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.
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).
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 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.
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
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 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......
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!
@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.
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.
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?
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.
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.
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.
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?
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.
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.
@@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.
@@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.
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.
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
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
@@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.
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.
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
@@MichaelDarrow-tr1mn enclave noun a portion of territory within or surrounded by a larger territory whose inhabitants are culturally or ethnically distinct.
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.
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.
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.
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
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 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.
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.
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.
@@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.
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.
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.
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!
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.
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 ;)
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)
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
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 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.
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.
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.
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
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.
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.
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.
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.
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.
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.
@@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).
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
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.
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.
@@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.
@@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
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
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
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.
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
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.
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?
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
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
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 :-).
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?").
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?
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
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
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.
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.
@@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.
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.
er.. prevents, not allows.
Interesting, never thought of that.
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).
That's such a terrific example! You just blew my mind!
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
Some maps ACTUALLY don't work with this
@@amaanali9525 example?
@@user-zz3sn8ky7z the ones made by Susan Goldstein.
@@amaanali9525 that was interesting, thanks for sharing! Although I'm not sure if it counts as a "map"
@@user-zz3sn8ky7z oh okay your welcome
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.
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
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 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......
@@error.418 April 1st. Think about it.
I like this comment.
h4 is the best first move in chess? This has to be a joke.
@@willyantowilly7165 April 1st
I enjoy stork theory of reproduction papers
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!
@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.
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.
Circular tyre 1 wont work?
everytime i take a test i imagine that he's looking over me and kinda guiding my way to success lol
Sounds like you envy him more than you admire him.
@@klaud7311 for me sounds like he just likes the attitude of this guy idk lel
Wouldn't that be great. His IQ must be off the charts.
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?
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.
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.
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.
I'm gonna assume it would be 8. I can't back this up . . . but I think it's related by 2^dimension.
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?
6:03 careful dude you're gonna summon the devil
Michael Darrow
Nah I just watch video’s upside down for fun
JuliasJulian. Cool!! So "Ontario" reads "JuliasJulian" when upside down? Wouldn't have expected that!!
6:14 Exclaves: am I a joke to you?
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.
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.
@@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.
@@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.
What about water?
Why is water, the "background color" of a world map, not considered a color that counts?
That's a cheap cop out given the first examples were political maps
0:04
"It's easy to state"
I see what you did there..XD
"Let's try making a map that requires five colors"
*second map drawn only has four sections*
Technically the space outside counts as a region as well (and can include lines that continue for eternity).
ikr
That was drawn to illustrate the network.
14:14 look above the o there are 2 yellows
There are six colors on that "map" so its not really meant to be an accurate example.
Gasp
culd have been pink
That's pretty bothersome given the video topic
@@brokenwave6125 I think he wanted to be colored with 4, but he gave up :D
I love this mans enthusiasm
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.
I think this is the "even-odd rule" in computer graphics.
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
Thought the same!
Four Color Theorem: Exists
Enclaves and Exclaves: I'm about to end this man's whole career
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
@@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.
@@williamchaney448 Would the dual graph be planar?
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.
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.
0:16 Dang foreigners colored Michigan two different colors lolol
@@yesno1498 that is true, but has nothing to do with the fact that Michigan is to large.
but then at 1:47 they have it right XD
@@yesno1498 So when factoring in enclaves and exclaves, how many do you need?
@@feli-the-sunfairy Take up all complaints with the state of Ohio on that one, regarding the Toledo War
Yoopers gettin no respect
Conjecture: Every video of Numberphile requires extensive recursion.
Yes! James Grime!
^^^^^
9th
He's the absolute best!
I met him :)
I am jealous
Duck He came to my college in January. Top guy!
0:15 Did they just colour Michigan wrong?
what?
The purple and blue state?
Yeah.
Unchi what?
Michigan was filled with two colors. (Being separated by the Great Lakes.)
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
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
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.
do you mean exclaves
@@MichaelDarrow-tr1mn an enclave is just an exclave of a different country.
