It's not just useful for drawing maps, either: the same principle allows cell towers from interfering with each other, by using four sets of frequencies. Using four sets of frequencies, no adjacent cells have to use the same frequencies.

@jimthompson3053

7 жыл бұрын

er.. prevents, not allows.

@Natalie-cd4mf

7 жыл бұрын

Interesting, never thought of that.

@iycgtptyarvg

7 жыл бұрын

Fantastic example of applied math. Personally, I like explaining people how the principle of GPS works (in simple terms with as little actual complicated math as possible).

@a24396

7 жыл бұрын

That's such a terrific example! You just blew my mind!

@IONATVS

7 жыл бұрын

Fester Blats And because real countries can have exclaves and enclaves--regions that are legally part of a country while not being connected to that country by any actual land. Such things violate the premise of the 4 color map theorem (regions are required to be contiguous in the theorem), and allowing them is the same as allowing intersecting edges in the equivalent networks--a map could be made to need as many colors as you arbitrarily want by using such territories

I love how all this started with some guy filling out a map with colors and noticing that he only needed 4

@amaanali9525

3 жыл бұрын

Some maps ACTUALLY don't work with this

@user-zz3sn8ky7z

3 жыл бұрын

@@amaanali9525 example?

@amaanali9525

3 жыл бұрын

@@user-zz3sn8ky7z the ones made by Susan Goldstein.

@user-zz3sn8ky7z

3 жыл бұрын

@@amaanali9525 that was interesting, thanks for sharing! Although I'm not sure if it counts as a "map"

@amaanali9525

3 жыл бұрын

@@user-zz3sn8ky7z oh okay your welcome

omg watched this mindlessly 3 years ago when i was in high school then here i am studying graph theory in college coming back to see how it actually works like an hour before midterm

@PrivateSi

3 жыл бұрын

The graph solution is much more complicated than mine... In 2D, the maximum number of nodes that can be connected to each other (each to each) without connectors crossing is 4.

I never thought I would say this of a mathematician, but I don't believe I could ever tire of listening to James Grime. I actually find myself smiling far more often than was likely ever the case back in my school days. Thank you Dr Grime.

I recall an issue of Scientific American back in about 1974 (more or less) that had an article that purported to show 7 amazing recent discoveries. One was that the best first move in chess was shown to be h4, another was a logical way to disprove special relativity, another was that Di Vinci invented the toilet and another was that someone came up with a map that required five colors. I can't recall the year of the publication, but I can recall the month. The magazine came out on April first......

@edsanville

5 жыл бұрын

@@error.418 April 1st. Think about it.

@galactica58

5 жыл бұрын

I like this comment.

@willyantowilly7165

5 жыл бұрын

h4 is the best first move in chess? This has to be a joke.

@XenophonSoulis

5 жыл бұрын

@@willyantowilly7165 April 1st

@expertoflizardcorrugation3967

5 жыл бұрын

I enjoy stork theory of reproduction papers

everytime i take a test i imagine that he's looking over me and kinda guiding my way to success lol

@klaud7311

3 жыл бұрын

Sounds like you envy him more than you admire him.

@solarean

3 жыл бұрын

@@klaud7311 for me sounds like he just likes the attitude of this guy idk lel

@nosuchthing8

2 жыл бұрын

Wouldn't that be great. His IQ must be off the charts.

0:04 "It's easy to state" I see what you did there..XD

6:03 careful dude you're gonna summon the devil

@capy9846

4 жыл бұрын

Michael Darrow Nah I just watch video’s upside down for fun

@dondeestaCarter

4 жыл бұрын

JuliasJulian. Cool!! So "Ontario" reads "JuliasJulian" when upside down? Wouldn't have expected that!!

@ValkyRiver

3 жыл бұрын

6:14 Exclaves: am I a joke to you?

I learned this in a book called "betcha can't" (which actually has a lot of the problems I've seen on Numberphile). But the story was that the father died and his five sons inherit his land. In the will it says they can divide it up however they want, but each plot needs to be all one piece and must share a border with all four other sons' plots.

@sheilakijawani2526

7 ай бұрын

Circular tyre 1 wont work?

Yes! James Grime!

@branflakes2600

7 жыл бұрын

^^^^^ 9th

@JM-us3fr

7 жыл бұрын

He's the absolute best!

