The Four Color Map Theorem - Numberphile
ฝัง
- เผยแพร่เมื่อ 19 มี.ค. 2017
- The Four Color Map Theorem (or colour!?) was a long-standing problem until it was cracked in 1976 using a "new" method... computers!
A little bit of extra footage from this: • Four Color Theorem (ex...
This video features Dr James Grime - jamesgrime.com
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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
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 never thought I would say this of a mathematician, but I don't believe I could ever tire of listening to James Grime. I actually find myself smiling far more often than was likely ever the case back in my school days. Thank you Dr Grime.
I recall an issue of Scientific American back in about 1974 (more or less) that had an article that purported to show 7 amazing recent discoveries. One was that the best first move in chess was shown to be h4, another was a logical way to disprove special relativity, another was that Di Vinci invented the toilet and another was that someone came up with a map that required five colors. I can't recall the year of the publication, but I can recall the month. The magazine came out on April first......
@@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
0:04
"It's easy to state"
I see what you did there..XD
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?
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.
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.
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?
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
I love this mans enthusiasm
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!
"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.
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?
Conjecture: Every video of Numberphile requires extensive recursion.
Back in the windows xp days, I'd make images in paint by making one arbitrary continuous path both ends on an edge of the image. The curve would intersect itself at many points, but never intersect itself multiple times at the same point. I found that you could always cover the "map" created using these restrictions with exactly 2 colors.
I think this is the "even-odd rule" in computer graphics.
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.
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
This is an excellent, clear presentation by Dr. Grime.
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
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?
@@ilfedarkfairy Take up all complaints with the state of Ohio on that one, regarding the Toledo War
Yoopers gettin no respect
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!
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 ;)
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!
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.
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%
11:51 paper shows 1936... Reading it out loud: "one thousand nine hundred and thirty _eight_ ". Eh well, close enough. :-p
🤣🤣🤣Close enough
Congratulations on 2,000,000 subscribers!!!
i love how hes always so happy
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.
"I don't know why, but he was" seems to be true for a lot of math
I know I'm late to this conversation, but it got me thinking. I think it can be put more simply, actually, although mathematicians might not like it as much. Here goes...
With all of the parameters already set (contiguous borders and the like), the question becomes, can you:
1. Create a theoretical map that requires 1 color? Yes, duh.
2. Create a map with 2 colors? Yes, you just need 2 touching areas.
3. Create a map with 3 colors? Yes, like a pie cut into 3 slices, each piece touches both of the others, so 3 colors required.
4. Create a map with 4 colors? Yes, take the pie from before and make the center its own area that touches all 3 of the original slices.
5. Create a map with 5 colors? No. Here's why. Imagine 5 squares arranged into a cross or +. One in the middle, and one each on the top, bottom, left, and right. Right now, you only need 2 colors, as the outside squares don't actually touch. So, let the outside shapes bulge a little and touch their neighbors (top now touches left, right, and center, left touches top, bottom, and center, and so on). Now, you need 3 colors. Why not 4? because right now, the shapes on opposite sides of the center square don't touch and can be the same color. Let's try to fix that! Take the top shape now, and stretch it around to touch the bottom shape. Awesome, now we need 4 colors, since the top and bottom cannot share anymore! Now, let's go for 5! Currently, the only shapes still sharing colors are the left and right shapes, so we need to get them to touch. But wait, to get the top and bottom to touch, we had to go around either the left side or right side (we'll say left, but it doesn't matter). The right shape has no way of getting to the left now! Well, what if we cut under the top? Oops, the top and center are not touching anymore! Well, what if we slice through the arm connecting top and bottom? Well, then we're back to where we just were with top and bottom not touching. Feel free to play with it and make the shapes weirder, but you cannot get all 5 shapes to touch every one of the other shapes without breaking a connection that you had previously made. Even if you add a sixth shape wrapping around the outside of the whole mess, it will still be separated from the center square and will be allowed to use that color, unless you break one of your earlier connections (at which point, what have you accomplished?).
