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Since a providence may build on a bridge, and since certain bridges may be placed as archways over land: it is an obvious fact that this Earth could be turned into a topological and territoried donut/torroid; therefore, l assume it is one, needing 7 colors to be properly colored-in as a globe. As a flat map however; the underside and underneath portions of the bridge [and their territories] would be invisible, unless we gave them something akin to transparency; so let's do that. We will add a color for every possible permutation of overlap between the seven types of colorful territory. A bridge has: a "top" (where people walk); an "underside" (that people look up at, and can hang from); and an "underneath" (which is a path that passes under, both: the top, and the underside). This means that there are atleast 3 positions that a color can be on, so we will give this model the variable: "K" = 3; and the variable "N" = 7. To get every permutation of every ordered color-overlap which can appear on the bridge, we simple do: 7^3, to get the answer: 343. Then we add: 7 to 343, because there is also the possibility of surrounding flat areas, which gives us the answer 350. Right? So does one need atleast 350 colors to adequately color-in the boundries of a map of any given real places on the surface our Earth? Am I being too generous? Does it need more?

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2 Answers 2

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This is not true. You can color any map on a torus of genus 1 (that is a donut, or a normal cup with one handle) with at most 7 colors. But the higher the genus, the more colors you need. Any bridge you add increases the genus, so there is no a-priori limit on the number of colors needed.

There is a formula giving a lower bound to the minimum number of colors for any map on an orientable surface (like Earth with $n$ bridges), which has been conjectured in 1890 and completely proven in 1968. This is a lower bound, which gives $4$ for a sphere. It doesn't say that the bound is strict, there might be surfaces where you can't attain the bound. Still the bound is increasing and unlimited, the higher the genus (number of bridges) the more colors you need.

This is the formula: $$ \left\lfloor \frac{7+\sqrt{1+48g}}{2} \right\rfloor $$

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  • $\begingroup$ I was reading the answers but now they all say "[math processing error]" is it doing that on your end too? $\endgroup$
    – user179283
    Jan 2, 2018 at 14:46
  • $\begingroup$ You seem to know your stuff. My gut says to click your answer $\endgroup$
    – user179283
    Jan 2, 2018 at 14:47
  • $\begingroup$ I'm bummed that I'm seeing this: " [Math Processing Error] " instead of your explanation though, I was in the middle of reading; and then everything flickered [flickered and then changed to something glitchy/unreadable] $\endgroup$
    – user179283
    Jan 2, 2018 at 14:49
  • $\begingroup$ The stuff is back, I see it; it's neat, and your answer is logical. I think my flaw was in imagining a world with only one bridge. Or perhaps I meant to describe a kingdom with only one... either way: I think I'm getting it. I am curious if I am right, for a kingdom with only one archway landmass, with subdivided states though; I'm not sure if I wanna go through the trouble of askimg a whole other question though... Am I? Or no? $\endgroup$
    – user179283
    Jan 2, 2018 at 15:07
  • $\begingroup$ If you have only one bridge, 7 colors are always enough. $\endgroup$
    – rewritten
    Jan 3, 2018 at 18:45
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I don't understand where the number $7$ is coming from. If you allow the bridges, then we are not colouring planar graphs anymore: we are colouring arbitrary graphs and, for that, no finite number of colours would suffice.

Imagine you have $n$ countries, and imagine there is a bridge between each two of them. (The bridges may be very long and very high, some bridges may need to go over other bridges.)

Think: how many colours do you need in that case? $n$, I guess, as a minimum, just to colour the territories, plus whatever else you want to add in order to colour the rest.

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  • $\begingroup$ I was reading the answers but now they all say "[math processing error]" is it doing that onyour end too? $\endgroup$
    – user179283
    Jan 2, 2018 at 14:46
  • $\begingroup$ @user179283 It is all fine on my end... $\endgroup$
    – user491874
    Jan 2, 2018 at 14:48
  • $\begingroup$ This semi-unrelated, but could you or someone you have good faith in, take a look at this " math.stackexchange.com/q/2580652/515443 " question of mine? it's been flying under the radar for awhile now, and it's eat'n me up inside... ...cos the answer relates to a creative project I'm working on, and I want tto maintasin as much accuracy as possible $\endgroup$
    – user179283
    Jan 2, 2018 at 14:54
  • $\begingroup$ Oh, then maybe it's just bad cell service on my end; if so, it should come back eventually (hopefully) $\endgroup$
    – user179283
    Jan 2, 2018 at 14:55
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    $\begingroup$ @user179283 I will have a look but cannot guarantee anything. Have you thought of offering a bounty on your question? $\endgroup$
    – user491874
    Jan 2, 2018 at 14:56

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