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Math Help - Topology Problems

  1. #1
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    Angry Topology Problems

    I started doing a Topology problem for fun, now I can't even solve it and its driving me CRAZY!!!! A father is dying and he wants to leave his land to his five sons. The sons want to divide the land up so that each plot of land touches every other plot of land. That picture shows what they thought would work, but then they realized it didn't so they tried to make another map. What was the other map? I've tried to solve it many times, but I just can't figure it out! These are some of my failed results so that you don't make the mistake of trying them: PLEASE HELP!
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  2. #2
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    Re: Topology Problems

    Depending on the definition of "touches," such map would contradict the four-color theorem.
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  3. #3
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    Re: Topology Problems

    I mean it must be adjacent. The points can't just touch at one point. The circles can also be changed to practically any closed shape.
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  4. #4
    A Plied Mathematician
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    Re: Topology Problems

    Do the regions have to be path-connected?
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  5. #5
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    Re: Topology Problems

    Not necessarily. As you can see in the first incorrect picture I posted, all the borders are touching each other (besides 3 and 5 that is) so that is all I need and what I'm trying to do.
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  6. #6
    Senior Member Tinyboss's Avatar
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    Re: Topology Problems

    Have you heard of the four color theorem?

    Edit: Oops, somehow I managed to miss that this was already mentioned.
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  7. #7
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    Re: Topology Problems

    If region 1, say, doesn't have to be path-connected, then just do this:

    1 2 5
    4 1 3
    2 5 4

    Here I have imagined boxes around each numbered region.
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