The Four Color Map Theorem - Numberphile 

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.

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

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

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......

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

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

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

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.

Yes! James Grime!

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!

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.

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?

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

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

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.

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.

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

Conjecture: Every video of Numberphile requires extensive recursion.

14:14 look above the o there are 2 yellows

0:16 Dang foreigners colored Michigan two different colors lolol

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.

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

3:40 Aww I was hoping for the Chrome logo

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.

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.

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.

0:15 Did they just colour Michigan wrong?

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

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

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 which requires 5 But the comment threads are too small to hold the answer!

i love how hes always so happy

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.

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

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

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 ;)

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

Congratulations on 2,000,000 subscribers!!!

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

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

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

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

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

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 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

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.

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.

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

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.

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

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.

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

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

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

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.

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.

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

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

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

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

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.

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

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

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

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

13:11 Wow! New haircut!

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

This has always been of interest to me.

What about exclaves? They could produce crossing lines.

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?").

The first thing you learn in crystallography is the number of tilings that can completely fill space. A tiling would be taking some shape and translating, rotating, and or flipping it so that it completely fills an infinite floor with the same pattern. There are only about 120 of them and there is a proof that involves 4 dimensional geometry. I always thought that the tilings problem with its 4 dimensional aspect somehow related to the 4 color map theorem.

Love your videos!

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

Until you start considering enclaves/exclaves. Then you can require more than 4 colours.

Moral of the story, not every mathematical proof is beautiful. In other words, not all theorems have a well understood underlying argument - sometimes they just happen to be true.

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 :-).

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.

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

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.

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

What if u have a country that is split ip

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

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.

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

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

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

Intuition/Conjecture:- 4 Colours/Dimensions of Identification would seem to be the Rule of Geometric Drawing and Perspective, mapped/substantiated in QM-TIMING Vanishing Point Vertices => CofG type relative Location, ie that's why 3D+T has these dominant characteristics, as Perceived.. In other words, the phase-states of QM-Time modulation proceed in frequency dominance 1, 2, 3, 4, etc such that the Unitary-Duality 0-1-2 ×3, 2i covers the map, or topology of the dominant probabilities of relatively coherent / observable scales.

Do other countries math courses prefer the term graph or the term network? Its interesting how he mentions the petersen graph and planar graphs.

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

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

I love these videos.

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

So it took a German guy to find the final solution?

The key to breaking this: Enclaves and exclaves

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

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