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.
@jimthompson30537 жыл бұрын
er.. prevents, not allows.
@edaedaedaedaeda-u2g7 жыл бұрын
Interesting, never thought of that.
@iycgtptyarvg7 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).
@a243967 жыл бұрын
That's such a terrific example! You just blew my mind!
@IONATVS7 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
@proxy10356 жыл бұрын
I love how all this started with some guy filling out a map with colors and noticing that he only needed 4
@amaanali95253 жыл бұрын
Some maps ACTUALLY don't work with this
@user-zz3sn8ky7z3 жыл бұрын
@@amaanali9525 example?
@amaanali95253 жыл бұрын
@@user-zz3sn8ky7z the ones made by Susan Goldstein.
@user-zz3sn8ky7z3 жыл бұрын
@@amaanali9525 that was interesting, thanks for sharing! Although I'm not sure if it counts as a "map"
@amaanali95253 жыл бұрын
@@user-zz3sn8ky7z oh okay your welcome
@timsullivan45662 жыл бұрын
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.
@a0258223694 жыл бұрын
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
@PrivateSi4 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.
@davidyoung63316 жыл бұрын
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......
@edsanville6 жыл бұрын
@@error.418 April 1st. Think about it.
@galactica585 жыл бұрын
I like this comment.
@willyantowilly71655 жыл бұрын
h4 is the best first move in chess? This has to be a joke.
@XenophonSoulis5 жыл бұрын
@@willyantowilly7165 April 1st
@expertoflizardcorrugation39675 жыл бұрын
I enjoy stork theory of reproduction papers
@iancopple56495 жыл бұрын
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!
@coleabrahams93314 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.
@bentleystorlie80737 жыл бұрын
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 Жыл бұрын
Circular tyre 1 wont work?
@mazingzongdingdong7 жыл бұрын
everytime i take a test i imagine that he's looking over me and kinda guiding my way to success lol
@klaud73114 жыл бұрын
Sounds like you envy him more than you admire him.
@solarean4 жыл бұрын
@@klaud7311 for me sounds like he just likes the attitude of this guy idk lel
@nosuchthing83 жыл бұрын
Wouldn't that be great. His IQ must be off the charts.
@thepip35997 жыл бұрын
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?
@thepip35997 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.
@MikeRosoftJH6 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.
@sergey15196 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.
@ethendixon46126 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.
@greysquirrel4046 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?
@ontario21645 жыл бұрын
6:03 careful dude you're gonna summon the devil
@capy98464 жыл бұрын
Michael Darrow Nah I just watch video’s upside down for fun
@dondeestaCarter4 жыл бұрын
JuliasJulian. Cool!! So "Ontario" reads "JuliasJulian" when upside down? Wouldn't have expected that!!
@ValkyRiver3 жыл бұрын
6:14 Exclaves: am I a joke to you?
@TheOfficialCzex6 жыл бұрын
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.4186 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.4186 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.
@TruthNerds5 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-dp8uy5 жыл бұрын
What about water? Why is water, the "background color" of a world map, not considered a color that counts?
@Rannos223 жыл бұрын
That's a cheap cop out given the first examples were political maps
@abidhossain80744 жыл бұрын
0:04 "It's easy to state" I see what you did there..XD
@aurelia655367 жыл бұрын
"Let's try making a map that requires five colors" *second map drawn only has four sections*
@o769235 жыл бұрын
Technically the space outside counts as a region as well (and can include lines that continue for eternity).
@dancrane38075 жыл бұрын
ikr
@lilyfm71525 жыл бұрын
That was drawn to illustrate the network.
@p.mil.11477 жыл бұрын
14:14 look above the o there are 2 yellows
@brokenwave61255 жыл бұрын
There are six colors on that "map" so its not really meant to be an accurate example.
@Joe-qm9cp5 жыл бұрын
Gasp
@uxleumas4 жыл бұрын
culd have been pink
@IHaveaPinkBeard4 жыл бұрын
That's pretty bothersome given the video topic
@janprevratil10154 жыл бұрын
@@brokenwave6125 I think he wanted to be colored with 4, but he gave up :D
@FreshBeatles5 жыл бұрын
I love this mans enthusiasm
@zombiedude3477 жыл бұрын
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.
