Magic Hexagon - Numberphile

Dr James Grime talking Magic Hexagons (and magic squares).
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[@SpeeDim]

I just love James Grime

[@numberphile]

SpeeDim so do we!

[@andrew_cunningham]

There's just something about him, isn't there...

[@uselesssanity]

Andrew Cunningham perhaps its his little professor

[@gavs928]

maybe it's just because he's British and I'm not, but he seems like he'd make a great doctor who

[@alfredomoreno8516]

Yes he conveys so much enthusiam

[@superj1e2z6]

At least it is not a Parker hexagon

[@taba1950]

too late?

[@threepointonefour607]

Too soon

[@mco-meowcatofficial8415]

haha xD

[@davecrupel2817]

lolololololololololololol

[@nomb3179]

GD MeowCat gd

[@elwynbrooks]

His enthusiasm makes me so happy :D

[@何幽-p5x]

me too

[@CompoundInterest-SG]

James Grime is so great. I always know it is going to be good when it is a video with him.

[@sethgrasse9082]

That size 1 magic hexagon blew my mind

[@TrimutiusToo]

Yeah,not to mention the rigorous proof that it is indeed magical

[@yusuf-5531]

He didn't mention that an n=0 hexagon also works

[@TrimutiusToo]

@@yusuf-5531 diagonals in n=0 hexagon aren't well defined so it is way too hard of a proof for this video

[@Triantalex]

??

[@AlanKey86]

Does anyone have wood? I'll give you 2 wheat for 1 wood...

[@CraftQueenJr]

AlanKey86 yep, do you have 1 sheep? I’ll give you two wood.

[@Maninawig]

Awkward for any guy to hear.... Odd glances everywhere

[@MisterHunterWolf]

*rolls seven*

[@jacobr7729]

But I have all the ore....

[@TrimutiusToo]

I have wood for sheep

[@swarm9582]

Thank you for being colorblind friendly in the animation because I had no idea what you were talking about with the shape grouping until that point.

[@wolfiksk123]

What does it look like. You can only see... Gray? Ha?! No? :(

[@steinardarri]

wolfiksk123 It he means that the red and blue ones look too similar

[@SnakeBiteScares]

steinardarri Not exactly, it depends on what type of colorblindness he has, I myself am colorblind and found it difficult to distinguish the blue and the pink. Colorblindness is where you find it difficult to distinguish between certain colors

[@Maninawig]

8:33 When you're a Maths teacher and your student asks you to prove why 1+1=2

[@vsauce4678]

This guy has so much passion for what he loves and it shows in his videos

[@firstnamelastname-oy7es]

Incredible! It looks like all the other Hexagons have Hexa... _Gone_!!! I'm sorry for that.

[@thisisrtsthree9992]

+Bungis Albondigas shame

[@thepip3599]

Sometimes I really wish there was a facepalm emoji. Just, so, so much.

[@davecrupel2817]

that was a parker square. You still get a cookie :3

[@wmconorbrown]

"Let's count that to make sure." Very difficult math I see it is to check the other 1 magic hexagon.

[@jacderida]

Numberphile2 would have been a nice place for the full solution :).

[@numberphile]

Chris O'Neil there are some small extras from this video coming to Numberphile2 - but not that solution I'm afraid.

[@EebstertheGreat]

Numberphile Is the solution really that tedious?

[@joaomelo7538]

EebstertheGreat Its just solving five variables system, nothing big...

[@EebstertheGreat]

João Melo There's a lot more to it than that, though. That just tells you the sum of each color.

[@joaomelo7538]

yes, that's my point, if haven't understood I was being sarcastic. A five equation system takes too much time for a video

[@8bit_pineapple]

James: "What I have here is..." --- Me: "A poorly designed Settlers of Catan Board?"

[@tommytomthms5]

YES THIS!

[@greekfire995]

THE poorly designed Settlers of Catan board.

[@Triantalex]

WeirdChamp

[@8bit_pineapple]

@@TriantalexmonkaW

[@ragibhasan5745]

1:44 Its the cutest "why" I have ever heard!

[@General12th]

Brilliant video! Brilliant explanation, brilliant subject, brilliant professor. Simply intelligent.

[@SaveSoilSaveSoil]

Nice! I never paid attention to these magic n-gons! Thank you for raising my awareness!

[@DouglasZwick]

Oh man I laugh out loud at 1:50 every time

[@BattousaiHBr]

i cant even understand what he's saying "if you want to edit and cut to xxxxxx" ?

[@EchoHeo]

BattousaiHBr thats the point

[@McJaews]

Thank you Brady:) It's always great hearing Dr Grime talk about math. I did, however, notice a distinct lack of prime numbers in this video, and was wondering if there were any interesting mathematical things going on with geometric shapes that have a prime number of sides. I find it hard to imagine that there isn't.