@bencouperthwaite6735

7 жыл бұрын

I met him :)

@duck6872

7 жыл бұрын

I am jealous

@bencouperthwaite6735

7 жыл бұрын

Duck He came to my college in January. Top guy!

What if it was in 3d? like, with colouring 3d spaces instead of 2d shapes. Maybe filling hollow glass chambers with coloured liquid. How many colours would that take? Would there be a limit?

@thepip3599

6 жыл бұрын

On second thought, I've realized it would almost certainly not have a limit. In 3D, you can have tunnels going through stuff to other stuff. That doesn't really work in 2D.

@MikeRosoftJH

6 жыл бұрын

It's even worse: even when we require that each region of space is a rectangular box and the boxes are orthogonally arranged, it's still possible to create a division which requires arbitrarily many colors.

@sergey1519

5 жыл бұрын

no because you just can take map, then get line going from first country to every other country at next z(if map is at level(z coordinate) 0 just connect first country to every else country on level 1) then connect second map to any other at level 2, then connect third map to any other at level 3 etc. You have infinite plane so you can connect every country to any other country if your lines are are small enough(so you can just say that they are have width of 0). I hope you understanded what i writen there cuz i don't really know this language.

@ethendixon4612

5 жыл бұрын

I'm gonna assume it would be 8. I can't back this up . . . but I think it's related by 2^dimension.

@greysquirrel404

5 жыл бұрын

Or in the other direction let's consider the problem in 1d. If you had a series of connected line segments and a line segment had to be a different colour to the one(s) connected to it. How many colours would you need?

11:06 I'm currently studying Actuarial Science at the University of Illinois (same awesome school as Appel and Haken). You wouldn't think the Four Color Map Theorem would show up in an insurance internship, but I showed this theorem to a few of my coworkers and they made a colorblind-friendly map of the U.S. for me to use in a project. Thank you Numberphile!

@coleabrahams9331

3 жыл бұрын

@Ian Copple OMG!! Actuarial science. I’m 17 and I would also like to study actuarial science as I’m tremendously interested in maths. Please tell me about it. I couldn’t do job shadowing during the school holidays (vacations) due to the coronavirus pandemic, but I would really like to know what it’s all about. People have been telling me that I should study actuarial science, but I don’t really know what it’s about. Please provide me with some idea of how it is like, etc.

"Let's try making a map that requires five colors" *second map drawn only has four sections*

@o76923

5 жыл бұрын

Technically the space outside counts as a region as well (and can include lines that continue for eternity).

@dancrane3807

4 жыл бұрын

ikr

@lilyfm7152

4 жыл бұрын

That was drawn to illustrate the network.

Enclaves and exclaves can not be considered as the theorem requires *contiguous* regions. The term "map" in the theorem refers to a physical map as opposed to a political map. This could be confusing to grasp after watching this video as they refer to real-world examples as well as abstractions.

@error.418

5 жыл бұрын

Yes, this video has a restricted problem space. But it's still interesting to then talk about an extended problem space and consider what the solution is to that new problem space. The four color theorem doesn't work in the new problem space because the country and its disconnected exclave must be the same color. Because you now have two areas that don't share a border that must be the same color, you've added a rule which can require more than four colors.

@error.418

5 жыл бұрын

@@carnap355 No, it doesn't work because the country and its disconnected exclave now must be the same color. Because you now have two areas that don't share a border that must be the same color, you've added a rule which can require more than four colors.

@TruthNerds

5 жыл бұрын

​@@carnap355 That's not what *exclave* means, you are confusing it with a specific type of *enclave* I guess. Exclave is an *additional* territory politically belonging to one country but completely surrounded by foreign territory. Enclave, on the other hand, is any country territory completely surrounded by another country. The theorem allows for any enclave that is not an exclave, otherwise you'd run into the problem mentioned by Anonymous User. Here are some real-life examples for all cases: US mainland - neither an enclave nor an exclave Vatican city - an enclave (of Italy) that is not an exclave (because it is the sole sovereign territory of this state) Nakhchivan Autonomous Republic - an exclave of Azerbaijan that is not an enclave of any state (i.e. not completely surrounded by any other state). Karki - an Armenian exclave *within* the Nakhchivan Autonomous Republic, so it's both an exclave of Armenia *and* an enclave of Azerbaijan. Featured in the movie "exclaveception". ;-) (West Berlin is another. historic, example of an exclave that was also an enclave, because it was an additional territory of the FRG aka West Germany, but was completely surrounded by the GDR aka East Germany.) The latter two would impose additional constraints (i.e. if Nakhchivan and Azerbaijan, or Karki and Armenia, rsp., have to have the same color) and therefore might "break" the four-color-theorem.