@tfae10 ай бұрын
I think this is the "even-odd rule" in computer graphics.
@XiaoyongWu4 жыл бұрын
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
@luizazappala35722 жыл бұрын
Thought the same!
@Lockjaw_Larry4 жыл бұрын
Four Color Theorem: Exists Enclaves and Exclaves: I'm about to end this man's whole career
@montanafisher89963 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
@williamchaney4483 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.
@williamjones33133 ай бұрын
@@williamchaney448 Would the dual graph be planar?
@Robertlavigne17 жыл бұрын
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.
@ragnkja7 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.
@patrickhodson87157 жыл бұрын
0:16 Dang foreigners colored Michigan two different colors lolol
@feli-the-sunfairy6 жыл бұрын
@@yesno1498 that is true, but has nothing to do with the fact that Michigan is to large.
@tandemrecruit6 жыл бұрын
but then at 1:47 they have it right XD
@theblackwidower6 жыл бұрын
@@yesno1498 So when factoring in enclaves and exclaves, how many do you need?
@leonhostnik95166 жыл бұрын
@@feli-the-sunfairy Take up all complaints with the state of Ohio on that one, regarding the Toledo War
@the_real_ch36 жыл бұрын
Yoopers gettin no respect
@shivpatel82884 жыл бұрын
Conjecture: Every video of Numberphile requires extensive recursion.
@branflakes26007 жыл бұрын
Yes! James Grime!
@branflakes26007 жыл бұрын
^^^^^ 9th
@JM-us3fr7 жыл бұрын
He's the absolute best!
@bencouperthwaite67357 жыл бұрын
I met him :)
@duck68727 жыл бұрын
I am jealous
@bencouperthwaite67357 жыл бұрын
Duck He came to my college in January. Top guy!
@homopoly7 жыл бұрын
0:15 Did they just colour Michigan wrong?
@jakec9047 жыл бұрын
what?
@hjorth33877 жыл бұрын
The purple and blue state?
@homopoly7 жыл бұрын
Yeah.
@optimist23017 жыл бұрын
Unchi what?
@lyndonhanzpernites58607 жыл бұрын
Michigan was filled with two colors. (Being separated by the Great Lakes.)
@richarddeese19916 жыл бұрын
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
@sophieeula6 жыл бұрын
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
@carsonianthegreat46722 жыл бұрын
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 Жыл бұрын
do you mean exclaves
@tylerbird9301 Жыл бұрын
@@MichaelDarrow-tr1mn an enclave is just an exclave of a different country.
@MichaelDarrow-tr1mn Жыл бұрын
@@tylerbird9301 no it's not. an enclave is a country entirely surrounded by a different country
@tylerbird9301 Жыл 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 Жыл бұрын
@@tylerbird9301 a portion can be 100%
@Wolfsspinne4 жыл бұрын
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.
@juanignaciolopeztellechea94012 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.
@teh1tronner7 жыл бұрын
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.
@vangildermichael17677 жыл бұрын
Awesome point. I hadn't thought on the 3 dimension thing. I like the brain treat. yum.
@ZayulRasco7 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.
@andreashofmann45567 жыл бұрын
But you lose the looping around bit, which i think is what he's going for?
@rubenras13997 жыл бұрын
/dev/zero ii
@Korcalius7 жыл бұрын
And what if a country has a colony or more? Its technically still the same country.
@AlbySilly7 жыл бұрын
3:40 Aww I was hoping for the Chrome logo
@grantcarrell7 жыл бұрын
Albin9000 I was too.
@ayanshah26217 жыл бұрын
I thought it to be a pokemon
@erikplayz81927 жыл бұрын
Albin9000 same
@jamesbatley1736 жыл бұрын
Me, too!
@douira5 жыл бұрын
"I don't know why, but he was" seems to be true for a lot of math
@GreRe97 жыл бұрын
Is there a reason why James says "network" instead of "graph"?
@lucashenry25567 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-us3fr7 жыл бұрын
Green Red The average person recognizes the word "network".
