1. Tristan R June 16, 2019 at 5:39 am

    you failed in the first example.. orange on two sides..

  2. Kevin Stubbersfield June 16, 2019 at 9:29 pm

    Personally if i was to do a map with multiple counties/states id probably use about 6-7 colors…..

  3. Tim Solnze June 17, 2019 at 2:59 pm

    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.

  4. mikeatyouttube June 18, 2019 at 12:39 pm

    At 1:48 the maps of the US states contains dark blue, light blue, green, red… and brown. Why is the background (ocean water) ignored? To draw that map 'properly' with only 4 colors you would have to swap CA with NV (so that Nevada was dark blue, CA light blue), and similarly with Ohio and Kentucky. Then the 'oceans/lakes/rivers' can be changed from brown to dark blue.

  5. Anomasiri June 20, 2019 at 2:16 am

    I've won. I made a map that needs 5 colours. Where can i collect my money?

  6. Bill Busen June 21, 2019 at 5:46 pm

    Happy Birthday to Wolfgang Haken!

  7. The Red Namalas June 22, 2019 at 8:33 pm

    I have an issue for this:
    Country borders are weird sometimes, and can therefore be split into two or more pieces while still remaining the same country. Take for example Hawaii. Let’s say you put it in the map on the thumbnail. Since it stretches across 4 colors, it would have to be a different color than all of them.

  8. Richard Donovan June 23, 2019 at 9:48 pm

    Hi James! Before the four colour theorem, were there maps with 5 colours? If so, do any exist? Thanks

  9. Brandon Haffeman June 24, 2019 at 1:38 pm

    What about with Enclaves and Exclaves? Russia, any one?

  10. Matt Carrell June 28, 2019 at 11:18 pm

    I suspect that this ha to do with reduction of common factors associated with evens and odds or exceeding the number of sides of a triangle or square, your most basic forms. a square could only touch four other colors, but its really only two because if its even number of neighbors, you only need two more colors an it odd, you need three…. 2 is 2, 3 is 3, 4 is 2, 5 is three, 6 is two. and that goes on forever. Do it mathematically, not with diagrams. There is only one scenario where you would need five colors but that scenario does not to my knowledge exist and that would be a three dimensional map where the territories of some countries pass over or under the territories of others in the third dimension. IN that case you would need at least one more additional color.

  11. Calmituron July 1, 2019 at 10:13 pm

    Aaand it all breaks when you try to color countries on a globe.

  12. ohmarin July 2, 2019 at 4:47 am

    angrily huffs at my map where sudan and south sudan are the same colour

  13. Unsettling threats July 4, 2019 at 3:50 am

    What about like a pint between 5? Like a pizza almost

  14. Christian Hoen July 4, 2019 at 5:17 pm

    just put a circle and 5 lines from center to edge. bam! They all neighbor each other in the centerpoint. needs 5 colours.

  15. rugby james July 4, 2019 at 10:08 pm

    Pretty sure I just drew one that needs 5

  16. Naman Gupta July 5, 2019 at 2:53 pm

    And technically, I can also color a chess board following the same condition by using only 4 different colors

  17. Yan Xishan 閻錫山 July 5, 2019 at 7:09 pm

    This entire video is defeated by exclaves.

  18. Daniel Houser July 8, 2019 at 6:30 am

    what if the map is on a ball, 4 shapes surrounding 1 but the 4 shapes all go around the ball and touch on the back, you need 4 colors for the outer 4 and a 5th color for the one that touches the 4 surrounding not sure if the problem is only with 2 dimensions or if it can be mapped in 3d like i stated

  19. MacGamer Media July 9, 2019 at 2:58 pm

    Does it count if two sections are connected by a single point. Like two circles connected by a single point, do they count as touching each other or adjacent?

  20. Lockrime Channel July 9, 2019 at 6:17 pm

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

  21. Michael Darrow July 10, 2019 at 7:14 pm

    A triangular tiling has 6 edges from each vertex.

  22. Adam Hannath July 11, 2019 at 10:16 am

    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.

  23. Lafeo 007 July 13, 2019 at 7:56 am

    india v bangladesh border will be the ultimate test

  24. - July 14, 2019 at 6:28 am

    How can I check if my map is true or not? I made one and every single combination I have tried it needs 5.