[@PC_Simo]

Well; the regular pentagon has a prime number of sides (5); and its diagonals bisect each other in the golden ratio, which is very much related to the Fibonacci numbers; and the Fibonacci numbers seem to me to contain relatively more primes, than any old random sequence; which, I guess, makes sense, given that the golden ratio is kind of like the most irrational number there is; so, if I expected primes to show up anywhere, it’s definitely in the Fibonacci sequence 🤔.

[@Azmarith]

What's got 6 sides and isn't here any more? A hexagone.

[@Triantalex]

??

[@Azmarith]

@@Triantalex A hexagon has six sides. But it's gone. So it's a hexa-gone.

[@neelmodi5791]

Exponentiation of each number in the hexagon leads to a magic multiplicative hexagon!

[@AuroCords]

+Neel Modi please explain 0.0

[@pokestep]

+Auro Cords I believe what they meant is that if you had a magic hexagon (or a square, works there too) with any number to the power of numbers in the magic square (or a hexagon) and you multiplied them within rows, you'd get the same number! Observe: For the usual 3x3 magic square, with rows of (6,7,2) (1,5,9) (8,3,4), if instead you had numbers like (2^6, 2^7, 2^2) (2^1, 2^5, 2^9) (2^8, 2^3, 2^4), which equals (64, 128, 4) (2, 32, 512) (256, 8, 16) and multiplied them (rows, columns, diagonals), they'd give you the same number! (2^15 or 32 768). The reason this works is because of the way exponentiation works - if you multiply numbers, such as a^b and a^c, the result is a^(b+c), you get the sum of the powers! (Observe: 2^2*2^3 = 4*8 = 32 = 2^5.) This works for any base number (i.e. you can have 3^x, 10^x is especially nice because you only add 0s, e^x... it's up to you!). Hope that helps and answers your question!

[@AuroCords]

Amazing! I had forgotten about that property, I guess the original comment should have said "Exponentiation *to* each number in the hexagon..." I didn't quite get the last part of what you said: " (i.e. you can have 3^x, 10^x is especially nice because you only add 0s, e^x... it's up to you!)." Thank you =]

[@pokestep]

+Auro Cords You're welcome! What I meant by that part is that it doesn't need to be powers of 2 like I showed you, but it can also be powers of 3, powers of 10 (especially nice because then you're only adding 0s to the numbers, i.e. you get (100,1000000000,10000) (10000000,100000,1000) (1000000,10,100000000) I think), it can be powers of e - that is totally up to you! The sum of exponents during multiplication applies to any number. :)

[@AuroCords]

Ah yes, that's what I understood but wasn't sure. This is why I love maths, gotta get some practice tho to keep the brain slick. tx again!

[@abigailcooling6604]

I'm sure Matt Parker will create another magic hexagon that *almost* works. You've always got to give things a go!

[@svz5990]

You mean a Parker hexagon?

[@LLHLMHfilms]

Now I want to play the settlers of catan

[@MMorangatang]

Same

[@thepip3599]

Me too.

[@thepip3599]

+The Pip Man, I love that game.

[@mydemon]

That edit at 1:51 is one of the funniest things I've ever seen.

[@hiwadwardak2410]

8:35 to 8:44

[@NoriMori1992]

My favourite part. XD

[@whitherwhence]

#HardcoreMaths

[@awaiskhan_47]

Another James Grime classic!

[@najiali1068]

I love the way counting the sum of all numbers in one hexagon. Very nice video. I like your way of clearing up things. Thank you.

[@jsnadrian]

Having Dr Grime must be such a fun lecturer to have

[@johnkat4391]

Numbers that can't be in the same row in a 3x3 magic square: 1,2 1,3 2,3 7,8 7,9 8,9 7,4 3,6 Also, 4 needs to be in a row with 5 or 6. 5 with 4 or 6. 6 with 4 or 5. There are probably other numbers that can't be together or have to be together, but this is what I've found so far.

[@johnkat4391]

Sorry, let me correct that. (I am on mobile so I can't edit it.) A 3x3 magic square where you can only use numbers 1-9 and the answer needs to be 15.

[@johnkat4391]

Another correction! You can only use each number once.

[@7777stine]

8:22 That Smile!!! LOL! This guy loves numbers clearly

[@iisnothere]

I do love that this mathematical phenomenon created the entire genre of “hex bingo”

[@creature_from_Nukualofa]

I get the same feeling as reading a chapter by Martin Gardner. thanks Brady, thanks James for the wonderful content! ND

[@burpie3258]

1:52 can't stop laughing

[@michaelgerda494]

awsomm

[@minnermin]

Lol

[@LesMiserables999]

You're my magic hexagon James...

[@daggawagga]

10:00 are Grime's birthmarks the vertices and center of an equilateral triangle?