@@tylerbird9301 no it's not. an enclave is a country entirely surrounded by a different country
@@MichaelDarrow-tr1mn
enclave
noun
a portion of territory within or surrounded by a larger territory whose inhabitants are culturally or ethnically distinct.
@@tylerbird9301 a portion can be 100%
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.
The theorem only couts CONTIGUOUS countrys.
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.
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.
Awesome point. I hadn't thought on the 3 dimension thing. I like the brain treat. yum.
/dev/zero You can map a sphere to a 2d surface and preserve the properties we care about regarding the 4 color theorem.
But you lose the looping around bit, which i think is what he's going for?
/dev/zero ii
And what if a country has a colony or more? Its technically still the same country.
3:40
Aww I was hoping for the Chrome logo
Albin9000 I was too.
I thought it to be a pokemon
Albin9000 same
Me, too!
"I don't know why, but he was" seems to be true for a lot of math
Is there a reason why James says "network" instead of "graph"?
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
Green Red The average person recognizes the word "network".
gwuaph. Just kidding. He is awesome!
because real life applications of this idea often come in the form of actual networking
Because he speaks the British language
This is an excellent, clear presentation by Dr. Grime.
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.
i love how hes always so happy
"So, Let's talk about the Four Colour Theorem!". James Grime video, After all this time!
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"
Congratulations on 2,000,000 subscribers!!!
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 found a map which requires 5
But the comment threads are too small to hold the answer!
Ghost of Fermat! 😜
images deathshadow
images
Same
The map at the end of the video...
I did too actually... but i cant post picture here :)
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.
Depends on what you mean by "nearby"... if it were to only apply to orthogonally adjacent, the answer would be 2. :-B
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.
@@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.
11:51 paper shows 1936... Reading it out loud: "one thousand nine hundred and thirty _eight_ ". Eh well, close enough. :-p
🤣🤣🤣Close enough
But what if you have 5 different counties/countries meeting at one point?
Saabrina Adan points have no area. While a convergence, the fact that the convergence has no area invalidates it.
Those 5 countries cannot all share a side with each other one; only a point.
Though such things are possible in the world, here they are simply not considered to be borders.
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.
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.
Easy to... “state.” *brings up and starts coloring map of US*
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!
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.
But here we are talking about mainland Europe
1. The islands are (i think) not a part of the countries
2. That is clearly yellow
@@manioqqqq neither france nor the netherlands are yellow, and saint martin is considerd thier territories
@@thommunistmanifesto either i am colorblind or you are, but they're clearly yellow.
And, in the problem ignore the island border.
You're allowed to color a country in two separate colors. It's just that each region individually has to be a single color.
At the end those *yellows touching* in the top right are annoying me
And the single purple section on the left
6:19 it makes sense if you have exclaves/enclaves. Right?
They are off in the quadracolor theorem
And,
🟥🟦
🟦🟥
Is valid.
In the real world, yeah, but not in the context of this problem.
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 ;)
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)
so this is assuming maps cant have non contiguous sections? Some coutries/gerrymandered districs/ect can get weird and have breaks .
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
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
Summoning Satan at 6:06, I see what you did there.
This is Numberphile. They were summoning Pythagoras.
surely, 5474N
They were summoning Fermat's "I have a proof of this..." proofs.
this shape is a pentagram, but not inverted as in satanism
I think it had to work @11:06 where it's actually 666 seconds, what did he do there...
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.
What about exclaves? Shouldn’t they be the same color with their mainland?
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.
*4 colors, not 2.
@@piguy9225 Yes there is you mongoloid
@@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
Yeayy james grime! james grime! james grime! Forget about Terence Tao, James Grime is the sexy mathmatician celebrity we need.
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.
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.
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
its an example showing how connected graph theory is to topology.
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.
Corners don’t count as borders
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.
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.
Lol
the 4 colors theorem is only for 2D maps, if you go to 3D, you can make maps that require infinite colors
What about enclaves and exclaves? This could create some of the situations you presented as "impossible"
Yes. And they should have mentioned this.