@AK-dp8uy

5 жыл бұрын

What about water? Why is water, the "background color" of a world map, not considered a color that counts?

@Rannos22

2 жыл бұрын

That's a cheap cop out given the first examples were political maps

I love this mans enthusiasm

Conjecture: Every video of Numberphile requires extensive recursion.

Back in the windows xp days, I'd make images in paint by making one arbitrary continuous path both ends on an edge of the image. The curve would intersect itself at many points, but never intersect itself multiple times at the same point. I found that you could always cover the "map" created using these restrictions with exactly 2 colors.

@tfae

3 ай бұрын

I think this is the "even-odd rule" in computer graphics.

14:14 look above the o there are 2 yellows

@brokenwave6125

4 жыл бұрын

There are six colors on that "map" so its not really meant to be an accurate example.

@Joe-qm9cp

4 жыл бұрын

Gasp

@uxleumas

4 жыл бұрын

culd have been pink

@IHaveaPinkBeard

4 жыл бұрын

That's pretty bothersome given the video topic

@janprevratil1015

4 жыл бұрын

@@brokenwave6125 I think he wanted to be colored with 4, but he gave up :D

While watching this, I thought at 3:23, you could leave the last quarter circle border unclosed and make a bigger circle around everything. With the existing coloring, it looks like that would need a fifth color. But, after more thinking, it's doable by some shifting on the colors used earlier

@luizazappala3572

Жыл бұрын

Thought the same!

Four Color Theorem: Exists Enclaves and Exclaves: I'm about to end this man's whole career

@montanafisher8996

3 жыл бұрын

Exclaves and non-contiguous countries might throw a wrench into the cogs, but I think you might just have to shift the colours used to make it work in four

@williamchaney448

3 жыл бұрын

@@montanafisher8996 But you could certainly conceive of a map rich in exclaves and enclaves such that you'd need more than 4 colors... If a map includes 5 countries, and each country has an enclave in literally every other country, they'd all need to be different colors.

0:16 Dang foreigners colored Michigan two different colors lolol

@ilfedarkfairy

5 жыл бұрын

@@yesno1498 that is true, but has nothing to do with the fact that Michigan is to large.

@starwarsman333

5 жыл бұрын

but then at 1:47 they have it right XD

@theblackwidower

5 жыл бұрын

@@yesno1498 So when factoring in enclaves and exclaves, how many do you need?

@leonhostnik9516

5 жыл бұрын

@@ilfedarkfairy Take up all complaints with the state of Ohio on that one, regarding the Toledo War

@the_real_ch3

5 жыл бұрын

Yoopers gettin no respect

3:40 Aww I was hoping for the Chrome logo

@grantcarrell

7 жыл бұрын

Albin9000 I was too.

@ayanshah2621

7 жыл бұрын

I thought it to be a pokemon

@erikplayz8192

7 жыл бұрын

Albin9000 same

@jamesbatley173

6 жыл бұрын

Me, too!

The system doesn't work for exclaves. In an infinitely complex map each country would have infinitely many exclaves, connecting it to each other country. 1) Make a map that has 7 countries, put them wherever you want on your map. 2) For each country create 6 exclaves, being enclaves to each of the other countries. 3) Color the map using only one colors for all territories each country.

@juanignaciolopeztellechea9401

2 жыл бұрын

The theorem only couts CONTIGUOUS countrys.

@LuKaSGLL

Жыл бұрын

I thought about that too, when I noticed on every map Greenland and French Guyana were coloured differently than Denmark and France, respectively. But another comment on here asked about three dimensional "maps" and the answers were obviously you could make objects on 3D touch infinitely more than on a 2D plane, and I came to the conclusion that exclaves essentially make the map "3D", since an exclave would basically mean a tunnel outside of the plane is joining two or more objects. This theorem only applies in 2D.

"I don't know why, but he was" seems to be true for a lot of math

What really bothers me is that countries exist on a spherical surface, but the four color map theorem only works in a Euclidean space. Theoretically if a country stretched around the planet, planar graphs that include k5 and k3,3 subgraphs become possible.