@1990rockefeller7 жыл бұрын
gwuaph. Just kidding. He is awesome!
@EMETRL6 жыл бұрын
because real life applications of this idea often come in the form of actual networking
@cheesebusiness6 жыл бұрын
Because he speaks the British language
@RalphDratman5 жыл бұрын
This is an excellent, clear presentation by Dr. Grime.
@noide18377 ай бұрын
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.
@malik_alharb7 жыл бұрын
i love how hes always so happy
@azimjaved32437 жыл бұрын
"So, Let's talk about the Four Colour Theorem!". James Grime video, After all this time!
@earth111165 жыл бұрын
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"
@mathtexas9687 жыл бұрын
Congratulations on 2,000,000 subscribers!!!
@martinwood7442 жыл бұрын
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.
@toonoobie7 жыл бұрын
I found a map which requires 5 But the comment threads are too small to hold the answer!
@tit-bits61975 жыл бұрын
Ghost of Fermat! 😜
@notquitehadouken5 жыл бұрын
images deathshadow images
@commenturthegreat29155 жыл бұрын
Same
@samwarren60085 жыл бұрын
The map at the end of the video...
@RamsesTheFourth5 жыл бұрын
I did too actually... but i cant post picture here :)
@riccardopratesi79437 жыл бұрын
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_noob6 жыл бұрын
Depends on what you mean by "nearby"... if it were to only apply to orthogonally adjacent, the answer would be 2. :-B
@relaxnation17736 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.
@DheerajAgarwalD5 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.
@irrelevant_noob6 жыл бұрын
11:51 paper shows 1936... Reading it out loud: "one thousand nine hundred and thirty _eight_ ". Eh well, close enough. :-p
@coleabrahams93314 жыл бұрын
🤣🤣🤣Close enough
@saabrinaadan31105 жыл бұрын
But what if you have 5 different counties/countries meeting at one point?
@man-qw2xj5 жыл бұрын
Saabrina Adan points have no area. While a convergence, the fact that the convergence has no area invalidates it.
@o769235 жыл бұрын
Those 5 countries cannot all share a side with each other one; only a point.
@CalifornianMapping5 жыл бұрын
Though such things are possible in the world, here they are simply not considered to be borders.
@zahidhussain2515 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.
@ProfessorX5 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.
@damamdragon735 жыл бұрын
Easy to... “state.” *brings up and starts coloring map of US*
@ajpdeschenes7 жыл бұрын
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!
@thommunistmanifesto3 жыл бұрын
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.
@MistahPhone2 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 Жыл 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.
@filipman5 жыл бұрын
At the end those *yellows touching* in the top right are annoying me
@danielsantos32543 жыл бұрын
And the single purple section on the left
@9thfromthestar5 жыл бұрын
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 Жыл бұрын
In the real world, yeah, but not in the context of this problem.
@richm66335 жыл бұрын
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 ;)
@foomark217 жыл бұрын
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)
@Pseudo___7 жыл бұрын
so this is assuming maps cant have non contiguous sections? Some coutries/gerrymandered districs/ect can get weird and have breaks .
@jimmyfitz-etc70313 жыл бұрын
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
@tiffanie50125 жыл бұрын
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
@DrSnap237 жыл бұрын
Summoning Satan at 6:06, I see what you did there.
@Squideey7 жыл бұрын
This is Numberphile. They were summoning Pythagoras.
@JimSteinbrecher7 жыл бұрын
surely, 5474N
@CH3LS3A7 жыл бұрын
They were summoning Fermat's "I have a proof of this..." proofs.
@gyrfalcon237 жыл бұрын
this shape is a pentagram, but not inverted as in satanism
@unity3037 жыл бұрын
I think it had to work @11:06 where it's actually 666 seconds, what did he do there...
@fladmus7 жыл бұрын
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.
@lagcom5 жыл бұрын
What about exclaves? Shouldn’t they be the same color with their mainland?
@piguy92255 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.
@piguy92255 жыл бұрын
*4 colors, not 2.