All of the nightmare with proofs and computers and whatnot may be needed for mathematical certainty, but if you cannot get a mere 5 or 6 shapes to need 5 colors, then adding additional shapes just aggravates the issue of fighting for connections.
I tried to keep that whole thing simple enough to sketch along if anyone cannot follow in their head. My apologies, and thank you for coming to my talk.
i don’t know why i’m watching these math videos at 3 am bc i truly don’t understand them but everyone in the vids seem to so i keep on comin back
I found a map that needs five colors but it's only in my mind, the map is too big for the observable universe.
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.
"So, Let's talk about the Four Colour Theorem!". James Grime video, After all this time!
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 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.
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 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"
Easy to... “state.” *brings up and starts coloring map of US*
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.
At the end those *yellows touching* in the top right are annoying me
And the single purple section on the left
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.)
so sad he had to use 5 colors to color the square space map.
Sashamanxyz 6
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!
Love your videos!
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 :)
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
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.
Awesome explanation.
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.
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...
I love these videos.
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.
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)
Yeayy james grime! james grime! james grime! Forget about Terence Tao, James Grime is the sexy mathmatician celebrity we need.
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
well explained. thank you.
you should talk how the 4 color map problem can be used for scheduling. examples like students time slots for exames.
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.
Excellent video
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 enclaves and exclaves? This could create some of the situations you presented as "impossible"
Yes. And they should have mentioned this.
13:11 Wow! New haircut!
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 :-).
I didn’t think it was possible.
Congratulations, you have made coloring complicated...
so this is assuming maps cant have non contiguous sections? Some coutries/gerrymandered districs/ect can get weird and have breaks .
This video and the problem were quite interesting! But the end was...somewhat unsatisfactory! :(
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
Just begin from the worst case scenario: where each of the countries touch , you cannot put in any other node which touches all other nodes without crossing a vertex.
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
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).
I actually had this question on my graph theory course thats so cool
What a regrettable choice not to mention Gonthier and Werner's work on establishing the correctness (and improving) of the proof.
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.
What about exclaves?
They could produce crossing lines.
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.
I think it would be interesting to have a video on coloring maps on surfaces with higher genus or to colour empires on a plane
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
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
Thank u for this sir.
If hypothetically you had a region divided entirely into two separate parts but must be the same color because they are considered the same region, then you would need 5 colors
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?
What if u have a country that is split ip
Irrational maps: subdivided maps that have no end
3D version: imagine it like a puzzle; how many colored puzzle pieces would you need to fill a given shape so that no two colors touch
Infinitely many in 3d.
*draws a pentagram
"now this is not going to be a valid map"
*Puts a few colored dots*
*joins the outer three together*
WAIT A MINUTE DO I SEE SOMETHING...
Illuminati confirmed.
In my class, we call that a proof by exhaustion (literally)
Do spherical maps have a different color theorem? Or do they still count?
Couldn’t help but notice that you you colored Michigan’s peninsulas different colors near the beginning
11:00 The final solution was done by significantly more than two guys :s
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.
me thinking i can easily disprove this theorem in MS paint at 5am. you don't know how many times i've thought "AH YES, I HAVE DONE IT" only to realize a couple seconds later that i have not, in fact, done it
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
I came to this conclusion myself when I was 6, trying to colour in camouflage on a drawing. Interesting to see there's such a big following for colouring in!
the conclusion, they probably did. You can try for a while and go "huh, maybe it only takes four?". the PROOF of course not, but they didn't say that...
@@Paul-zx8jf Were you there, jackass?
i want to see a map of the H.R.E. with only four colors...
Ouch my eyes
The thing I kept thinking for all of this is how this is a fundamentally geometric approach to the issue of mapmaking, and countries, counties, and other things represented by maps often don't follow geometric rules with things like exclaves. And don't think I didn't notice how most of your four color examples still reserved a fifth color for the sea.
You can use a fifth color for the sea if you try hard enough.
Can't believe this was the first mathematical problem solved in the computer, wow