@vangildermichael1767

7 жыл бұрын

Awesome point. I hadn't thought on the 3 dimension thing. I like the brain treat. yum.

@ZayulRasco

7 жыл бұрын

/dev/zero You can map a sphere to a 2d surface and preserve the properties we care about regarding the 4 color theorem.

@andreashofmann4556

7 жыл бұрын

But you lose the looping around bit, which i think is what he's going for?

@rubenras1399

7 жыл бұрын

/dev/zero ii

@Korcalius

7 жыл бұрын

And what if a country has a colony or more? Its technically still the same country.

Thanks for the nerd snipe numberphile. Every time I see this theorem stated I always end up taking a stab at finding a weird case to disprove it. Today I was so close to calling a math friend to show him my counter example, before realizing I had a colour wrong.

@ragnkja

7 жыл бұрын

Real maps can require more than four colours, if there are exclaves that need to be coloured the same as the main part of the country.

The problem with this is that not all countries are contiguous, and so enclaves can force a hypothetical map to use more than four colors.

@MichaelDarrow-tr1mn

11 ай бұрын

do you mean exclaves

@tylerbird9301

10 ай бұрын

@@MichaelDarrow-tr1mn an enclave is just an exclave of a different country.

@MichaelDarrow-tr1mn

10 ай бұрын

@@tylerbird9301 no it's not. an enclave is a country entirely surrounded by a different country

@tylerbird9301

10 ай бұрын

@@MichaelDarrow-tr1mn enclave noun a portion of territory within or surrounded by a larger territory whose inhabitants are culturally or ethnically distinct.

@MichaelDarrow-tr1mn

10 ай бұрын

@@tylerbird9301 a portion can be 100%

Really interesting video; great job! I myself spent a lot of 'doodling' time back in the 80s trying to find a counter example. I also don't like the computer proof for the same reason you stated: it doesn't teach us anything but that some result is true. We don't know WHY it's true. But to me it boils down to a topology problem, not a color problem. I state it thus: The greatest number of closed figures which can be drawn on any 2D surface such as a map or globe in such a way that every figure touches every other figure along a side, is four. You'd literally have to put another figure into the third dimension, making it go above or below the 'plane' to connect it to other figures, thus forcing a 5th color. You simply can't do it any other way. That is what makes the 4 color conjecture true... but, of course, that is not a proof in itself. But I can tell you that I'm done doodling with it. I'm satisfied that eventually, someone will prove it with geometry or more likely topology. Rikki Tikki

0:15 Did they just colour Michigan wrong?

@jakec904

7 жыл бұрын

what?

@hjorth3387

7 жыл бұрын

The purple and blue state?

@homopoly

7 жыл бұрын

Yeah.

@optimist2301

7 жыл бұрын

Unchi what?

@lyndonhanzpernites5860

7 жыл бұрын

Michigan was filled with two colors. (Being separated by the Great Lakes.)

Is there a reason why James says "network" instead of "graph"?

@lucashenry2556

7 жыл бұрын

I think it's because he's focusing on the importance of the vertices, not the edges. That's just my guess though. I would have though he'd have called them graphs too because they would all have Euler characteristic 2

@JM-us3fr

7 жыл бұрын

Green Red The average person recognizes the word "network".

@1990rockefeller

6 жыл бұрын

gwuaph. Just kidding. He is awesome!

@EMETRL

6 жыл бұрын

because real life applications of this idea often come in the form of actual networking

@cheesebusiness

6 жыл бұрын

Because he speaks the British language

This is an excellent, clear presentation by Dr. Grime.

I thought I'd broken this when I first heard of it. I imagined two concentric circles with the inner one being split into quarters using a line going from top to bottom (the whole diameter of the inner circle) and another line going from right to left across the circle(again, the whole diameter), forming a cross. The inner circle then resembles a cake cut into four roughly triangular sectors. My (erroneous) thinking was that all four quadrants of the inner circle touched at the centre and so would require four colours, and then you'd need a fifth one for the outer circle! BUT......although it may look as though the four quarters touch in the middle, they don't. They can't. If two diagonally opposite triangles touch, then they sever the connection between the other two diagonal areas.