0:15 did he count the U.P. of Michigan a state? Or is it just every separate landmass with borders?
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.
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.
TheWeepingCorpse For what language?
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.
Numberphile: It's possible to paint a map with only four colours.
Exclaves: *I am gonna end this man's entire career*
You don't understand the problem.
@@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).
Anyone else try to draw a counterexample is ms paint and miserably fail?
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
Throbbin So Hard Frontiers cannot be made only by one point, so the center of the circle doesn't count as a valid frontier
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.
I succeeded in makng one on my channel..
I found one, but I don't know where to submit it
As a computer science student currently learning Boolean algebra, de Morgan’s name sends me into a fiery rage
I don’t know why. His theorems are so straightforward. It’s like basic knowledge that every programmer should have absorbed into their DNA.
I tried to draw a counterexample for ten minutes then decided, I’ll take his word for it 😂
Umm, originally it took 120 years. So, get back in there!
Try using enclaves or exclaves you can easily get a map that needs 5 colors
@Michael Darrow no but they may cause two countries that border each other the be the same color
so sad he had to use 5 colors to color the square space map.
Sashamanxyz 6
What a regrettable choice not to mention Gonthier and Werner's work on establishing the correctness (and improving) of the proof.
What about maps on other surfaces, like torus?
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.
ffggddss Links?
en.wikipedia.org/wiki/Four_color_theorem#Generalizations
goes into the case of the torus, e.g.
Thanks!
IIRC, the plane and sphere are technically only homeomorphic if you remove a singular point from the sphere.
4:29 The Deathly Hallows... J.K. Rowling was secretly a mathematician
"After all this time?"
Always.
It's a fake quote! He never said that
@@Irondragon1945 the actor didn't say it but Snape did if I remember correctly
@@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.
@@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
Irondragon1945 it definitely says it in deathly hallows
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
I found a map that needs five colors but it's only in my mind, the map is too big for the observable universe.
This video and the problem were quite interesting! But the end was...somewhat unsatisfactory! :(
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
11:00 The final solution was done by significantly more than two guys :s
What if u have a country that is split ip
Do spherical maps have a different color theorem? Or do they still count?
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.
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
WOw i think someone has made a mistake 14:14 look between the first letters S and Q
Actually he used 6 colors ;)
Wow i'm colorblind :D
Also there is a yellow connected with yellow under the letter "O".
you mean above the letter "O"
It's a Parker Coloring Session.
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.
0:16. Michigan is two different colors.
In the map, the disconnected are disallowed. So, Michigan is 2 countries.
@@manioqqqqthey both can be blue while still following rules
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.
The key to breaking this:
Enclaves and exclaves
and draw three lines that intersect at one point, his law is broken.
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?
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
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
0:05 'It's very easy to state'
*shows a map of the United States*
STATE
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 :-).
13:11 Wow! New haircut!
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?").
I didn’t think it was possible.
Congratulations, you have made coloring complicated...
"so the final solution..." 10:59
You deserve to wake up at 3am then fall asleep all warm in bed.
11:02 Imagine being named WolfGang, what a badass
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?
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
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
0d maps use 1 colour, 1d maps use 2 colours, 2d maps use 4, 3D uses 8. 2 to the power of n?
3d uses infinitely many.
what if the map is on a globe and goes around?
Then you have a three-dimensional shape, and the rules are different in three dimensions .....
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.
yea thats why i asked.. if earth had no sea you would ned more than 4 colors, im I right?
Actually no, as long as you stay in 2D (even on a globe's surface) it can all be done with 4 colours.
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.
Do other countries math courses prefer the term graph or the term network? Its interesting how he mentions the petersen graph and planar graphs.
Yes, and then someone said: "Hey, what about exclaves?" - And the theorem was rendered useless.
And then someone reminded that person that this theorem is about mappings of contiguous spaces, not political ones. And then the theorem worked again.
@@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.
@@TheJaredtheJaredlong but it happens to work for all countries on earth, so again it's fine :)