I remember when I was younger, I tried hard to disprove the four color theorem. I remember seeing cases where it didn't work on a donut or a mobius strip. But apparently it works on a sphere.

Now, this confused me, because I was able to create a sphere with 5 different countries, all of which touched each other. Therefore, we would need 5 colors.

Here each of (red, grey, orange, blue, green, brown) seems to touch each other, with orange and blue wrapping vertically and brown and grey wrapping horizontally.

What bad assumptions am I making about the four color theorem or its constraints?enter image description here

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    $\begingroup$ If you can do it on a sphere, you should be able to do it on a plane, by removing a point on the sphere. $\endgroup$ – Thomas Andrews Jun 5 '19 at 19:39
  • $\begingroup$ @ThomasAndrews oh. I think I get it now. I'd read this argument before, but I didn't 100% understand it. I realize what it means now. I could put the sphere on a table, cutting off a part that didn't contain a country. Then I could stretch or shrink parts of the sphere as needed, to convert it into a similar flat map where the borders are preserved. $\endgroup$ – aschultz Jun 5 '19 at 19:48
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    $\begingroup$ You can even cut out part of a country, as long as you cut out an interior part of exactly one country (so it still remains connected.) @aschultz $\endgroup$ – Thomas Andrews Jun 5 '19 at 19:51
  • $\begingroup$ @ThomasAndrews on reflection, I suspect when I thought this up a long time ago I eventually saw the countries couldn't all touch each other, but the example would work on a torus. Then with the passage of time I blended together "oh, 4 color theorem doesn't work in 3d spaces" + "sphere is a 3d space." Oops. Thanks to you and SmileyCraft for shining a light on things. $\endgroup$ – aschultz Jun 5 '19 at 20:59

You have colors wrapping horizontally and colors wrapping vertically, then they have to pass through each other if you had an actual sphere.

EDIT: The four color theorem on a sphere is actually a direct consequence of the four color theorem on the plane. For any map on the sphere, you can poke a hole inside some region and then stretch the surface into a plane.

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    $\begingroup$ Yep, you are hiding the error in the back of the sphere. $\endgroup$ – Thomas Andrews Jun 5 '19 at 19:40

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