i love how hes always so happy

I know I'm late to this conversation, but it got me thinking. I think it can be put more simply, actually, although mathematicians might not like it as much. Here goes... With all of the parameters already set (contiguous borders and the like), the question becomes, can you: 1. Create a theoretical map that requires 1 color? Yes, duh. 2. Create a map with 2 colors? Yes, you just need 2 touching areas. 3. Create a map with 3 colors? Yes, like a pie cut into 3 slices, each piece touches both of the others, so 3 colors required. 4. Create a map with 4 colors? Yes, take the pie from before and make the center its own area that touches all 3 of the original slices. 5. Create a map with 5 colors? No. Here's why. Imagine 5 squares arranged into a cross or +. One in the middle, and one each on the top, bottom, left, and right. Right now, you only need 2 colors, as the outside squares don't actually touch. So, let the outside shapes bulge a little and touch their neighbors (top now touches left, right, and center, left touches top, bottom, and center, and so on). Now, you need 3 colors. Why not 4? because right now, the shapes on opposite sides of the center square don't touch and can be the same color. Let's try to fix that! Take the top shape now, and stretch it around to touch the bottom shape. Awesome, now we need 4 colors, since the top and bottom cannot share anymore! Now, let's go for 5! Currently, the only shapes still sharing colors are the left and right shapes, so we need to get them to touch. But wait, to get the top and bottom to touch, we had to go around either the left side or right side (we'll say left, but it doesn't matter). The right shape has no way of getting to the left now! Well, what if we cut under the top? Oops, the top and center are not touching anymore! Well, what if we slice through the arm connecting top and bottom? Well, then we're back to where we just were with top and bottom not touching. Feel free to play with it and make the shapes weirder, but you cannot get all 5 shapes to touch every one of the other shapes without breaking a connection that you had previously made. Even if you add a sixth shape wrapping around the outside of the whole mess, it will still be separated from the center square and will be allowed to use that color, unless you break one of your earlier connections (at which point, what have you accomplished?). All of the nightmare with proofs and computers and whatnot may be needed for mathematical certainty, but if you cannot get a mere 5 or 6 shapes to need 5 colors, then adding additional shapes just aggravates the issue of fighting for connections. I tried to keep that whole thing simple enough to sketch along if anyone cannot follow in their head. My apologies, and thank you for coming to my talk.

i don’t know why i’m watching these math videos at 3 am bc i truly don’t understand them but everyone in the vids seem to so i keep on comin back

Easy to... “state.” *brings up and starts coloring map of US*

I found a map that needs five colors but it's only in my mind, the map is too big for the observable universe.

But what if you have 5 different counties/countries meeting at one point?

@man-qw2xj

5 жыл бұрын

Saabrina Adan points have no area. While a convergence, the fact that the convergence has no area invalidates it.

@o76923

5 жыл бұрын

Those 5 countries cannot all share a side with each other one; only a point.

@CalifornianMapping

5 жыл бұрын

Though such things are possible in the world, here they are simply not considered to be borders.

@zahidhussain251

5 жыл бұрын

Actually this leads to the answer. In whatever way you draw four areas where each one is touching all the three, there is no way you can draw a fifth one which touches all four.

@ProfessorX

5 жыл бұрын

Zahid Hussain are you sure? What about a circle with four divisions nested inside a larger circle (like a ring)? Edit: I redact the above. I just learned about enclaves and enclaves.

11:51 paper shows 1936... Reading it out loud: "one thousand nine hundred and thirty _eight_ ". Eh well, close enough. :-p

@coleabrahams9331

3 жыл бұрын

🤣🤣🤣Close enough

This one took me nearly the whole video to wrap my mind around. Just trouble visualizing. But it just shows how amazing these videos are that before the end they got me there ;)

Congratulations on 2,000,000 subscribers!!!

At the end those *yellows touching* in the top right are annoying me

@danielsantos3254

3 жыл бұрын

And the single purple section on the left

Thank you. I'm writing a paper that involves this, and I was really struggling to explain it. This video will be added to my citations.

Hmm wow it really is impossible. I tried it for a while, and after a minute, I realized that once you get to the fourth color, no matter how you draw the last section, it either cuts across one section(which means that the cut off section can be changed) or it doesn't touch all other sections, meaning I still use four colors.

I always thought of this when looking at maps of me in class. Like "hmmm i wonder if i could force two of the same color to be beside each other with only 4 colors"

I found a map which requires 5 But the comment threads are too small to hold the answer!