@jako09814 жыл бұрын
@@piguy9225 Yes there is you mongoloid
@RazvanMaioru4 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
@lolsluls9957 жыл бұрын
Yeayy james grime! james grime! james grime! Forget about Terence Tao, James Grime is the sexy mathmatician celebrity we need.
@adamhannath14175 жыл бұрын
What about portions of land held by a country to which the portion is not connected, or exclaves. For example, 3:25 say there is a small country between pink yellow and blue but the country also owns a small sliver of land between yellow blue and purple. Both portions of this country would have to be the same colour to show dominion but together touch all four colour of the wheel, it must therefore be colour five. There are real instances a country being separated by other land masses (eg. Nakhchivan). I realise this is pedantic as the problem is supposed to only work with solid land masses.
@emilyp73624 жыл бұрын
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.
@TheThunderSpirit7 жыл бұрын
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
@marios18615 жыл бұрын
its an example showing how connected graph theory is to topology.
@trexaz194 жыл бұрын
Am I missing something here? The example he gave for a network that couldn't be a map very much could exist. What if you had a map of say 65different countries separated like a pie? All 5 countries would be touching every other country; therefore, you would need 5 colors. A situation like this exists between Arizona, New Mexico, Colorado, and Utah (only with 4, all states are touching each other). So this would be a similar situation just with 5 states (or countries). Like @10:47 Fig 6 only where b and e are touching as well.
@arnouth52604 жыл бұрын
Corners don’t count as borders
@richarddeese19913 жыл бұрын
A 'network' can be drawn such that more than 4 colors are needed. But it doesn't correspond to a real-world map. I tried this for years, just doodling to amuse myself. You can't draw more than 4 closed shapes that all touch each other along a side! You'd literally have to veer into the 3rd dimension to do it. But I still think that computer proof is pretty rotten. What did it teach us, except "Beep. Here's your answer. Beep." tavi.
@archiehellshire10816 жыл бұрын
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.
@TheOfficialCzex6 жыл бұрын
Lol
@hevgamer60876 жыл бұрын
the 4 colors theorem is only for 2D maps, if you go to 3D, you can make maps that require infinite colors
@alex_meli4 жыл бұрын
What about enclaves and exclaves? This could create some of the situations you presented as "impossible"
@OleTange3 жыл бұрын
Yes. And they should have mentioned this.
@macaroni94964 жыл бұрын
0:15 did he count the U.P. of Michigan a state? Or is it just every separate landmass with borders?
@coling12585 жыл бұрын
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.
@TheWeepingCorpse7 жыл бұрын
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.
@UltimatePerfection7 жыл бұрын
TheWeepingCorpse For what language?
@TXKurt5 ай бұрын
7:50, if there must be a country connected to five or fewer others, that means we must be excluding infinite maps, right? Does the four-color theorem also hold for those? What about maps on a sphere or torus or... ? Edit: Lots of information on these cases on Wikipedia.
@lockrime5 жыл бұрын
Numberphile: It's possible to paint a map with only four colours. Exclaves: *I am gonna end this man's entire career*
@EricTheRea5 жыл бұрын
You don't understand the problem.
@OleTange3 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).
@CoolExcite7 жыл бұрын
Anyone else try to draw a counterexample is ms paint and miserably fail?
@erichiguera7 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
@orionmartoridouriet68347 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
@skyr84497 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.
@CraftQueenJr7 жыл бұрын
I succeeded in makng one on my channel..
@JohanBregler6 жыл бұрын
I found one, but I don't know where to submit it
@ThankNephew5 жыл бұрын
As a computer science student currently learning Boolean algebra, de Morgan’s name sends me into a fiery rage
@lawrencedoliveiro91043 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.
@nextlevelnick93395 жыл бұрын
I tried to draw a counterexample for ten minutes then decided, I’ll take his word for it 😂
@dancrane38075 жыл бұрын
Umm, originally it took 120 years. So, get back in there!
@permafrost01365 жыл бұрын
Try using enclaves or exclaves you can easily get a map that needs 5 colors
@permafrost01363 жыл бұрын
@Michael Darrow no but they may cause two countries that border each other the be the same color
@SashaPersonXYZ7 жыл бұрын
so sad he had to use 5 colors to color the square space map.