  25. commentor silensor July 14, 2019 at 9:12 pm

    I took graphic theory in 1989. The math professor told me the same thing. He mentioned unlike other proof, this was done by computer. After 6 months, these two people couldn't find a map that used more than 4 colors. Since no one had time n computing was so expensive, people couldn't check work, so people just believe. The professor got his PhD from Harvard.. he was very knowledgeable. He couldn't do the proof. He just had to believe. We were all laughing in class. The proof happened in 1970s. Its 201miles. Any smart phone computing power is better than the most expensive one in 70s. The super computer in this day can do much more. people still believe without finding real one. Er just believe. This is one theory in my life that we don't a math paper to prove.

    Ok I am not a mathematician. After getting my computer science degree, I don't care about proof anyone

  26. KliqxTV July 14, 2019 at 10:08 pm

    Draw two circles, one bigger than the other, divide that ring into 4 sections + the one on the inside. Inside is connected to all sides, theorem broken?

  27. Tomás Frazer July 15, 2019 at 1:43 pm

    I wonder what happens when you introduce the concept of an Alaska, ie where a country consists in more than one surface

  28. MarbleSwan666 July 15, 2019 at 3:42 pm

    "1970's, final solution."

  29. Philipp Doe July 16, 2019 at 9:23 pm

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

  30. Giorgos Tridimas July 16, 2019 at 10:41 pm

    Colourblind people can't see the point for this theorem.

  31. Ayaan Naha July 22, 2019 at 2:56 am

    my globe has 6 different colors

  32. Stella Cat July 26, 2019 at 12:20 pm

    If you can invent maps, you could do five countries that meet at one point they're all bordering and there are more than 4. I don't know why this is so hard for mathematicians.

  33. Jonathon Whitby August 6, 2019 at 10:32 pm

    What if all the countries were hexagons? Would that not be a case where each country has 6 countries connected to it in a network?

  34. TituroFox August 8, 2019 at 3:06 am

    In 1 Dimensoin (a line), you'd logically need TWO colors to color all the segments of a longer straight line. If in a 2 Dimensoins plane you need FOUR colors to separate regions like in this video, would that mean you'd need EIGHT colors to separate a 3 Dimensionnal object in smaller pieces and not have any pieces of the same color touching each other ? Hmmm… (Hope i explained it well enough…)

  35. Niels van Crugten August 8, 2019 at 11:17 pm

    Wouldnt it be a problem if a country would wrap around itself on a map. looking at the world it is a sphere, so if you would use a map one country will be cut msot of the time. If you would cut it just right it wouldn't be possible.

  36. Xuraiis August 10, 2019 at 10:16 pm

    Graphs, not networks T-T

  37. Michael Roach August 13, 2019 at 3:27 pm

    So I'm looking at this and I'm thinking that in order for a map to require c number of colors, there would have to be a network with n number of nodes such that n=c.

    If some larger map satisfied the requirement where n > c, then there would be nodes that duplicate colors. In such a case there would be a duplicate that could be removed that would reduce that map to one where n=c.

    In order to require c colors, no node can be isolated. That is, every node must be connected to every other node in the map where c=n. If two were not connected, they could be the same color, reducing the number of colors necessary.

    If n=c=3, the network can be represented as a triangle. in order to make a network where n=c=4, a node must be added to the n=c=3 network such that it connects to all three.

    If node 4 is inside the triangle …
    no node 5 added outside the triangle can connect to it.
    a node 5 added inside the triangle that can connect to it must be in one of the sub-triangles formed when node 4 was added
    a node 5 inside one of the sub-triangles is isolated and cannot connect to points outside that sub-triangle, meaning it
    can only connect to 3 out of the 4 nodes and therefore there would not be able to connect to the fourth node and
    force it to be a different color.

    If node 4 is outside the triangle …
    you can connect it to the first 2 nodes without isolating any node
    the third connection between node 4 and the third node in the triangle to be connected must either go left or right around
    the triangle.
    the third connection isolates either node 1 or node 2, leading to essentially the same structure as you would have if node
    4 were inside the triangle (just with a different node isolated)

    If a fifth node can't be connected to all nodes in a 4 node network then no larger network where n=c can exist either. Because if such a network could exist such that all nodes are connected to all other nodes, then removing one node would result in network one smaller where all nodes are connected to each other. Any larger network could remove nodes until it had only 5 nodes where every node were connected. Because the 5 node network where n=c cannot exist, all larger networks can be dismissed.

  38. Aries Lasagna Boy / August 14, 2019 at 2:05 pm


  39. Jörg Asmussen August 15, 2019 at 9:06 pm

    What if countries could be 3D. Would 9 colours suffice?