@tit-bits6197

4 жыл бұрын

Ghost of Fermat! 😜

@notquitehadouken

4 жыл бұрын

images deathshadow images

@commenturthegreat2915

4 жыл бұрын

Same

@samwarren6008

4 жыл бұрын

The map at the end of the video...

@RamsesTheFourth

4 жыл бұрын

I did too actually... but i cant post picture here :)

Thanks for the video this was really interesting, especially about the first case wich has used computer assistance as a proof, and just as a remark the guy in the video seems very passionate that gave more value to the video

i tried coming up with a counterexample with some very strange shapes and i found that no matter how you shift around the shapes and borders, every door you shut will open another. it reminds me of that impossible puzzle where you have three houses and a source of oil, electricity, and water and you have to try and connect all three sources to each house without interesting pipes

so sad he had to use 5 colors to color the square space map.

@parad0x448

7 жыл бұрын

Sashamanxyz 6

"So, Let's talk about the Four Colour Theorem!". James Grime video, After all this time!

The picture of the map at 1:40 is actually in correct, you can see that both the netherlands and france are colored green but on the island of saint martin they border. The island is split between them both.

@MistahPhone

Жыл бұрын

But here we are talking about mainland Europe

@manioqqqq

Жыл бұрын

1. The islands are (i think) not a part of the countries 2. That is clearly yellow

@thommunistmanifesto

Жыл бұрын

​@@manioqqqq neither france nor the netherlands are yellow, and saint martin is considerd thier territories

@manioqqqq

Жыл бұрын

@@thommunistmanifesto either i am colorblind or you are, but they're clearly yellow. And, in the problem ignore the island border.

@isavenewspapers8890

7 ай бұрын

You're allowed to color a country in two separate colors. It's just that each region individually has to be a single color.

I love the fact that this was a problem that I loved to do in my school books since the age of 9 maybe, not knowing that it was a well known mathematical problem! Reaching 30... I realise that I had deep questions about many things in science, like the prime numbers sequence, the problem of perception vs. attention in psychology, the philosophical question of time and other type of questions that if I had spend time on it... who knows what I would have found!

What is the least number of different texts for students at an exam, so that two nearby students don't have the same text? Obvious: 4. It's an application of this theorem.

@irrelevant_noob

5 жыл бұрын

Depends on what you mean by "nearby"... if it were to only apply to orthogonally adjacent, the answer would be 2. :-B

@relaxnation1773

5 жыл бұрын

And if they are in a star formation this is even more wrong. Students don't have borders like counties do, so it is how you decide what their "borders" are.

@DheerajAgarwalD

5 жыл бұрын

@@relaxnation1773 I think the statement holds. It's the same as map coloring. You can fill star formation or any planar formation with less than four colors, so "at most" you'd need 4, no matter how you choose your seating arrangement.

Summoning Satan at 6:06, I see what you did there.

@Squideey

7 жыл бұрын

This is Numberphile. They were summoning Pythagoras.

@JimSteinbrecher

7 жыл бұрын

surely, 5474N

@CH3LS3A

7 жыл бұрын

They were summoning Fermat's "I have a proof of this..." proofs.

@gyrfalcon23

7 жыл бұрын

this shape is a pentagram, but not inverted as in satanism

@unity303

7 жыл бұрын

I think it had to work @11:06 where it's actually 666 seconds, what did he do there...

If the map is such that in any point where more than 2 countries meet an even number of countries meet, you can always colour it with 2 colours.

so this is assuming maps cant have non contiguous sections? Some coutries/gerrymandered districs/ect can get weird and have breaks .

you should talk how the 4 color map problem can be used for scheduling. examples like students time slots for exames.

What about enclaves and exclaves? This could create some of the situations you presented as "impossible"

@OleTange

2 жыл бұрын

Yes. And they should have mentioned this.

Just begin from the worst case scenario: where each of the countries touch , you cannot put in any other node which touches all other nodes without crossing a vertex.

Love your videos!

Small point: Dr. Kenneth Appel is pronounced Dr. Ah-pel not Dr. Apple. (Source: he was my independent study teacher in high school - he had retired by that point)

What about exclaves? Shouldn’t they be the same color with their mainland?

@piguy9225

4 жыл бұрын

I was just thinking about that. If you color enclaves the same as the mainland, you could forces a situation where you would need more than 2 colors. I don't think there is any case a something like that happening on real life, but it is possible.

@piguy9225

4 жыл бұрын

*4 colors, not 2.