@parad0x4487 жыл бұрын
Sashamanxyz 6
@philipphaselwarter22875 жыл бұрын
What a regrettable choice not to mention Gonthier and Werner's work on establishing the correctness (and improving) of the proof.
@luck39497 жыл бұрын
What about maps on other surfaces, like torus?
@ffggddss7 жыл бұрын
The question had been long-settled on all the other 2D topologies, which was another thing that made the 4-color problem so galling. For the torus, the maximum number of required colors is 7. For the Möbius strip, the Klein bottle and the projective plane, the number is 6. And for other, more complicated surfaces, the problem was solved long ago. But for the plane/sphere (which are topologically equivalent), it remained unsolved for ages.
@JohnnyDoeDoeDoe7 жыл бұрын
ffggddss Links?
@ffggddss7 жыл бұрын
en.wikipedia.org/wiki/Four_color_theorem#Generalizations goes into the case of the torus, e.g.
@JohnnyDoeDoeDoe7 жыл бұрын
Thanks!
@Necroskull3887 жыл бұрын
IIRC, the plane and sphere are technically only homeomorphic if you remove a singular point from the sphere.
@fornkly6 жыл бұрын
4:29 The Deathly Hallows... J.K. Rowling was secretly a mathematician "After all this time?" Always.
@Irondragon19455 жыл бұрын
It's a fake quote! He never said that
@Feuerpfeil3695 жыл бұрын
@@Irondragon1945 the actor didn't say it but Snape did if I remember correctly
@Irondragon19455 жыл бұрын
@@Feuerpfeil369 I think a fan on tumblr said it about himself, and some people then attributed it to the Snape/actor persona. I am 80% sure, but I wouldnt know what to quote.
@Feuerpfeil3695 жыл бұрын
@@Irondragon1945 at least in the German Version it is like that. When Harry watches Snape's Memory's in DH. After Dumbledore tells him Harry has to die
@JG-ld6cf4 жыл бұрын
Irondragon1945 it definitely says it in deathly hallows
@jankisi4 жыл бұрын
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
@איתןגרינזייד4 жыл бұрын
I found a map that needs five colors but it's only in my mind, the map is too big for the observable universe.
@schonerwissen20135 жыл бұрын
This video and the problem were quite interesting! But the end was...somewhat unsatisfactory! :(
@baileymendel29796 жыл бұрын
at 6:03: you can draw a map of the five countries if you imagine the polyhedron from the top. It would be a bit weird, but the countries could theoretically intersect at a single point
@shinjinobrave4 жыл бұрын
11:00 The final solution was done by significantly more than two guys :s
@laurencewilson61635 жыл бұрын
What if u have a country that is split ip
@thomasmiller82895 жыл бұрын
Do spherical maps have a different color theorem? Or do they still count?
@timsolnze73005 жыл бұрын
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.
@MOHAMMADALAHDAB5 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
@Vjdkgaming7 жыл бұрын
WOw i think someone has made a mistake 14:14 look between the first letters S and Q
@janmalek97947 жыл бұрын
Actually he used 6 colors ;)
@Vjdkgaming7 жыл бұрын
Wow i'm colorblind :D
@otlat7 жыл бұрын
Also there is a yellow connected with yellow under the letter "O".
@redbeam_7 жыл бұрын
you mean above the letter "O"
@Kourindouinc7 жыл бұрын
It's a Parker Coloring Session.
@heimdall19735 жыл бұрын
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.
@iotashift7 жыл бұрын
0:16. Michigan is two different colors.
@manioqqqq Жыл бұрын
In the map, the disconnected are disallowed. So, Michigan is 2 countries.
@microwave856 Жыл бұрын
@@manioqqqqthey both can be blue while still following rules
@diegomo141315 күн бұрын
Same with Maryland and Virginia. The part under Delaware, which contains a portion of Maryland _and_ Virginia, is colored blue, while Maryland and Virginia are colored yellow and purple respectively.
@therealdave065 жыл бұрын
The key to breaking this: Enclaves and exclaves
@seanleith53124 жыл бұрын
and draw three lines that intersect at one point, his law is broken.