  40. Ethan Harr August 16, 2019 at 12:13 am

    does it count if the two regions meet at a point, and not a line, for example the four corners in the US, could those be done with only two colors

  41. Cerugona August 16, 2019 at 11:02 am

    Curious to see how many you would need in 3 dimensions

  42. light August 16, 2019 at 1:56 pm

    I love this mans enthusiasm

  43. Amazing Leomar August 17, 2019 at 12:03 am

    But if you extend an outer region of the map at 3:26 around to the other side and touch that same color you now have a region that touches pink, yellow, blue, and purple already, so you would need atleast 5 colors. What am I missing here?

  44. starcrafter13terran August 19, 2019 at 8:43 pm

    wait, if the map is covered with countries all shaped by hexagons, then can you do it with less than 7 colors?

  45. Thomas Africa August 26, 2019 at 7:52 pm

    Anyone else see the person standing in the window?

  46. Snootiest Turtle August 27, 2019 at 4:15 am

    I created a map that needs 5 colors

  47. Wiebejamin August 27, 2019 at 9:43 am

    What if some terrorists are discontinuous? Like how Michigan is in half, or how Alaska and Hawaii are USA but aren't connected? One of those, but with more connections? Could they require 5 colors then?

  48. SomeDawidGuy August 27, 2019 at 12:13 pm

    The key to breaking this:
    Enclaves and exclaves

  49. Science 2020 August 28, 2019 at 1:22 pm

    Wait until he finds out about enclaves.

  50. Alex Thielke August 29, 2019 at 7:53 am

    It feels weird but i think i have a 5 sections map wich cannot be coulord with only 4 you need 5 differend types for sure

  51. JustJad3n September 2, 2019 at 5:03 am

    What if I have this
    The a curved territory that curves between the start of blue and end point of purple, maybe I’m breaking a rule or something, could someone explain?

  52. Daniel September 4, 2019 at 7:20 am

    hey what if Oklahoma's panhandle was a seperate state

  53. Kestalami September 4, 2019 at 7:56 am

    i just made a map that needs five colors

  54. Michael Scott September 7, 2019 at 9:15 am

    A world map could need 5 colours if you colour the seas as well.

  55. evin tyler September 9, 2019 at 9:19 am

    Love the videos, but I imagine split territories throws a wrench in that first statement. You could use different colors for each part, but that would be more confusing I'd imagine.

    It is usual for territories to be contained to one shape, but a map of Michigans could require infinite number of colors.

  56. Busty Bonnie September 10, 2019 at 7:28 am

    What rivers though? all the geographic regions but then there's the rivers & oceans and lakes

  57. Nate Connell September 12, 2019 at 1:10 am

    This is only true if you don't count countries that have disconnected parts.

  58. Lukedub/Fubudub Productions September 14, 2019 at 5:04 am

    You look like projared, bill nye, and Reportoftheweek fused together and you were the result

  59. MadTamB September 15, 2019 at 4:22 pm

    Real maps might not because of enclaves and exclaves.

  60. Jakob butzbach September 15, 2019 at 10:03 pm

    Here's a proof. try to get all your fingers on one hand to touch each other… you can with 4… not 5… Done

  61. Jakob butzbach September 15, 2019 at 10:04 pm

    Here's some logic. try to get all your fingers on one hand to touch each other on a 2-dimentional plane…. you can with 4… not 5… beats the 1400figures QED

    sometimes you have to think inverted, to solve.

  62. James KNIGHT September 17, 2019 at 9:23 am

    This falls apart when countries have exclaves and enclaves

  63. Harry Heller September 21, 2019 at 3:34 am

    What if you do the world, but count the oceans as one tile

  64. Alexander Kelly October 2, 2019 at 3:08 pm

    France is the same color as germany

  65. Tony Hakston October 5, 2019 at 9:03 pm

    Does this work for non-contiguous territories?

  66. RamsesTheFourth October 7, 2019 at 3:40 pm

    What if a country does not have to be uniform? Like Alaska to USA or Kalinigrad for Russia. They are detached regions which you would want to color same way. With this in mind, you can create network that does have crossings in it. And also it makes sense. At least to me. And then it might happen you will need 5 colors.

  67. GLG Locobird00 October 8, 2019 at 2:36 am

    It hurts me that Michigan was colored two separate colors in the beginning

  68. Benjamin Vanderhoff October 8, 2019 at 7:00 am

    What if one country was next to four other countries???

  69. Orange Man October 8, 2019 at 8:00 pm

    Enclaves and exclaves kinda mess this up though, but if we lived in an ideal world then this would work

  70. Coolexcitingname October 10, 2019 at 2:51 am

    You can actually color any map with 3 colors. Just don't color all the territories.

  71. Jakub Braun October 10, 2019 at 3:52 pm

    what about exclaves and enclaves?

