIP John Conway.

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.

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

I FUCKING LOVE DR. STANKOVA'S VIDEOS.

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

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.

Checker has been disappeared

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

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.

Excellent!

I would drink Prof Stankova's bath water.

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

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

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

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

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

She's back! She even tops Grime!

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

Cool Zvezda, Sending regards to from UK

Omg this is so addictive!

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

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.

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

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.

This is the first numberphile video that I even vaguely understood

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.

I really like this lady'a personality. 🙂

The return of the Parker square.

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

Saw Prof Stankova, clicked in .0000001 seconds.

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

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

You didn't fail. You ParkerSquared!

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

Brady, you are frickin awesome.

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.

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.

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.

wow she is awesome

Mooie Zvezdelina!

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.

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

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

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

Row 2: The bottom piece jumps over the piece above it, the leftmost piece jumps over the one to the right of it, and lastly that same piece jumps over the piece now above it. Was that supposed to be hard? __________ ooo o

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?

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

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?

Holy crap I've missed her.

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

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

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

And now to something completely different: What is the least number of moves necessary to win a match of pool?

I am in love with Zvezdelina

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.

I love that Bulgarian accent! 😍😍😍

Always knew Brady has the spirit of a mathematician

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

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.

My question is; how far could you get with 3D Conway checkers?

I want to try this but i don´t have a checker board at home and playing it in my head is too difficult. After getting to the third row i have to check the configuration and the reality of the chips being there destroys the picture im building inside my head. And of course i can´t see my own errors.

What kind of shape would 2^(-x)+2^x=y be?

What if we used an extra dimension? The third one?

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

"inductive" yeah you better put that between quotes!!

So we have this board game called dominion(+2 expansions) they have 71 unique cards, in a game you choose 10 of these 71 cards to play with. so I wanted to know how many combinations of cards you can play the game with. I think i did my math right here it is: 71X70X69X68X67X65X64X63X62=1.67555505E+18 or 1 quintillion 675 quadrillion Is that correct?

Freeing the clowns 2.0