@jako0981

4 жыл бұрын

@@piguy9225 Yes there is you mongoloid

@RazvanMaioru

3 жыл бұрын

@@jako0981 I'm sure I don't need to tell you how racist using "mongoloid" as an insult is... you're lucky more people didn't see that

When I was in middle school (Year five or six) I thought it was the three colour theorem and proved on a map in the back of my exercise book that it wasn't possible to colour it with only three colour

As a computer science student currently learning Boolean algebra, de Morgan’s name sends me into a fiery rage

@lawrencedoliveiro9104

3 жыл бұрын

I don’t know why. His theorems are so straightforward. It’s like basic knowledge that every programmer should have absorbed into their DNA.

This wouldn't work in Terry Pratchett's Discworld (completely flat planet sitting on the backs of 4 elephants, standing on top of the giant turtle, A'tuin), in which borders also have height and depth. The Dwarf Kingdom is entirely subterranean and runs underneath Ahnk-Morpork, Sto-Lat, Borogravia, Uberwald, Lancre, et al. Because their map is three dimensional (four dimensions if you count the Fair Folk, let's not) you couldn't swing it with just 4 colors.

@TheOfficialCzex

5 жыл бұрын

Lol

@hevgamer6087

5 жыл бұрын

the 4 colors theorem is only for 2D maps, if you go to 3D, you can make maps that require infinite colors

this problem is all about 'graph' theory in particular colouring of 'planar' graphs but u never mentioned any of these terms and the Euler's famous formula R=e-v+2

@marios1861

5 жыл бұрын

its an example showing how connected graph theory is to topology.

Awesome explanation.

well explained. thank you.

13:11 Wow! New haircut!

im writing a compiler and this reminds me of cpu register coloring. @QVear for some reason I cant reply to your comment, I've created a language that mixes together parts of C++ with BASIC.

@UltimatePerfection

7 жыл бұрын

TheWeepingCorpse For what language?

I think it would be interesting to have a video on coloring maps on surfaces with higher genus or to colour empires on a plane

I didn’t think it was possible. Congratulations, you have made coloring complicated...

What about exclaves? They could produce crossing lines.

Yeayy james grime! james grime! james grime! Forget about Terence Tao, James Grime is the sexy mathmatician celebrity we need.

Do spherical maps have a different color theorem? Or do they still count?

Irrational maps: subdivided maps that have no end 3D version: imagine it like a puzzle; how many colored puzzle pieces would you need to fill a given shape so that no two colors touch

@themobiusfunction

2 жыл бұрын

Infinitely many in 3d.

What if u have a country that is split ip

Numberphile: It's possible to paint a map with only four colours. Exclaves: *I am gonna end this man's entire career*

@EricTheRea

4 жыл бұрын

You don't understand the problem.

@OleTange

2 жыл бұрын

@@EricTheRea @Lockrime understands the problem that Numberphile stated. You can blame Numberphile for not stating the problem they look at correctly. (Hint: They are not looking at political maps).

I actually had this question on my graph theory course thats so cool

Does this rule also apply to vertex/edge networks? Could I take it as given there are only four colors of vertex?

6:19 it makes sense if you have exclaves/enclaves. Right?

@manioqqqq

Жыл бұрын

They are off in the quadracolor theorem And, 🟥🟦 🟦🟥 Is valid.

@isavenewspapers8890

7 ай бұрын

In the real world, yeah, but not in the context of this problem.

I would imagine that if you wanted to make a map that requires 5 colors you might wanna try drawing it on a donut because of the specific priorities that the toroidal shape has this making it possible to make said map

This one guy, on this channel. Is making me care about more interesting aspects of math. Seems his name is James Grime. Man I hope he's a teacher.

Tbh, whole this falls apart when you realize you can have a small area inside country A which belongs to a country B which does not even border country A on the outside (exclaves). So basically once we allow that country can have an area detached from itself and such country has to be still coloured with the same colour, we can produce a graph with crossed lines.

@OleTange

2 жыл бұрын

The theorem breaks down for enclaves and exclaves. And I find it a serious error that they do not even mention this. Example: Assume you have 7 countries each with 6 enclaves from the remaining countries. You now have 7 countries that all share at least one border.

This video and the problem were quite interesting! But the end was...somewhat unsatisfactory! :(

A lot of commenters think they've found a counterexample, when what's really happened is they either didn't examine what the theorem considers "adjacent" ("What about five countries that meet at a point?"), or they tricked themselves into thinking their map needs more than four colours when it doesn't ("What about one country surrounded by four countries?").