  72. Rich Grogan October 11, 2019 at 12:13 pm

    What about enclaves and exclaves

  73. TheJaredtheJaredlong October 12, 2019 at 4:26 am

    Does is hold true for coloring a globe?

  74. Zombles Allegoy October 12, 2019 at 1:05 pm

    What about enclaves and exclaves that need to be the same color as their main country? This assumes that countries are solid blocks, which is not true, like the Russian exclave of Kalingrad, the Dutch-Belgian city of Baarle and other examples.

  75. William October 13, 2019 at 3:18 am

    what about 3d maps? in 1d map its 2 colour, 2d 4, 3d would require more?

  76. Ontario October 14, 2019 at 2:12 am

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

  77. SDestySD October 15, 2019 at 4:36 am

    Is it just me or is a map with 5 easy

  78. Tyler Helm October 15, 2019 at 4:42 am

    Theoretically, if a state were to be under the same name but in separate chunks such as Michigan, would you color them the same color as they are the same state?

  79. shadowatom October 17, 2019 at 4:07 am

    It seems like this has to do with the fact that the maps are drawn on a flat piece of paper. If you draw a map on a torus, you can force the need for a 5th color.

  80. Jay Moreau October 18, 2019 at 2:07 am

    12:30 mathematicians don’t like the real world

  81. Slim_Shells October 20, 2019 at 7:34 pm

    You forgot about country exclaves

  82. Oscar Zabel October 23, 2019 at 7:12 am

    Inclave and exclave

  83. Scott Happy October 23, 2019 at 9:22 pm

    I imagine people scribbling on a piece of paper then trying to color each closed off section with only four

  84. CaTastrophy427 October 24, 2019 at 10:00 am

    1:45 Now take that, and look. There's a fifth color: Brown. Add in the uncolored space as another part of the map needing a color. I count all four colors in contact with that.

  85. Bob Thelob October 26, 2019 at 4:05 pm

    This was a problem in my school
    Thanks 🙂 I’m gonna get a free slice of pizza!

  86. Jordan Smith October 27, 2019 at 12:26 am

    When you have a fictitious world with strange exlaves you must use more than 4 , if you want the exclaves to match the color of the mainland.

  87. Lucas B October 27, 2019 at 9:31 am

    i'm pretty sure the Irish don't consider themselves counties of Britain lol

  88. Dan Doperganger October 29, 2019 at 3:52 pm

    1:49 Arizona and Colorado touch and are same color

  89. Fennec Besixdouze October 30, 2019 at 2:28 am

    The map of the world most certainly has 5 mutually bordering countries, due to enclaves and exclaves. So no, the map of the world is not colorable by four colors such that every bordering COUNTRY gets a different color, as stated in the video. COUNTRIES are not contiguous regions in real life.

  90. nonam nonam October 30, 2019 at 8:19 pm

    Well… ships count as foreign ground… the practical way of solving it…

  91. Black Templar November 2, 2019 at 8:12 pm

    Ever since the computer solved this problem the computer has been trying to "SOLVE" the colour problems with our real maps…..one day there will be zero difference between any of us because the "problem" of humanity will of been solved by computers……

  92. Jesse Kipp November 8, 2019 at 3:18 am

    So if you take a map like he has in 3:56 and put a country around it with a section of that country nested in side the inside circle you need five colors don't you? It be like how Russia or the US isn't all self contained but it's colored the same because it's the same place politically

  93. Mark Hoover November 10, 2019 at 3:16 am

    Tyro question: does not this subject have correlations in topology?

  94. cody the programmer November 10, 2019 at 8:40 am

    Actually I my maths teacher told us to sit for 3 hours in class trying to design a map that needs 5 colors. I did manage to do one, with 12 over regions. However, one person in my class did it with only 5 regions…

  95. Zakimals November 10, 2019 at 9:41 pm

    exclaves create a problem

  96. Durv2 November 11, 2019 at 2:32 pm

    is it just me or is it really easy to draw a map that needs 5 colours. If you had 4 enclaved countrys (a country within another country) all touching that would need 5 colours right?

  97. akshat mehta November 11, 2019 at 6:42 pm

    Awesome explanation.

  98. Ben B November 11, 2019 at 10:08 pm

    A map is only a 2D surface. How many colors would we need if we brought this into the Third Dimension and why?

  99. sadas November 12, 2019 at 7:07 pm

    If a country can be split up like the USA and still require the same colour then you aboslutely need 5 colours.

  100. Ralph Dratman November 14, 2019 at 3:13 am

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

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