[@aves8964]

Illuminati confirmed.

[@daggawagga]

***** if it was an equilateral triangle, it would be all of them! (I loved that one video)

[@NoriMori1992]

+Daggawaggaboof It looks like it _is_ an equilateral triangle!

[@jacecockayne2054]

What an awesome birth mark

[@mclam168]

3 zeros in the time stamp. 3 side in a triangle. Illuminati confirmed.

[@gregotox]

he`s so happy about it! :D

[@lejink]

Dr James Grime is my favorite :)

[@leonhrad]

I want a t-shirt with a magic hexagon on it

[@dominikf.1409]

I love the singing banana

[@frawding9438]

Any Parker hexagons?

[@rewrose2838]

120 of them all got rejected at the end in the favour of the correct one 😂

[@DreamFreeFPV]

Not going to lie. My interest in watching this was to get better at settlers @numberphile

[@BiffTech05]

8:31 The Highlander magic hexagon

[@KunamaElgar]

While I enjoy numberphile videos they usually go right over my head! I actually understood this video and followed his thinking all the way through so I really liked it.

[@liamogrady5868]

"And the diagonals too!" Matt Parker: what.

[@kattay11]

This went whoosh, over my head. But I love his dimple

[@Richard_is_cool]

Thank you! Also, will there be a Mandelbrot Set continuation? It's been more than a month. :) Keep up the good work!

[@Ollervo100]

James you are awesome! Keep up the good work!

[@zerid0]

Poor empty hexagon, he didn't even get mentioned :'(

[@ZardoDhieldor]

Yeah! And what about n=-3! :(

[@louisng114]

Zardo Schneckmag n = -3 would make the denominator 0; better make it n = -2.

[@ZardoDhieldor]

louisng114 n=-3 would make a denominator of -5. A denominator of zero never appears.

[@louisng114]

Zardo Schneckmag Oops, I mean "makes the denominator -7."

[@ZardoDhieldor]

louisng114 Yeah, whatever! :D I'm used to calculate with 2n+1 more than 2n-1.

[@boilpoil]

I like how Numberphile finally touched on Magic Squares :) Also, every comment (except Brady's) below me has nothing to do with the video. Lol.

[@numberphile]

boilpoil we've done magic squares before!!!

[@boilpoil]

Numberphile Really? I only subscribed a few months ago, at the video about -1/12 xD

[@numberphile]

boilpoil better get into the back catalog!!!!

[@Kram1032]

boilpoil boy got you some work ahead of you

[@grandexandi]

There already is another video about magic squares.

[@robzwolf]

Really good video. Great chromakeying with the blue writing too, and very interesting to watch. Love it! :-)

[@thrillscience]

Sorry, James! That's not the only magic hexagon. I have one just like it here!

[@venweera4516]

@8:33 there is a slight addition error, happens to the best of us

[@CraftQueenJr]

Ven Weera It has to have all unique numbers

[@billymatthews4150]

3:12

[@Radii_DC]

0:58 NICE! :D

[@MKWKezer]

Very nice and not too hard either! You should do more videos, I love them, James :)

[@TheSleepingAsian]

Kept seeing the "Settlers of Katan" board when I saw the Hexagons, haha.

[@natereniger8773]

Dr. Grime is so fun to listen to... I wish I could do my whole undergrad over again where he teaches every class.

[@YindiOfficial]

James' videos are my favorite tbh.

[@christianavery5518]

Reminds me of a game we used to play in maths class called 'Nubble'. The maths of the game has nothing in common with the video but there were numbers in hexagons which formed a large hexagon.

[@moppop275]

when he checked the magic hexagon of n=1 I died.

[@CodyBenson13]

Very interesting. I have to say though, I only really watch when James is in the videos.

[@brandonhorvath5881]

Thank you for blowing my mind once again.

[@acediamond5399]

Amazing! Great video. Seems a bit miraculous that even the 3-layer hexagon works.

[@leonardomona9376]

+Ace Diamond theres nothing miraculous about it, its just a coincidence, things would be different if the numbers used is base 6 not base 10

[@acediamond5399]

Well yeah, that's kinda what I meant, not a literal miracle, lol.

[@acediamond5399]

But, besides the point, this concept is base-independent.

[@MrCyanGaming]

In the video you never mentioned that the Magic Hexagon must be made of consecutive numbers. Since you can just multiply all of the Numbers in the Hexagon shown in this video by 2 and get a new Hexagon that Works. (MAGIC NUMBER: 76) If you want a Magic Hexagon in it's simplest form, you can take the Hexagon shown in the video and Add 8 to the Pink, then Add 16 to the Blue and Center. This will give you a new Magic Hexagon in it's simplest form. (MAGIC NUMBER: 70)

[@migorpresents8991]

Thanks for the explanation. I couldn't figure out what was going on at 2:34

[@yatra6110]

Magic number 76 works as well as other multiples of 38 . The array of numbers for M 76 are consecutive even numbers! These magic hexagon 1-19 numbers x3 =114 should work but they r not consecutive numbers.. and for 70, I tried but it’s not working for all rows..