This only applies to plane surfaces or the surface of a sphere. I seem to remember that a map on the surface of a torus requires seven colours.

What a regrettable choice not to mention Gonthier and Werner's work on establishing the correctness (and improving) of the proof.

11:00 The final solution was done by significantly more than two guys :s

Hi, I want to make friends. I am interested in math, especially in geometry. I found myself alone. No one want to talk with me about math. And I interested not only in math but also in education system and science. That was vague, I have nothing to say more. I'm just a little bit sad now.

@MOHAMMADALAHDAB

4 жыл бұрын

There are a lot of people who like to talk about math :D Check the facebook group: >implying we can discuss mathematics and the group: Mathematical Mathematics Memes for some quality grad level mathematical memes :3

I have not watched the video so I do not know if my idea violates a rule but if you have some sort of closed loop (circle, square......) and you cut it like a pizza (every dividing line goes through one specific point) that one point could arouse a problem

If hypothetically you had a region divided entirely into two separate parts but must be the same color because they are considered the same region, then you would need 5 colors

6:09 but what about enclaves? It would work out if you use them.

@DeathlyTired

7 жыл бұрын

Yes, my immediate thought was, "But what about a map with many a Kaliningrad'"

@screw0dog

7 жыл бұрын

Yep, the four colour theorem doesn't apply to maps with arbitrary enclaves. I suspect you can create maps that require an arbitrarily high number of colours if enclaves are allowed.

@daande97

7 жыл бұрын

The 4 colour theorem does not apply for the network James has drawn @6:09. If you insist that every piece of land which is surrounded by a border has got its own colour the 4 colour theorem is true.

@stevethecatcouch6532

7 жыл бұрын

If all of the territory of a country must be colored the same, an enclave could force the need for a 5th color. Look at his map at 3:58. Add a 5th country on the outside that touches all three of the countries not in the center. Now imagine an enclave of that country situated within the center country. What color can the 5th country be colored? It can't be the same as the outer counties, all of which it touches. It can't be colored the same as the middle country, because the enclave would be that same color. A 5th color would be needed.

@migueldp9297

7 жыл бұрын

Yeh, I also was thinking on enclaves, but anyways, is a cool theorem, right?

I tried to draw a counterexample for ten minutes then decided, I’ll take his word for it 😂

@dancrane3807

4 жыл бұрын

Umm, originally it took 120 years. So, get back in there!

@permafrost0136

4 жыл бұрын

Try using enclaves or exclaves you can easily get a map that needs 5 colors

@permafrost0136

2 жыл бұрын

@Michael Darrow no but they may cause two countries that border each other the be the same color

I love these videos.

The thing I kept thinking for all of this is how this is a fundamentally geometric approach to the issue of mapmaking, and countries, counties, and other things represented by maps often don't follow geometric rules with things like exclaves. And don't think I didn't notice how most of your four color examples still reserved a fifth color for the sea.

@r3ked272

2 жыл бұрын

You can use a fifth color for the sea if you try hard enough.

In my class, we call that a proof by exhaustion (literally)

Anyone else try to draw a counterexample is ms paint and miserably fail?

@erichiguera

7 жыл бұрын

the network at 6:08 can actually be drawn as 5 countries. just make a circle and divide the circle into 5 parts. since all 5 touch in the middle, you need more than 4 colors

@orionmartoridouriet6834

7 жыл бұрын

Throbbin So Hard Frontiers cannot be made only by one point, so the center of the circle doesn't count as a valid frontier

@skyr8449

6 жыл бұрын

yeah, I have basically shown to myself how things need to warp to do it, and as I knew it would be impossible, I have found something so sadly close that it kept cutting off the strands of color of other things.

@CraftQueenJr

6 жыл бұрын

I succeeded in makng one on my channel..

@JohanBregler

6 жыл бұрын

I found one, but I don't know where to submit it

Couldn’t help but notice that you you colored Michigan’s peninsulas different colors near the beginning

I would say I'm also a bit numberphile (that's why I study computer science), but I'm far from being as numberphile as you are. When I watch you're videos I get more numberphile, but I can't keep up that level. If I could get near to being as numberphile it would really help me at university, but although I can't I really enjoy watching you're videos :-).