@michaelgoddard1435 Жыл бұрын
So what about at 1:48 where the 4 corner states touch point in the USA specifically Colorado and Arizona are the same color green and touch... seems off a bit doesn't it?
@TomasAragorn Жыл бұрын
We don't consider areas that only border in a single point to be neighbors. Otherwise, you can let a 100 areas border each other and you would need 100 colours
@mihailazar24877 жыл бұрын
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
@madlad2554 жыл бұрын
0:05 'It's very easy to state' *shows a map of the United States*
@themobiusfunction3 жыл бұрын
STATE
@Nilslos7 жыл бұрын
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 :-).
@DemianNuur7 жыл бұрын
13:11 Wow! New haircut!
@NoriMori19925 жыл бұрын
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?").
@whywelovefilm70795 жыл бұрын
I didn’t think it was possible. Congratulations, you have made coloring complicated...
@missingno-xk7kp6 жыл бұрын
"so the final solution..." 10:59
@Reeceeboy5 жыл бұрын
You deserve to wake up at 3am then fall asleep all warm in bed.
@hadhave79616 жыл бұрын
11:02 Imagine being named WolfGang, what a badass
@gaberockmain4 жыл бұрын
I’m slightly confused about this one here. We’re there certain parameters I missed that limited what countries could look like on maps? If you have a circle with a cross in the middle, evenly splitting the area of the circle into four sections, then have a country surrounding the entire circle that looked like a ring, would you not need 5 colors for that?
@genericusername42064 жыл бұрын
all simply connected maps can be coloured with 4 colours or less also i think your example cannot be made into a graph/network you have not discovered a new discovery or something, 5 colour theorem has been a thing for a while
@genericusername42064 жыл бұрын
i think i figured out how your example can be solved using 4 colours or less the hint is that the countries can be the same colour if they shâre a corner but not if they share an edge
@danielboland65785 жыл бұрын
0d maps use 1 colour, 1d maps use 2 colours, 2d maps use 4, 3D uses 8. 2 to the power of n?
@themobiusfunction3 жыл бұрын
3d uses infinitely many.
@manta8947 жыл бұрын
what if the map is on a globe and goes around?
@bluerizlagirl7 жыл бұрын
Then you have a three-dimensional shape, and the rules are different in three dimensions .....
@gencshehu7 жыл бұрын
The surface of is a two-manifold, that is, it is two dimensional. If the 'surface' was three dimensional the rules would be different. But this really points to the thing itself: what's the relation between dimensions and is there a continuous translation between dimensions? Fractals, for example, can be described as non-discrete dimensional objects. They would be the answer to the question of what does it mean for something to be 1.5834... dimensional, that is, non-discrete.
@manta8947 жыл бұрын
yea thats why i asked.. if earth had no sea you would ned more than 4 colors, im I right?
@Deadfunk-Music7 жыл бұрын
Actually no, as long as you stay in 2D (even on a globe's surface) it can all be done with 4 colours.
@sk8rdman7 жыл бұрын
You could color all of the oceans as one blue section, and no country adjacent to the ocean would need to be blue. One thing to note about coloring something like countries, specifically, is the fact that a single country might sometimes be divided by other countries or a body of water. In this case, it won't always be possible to color all isolated sections of a country the same color. This is purely a topology problem, and sometimes political dynamics can break the rules.
@jsax4heart4 жыл бұрын
Do other countries math courses prefer the term graph or the term network? Its interesting how he mentions the petersen graph and planar graphs.
@jmkbartsch7 жыл бұрын
Yes, and then someone said: "Hey, what about exclaves?" - And the theorem was rendered useless.
@brianbethea30695 жыл бұрын
And then someone reminded that person that this theorem is about mappings of contiguous spaces, not political ones. And then the theorem worked again.
@TheJaredtheJaredlong5 жыл бұрын
@@brianbethea3069 But the gensis of the problem was a real world application. But I guess it is easier to solve a completely different question than the one actually asked.
@rednecktash5 жыл бұрын
@@TheJaredtheJaredlong but it happens to work for all countries on earth, so again it's fine :)