[@MrCyanGaming]

@@yatra6110 Add 16 to the center instead of 8, that's mb

[@thermotronica]

Beautiful video!

[@Tobis0x00]

Dr James Grime said "There can be only one..." He's my new hero. The Highlander of Hexagons!

[@quaironnailo]

This is entirely based on the fact that it has to use every single number from one to the number of hexagons (19 in this case), which is not a condition for a magic shape.

[@BLiZIHGUH]

5:24 and then James Grime sets the hexagons on fire

[@aurabozzi228]

This is a beautiful proof!

[@jlolme]

I'M CRYING AT BRADY'S EDITING

[@AlbinosaurusR3X]

Love your show, Numberphile.

[@maxor669]

No one tell Matt this is the only one and let him "have a go at it" XD

[@nonitta]

This is Amazing, I find this so interesting! Thank you for teaching me something new!

[@TheConnor12500]

Could you perhaps do a video on slide puzzles? I've been doing lots recently and can never do a scrambled 4x4 in less than around 60 moves, is there a number of moves all can be completed in like Gods Number, and how about for a nxn puzzle? Love the videos!

[@MrSaemichlaus]

5:36 sponsered by Google Chrome.

[@dans5606]

Sitting here watching videos about magic squares, and just noticed Matt Parker's "Magic Square Party Trick" for the 34 magic square! Oh, you guys.

[@brankodimitrijevic382]

Great videos watching from Serbia!

[@jimbo-fk4dq]

Numberphile should get a show on PBS. Preferably PBS, because it's too smart for cable TV. I'd love watch it.

[@TheAAMoy]

FYI, this fucking ROCKS!

[@carlb.9518]

Numberphile Nice! What about magic cubes and hypercubes?

[@Zack0ry]

its cool how you just added his handwriting instead of a preset font :)

[@drorfrid]

At the end, I have checked why it is the only magic hexagon, and I didn't understand why Y is even, smallest possible

[@xvipes]

Welcome back James

[@karlkastor]

That's great. Magic Squares are to mainstream, so it's good to have this. Very interesting that there's only one possible way to do it.

[@JoshDan12]

YES!

[@rywilk]

Very cool! I'll have to give this a go in my spare time =p

[@Ahov]

Wow, grats on 1m subs!

[@TimmahDee]

Not sure if editing humor at 1:52... or just mistake during editing...

[@choco_jack7016]

I think it says "sort of edit and cut to hoint (idk) with theee so..."

[@Nami-x]

Numberphile If I get this right, the definition of a magic hexagon is to use each number once. If you allowed that rule, you could create infinite magic hexagons by simply adding 2 to all outer numbers, 1 to all middle ring numbers and 0 to the central number (in this example they all add up to 44 then (you take 3x the number you added to the outer ring)), however in my workings you do get the number 5 three times.

[@Ajax-0137]

Lol, when Brady edited. That was hilarious

[@MartinMllerSkarbiniksPedersen]

Fantastic. Thanks a lot.

[@Adamantium9001]

What if you remove the requirement that the numbers in the cells have to be 1 through n?

[@sevilus7812]

Thanks

[@dr-baboul3077]

Good work !!

[@all_the_moga]

sure glad he cleared up the 1 hex for me :D :D :D

[@robertvdhill367]

Step 1: Make hexagons Step 2: Make honeycomb Step 3: Magic Step 4: Profit! ;)

[@ultravidz]

Great one!

[@NoriMori1992]

9:55 - …Yellow? _Yellow?!_

[@ch48_]

Yeah, that's clearly yellow. Are you colorblind or something?

[@NoriMori1992]

+Corny Can't tell if you're trolling, joking, or simply colourblind yourself.

[@KessaWitdaFro]

I'm not joking at all it really looks greenish yellow to me :/. they thing is I've taken color discrimination tests and usually do really well so I'm currently taking a poll of all my friends to get a consensus

[@ch48_]

TBH, it looks yellowish green to me.

[@qwertyTRiG]

It will depend somewhat on your monitor settings and on the ambient light as you watch. TRiG.

[@Eliina552]

James is my fave guest :)

[@anon8109]

Lovely. It's surprising that it's possible to tease out 3 independant equations by adding up rows in different ways. Are there any other magic shapes?

[@whydontiknowthat]

SONG FROM THE MANDELBROT SEQUENCE VIDEO! AT THE END!

[@Odood19]

Equilateral hexagons are my favorite polygon! They occur in nature.