Who can see who

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Tomolino

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Hello, I am currently stuck on a following problem: Imagine a group of children on a playground, where each child represents a point and they stand in such position, that none of 4 children stands in one line(abscissa). Then if there is no other child standing on abscissa AB, both A and B can see each other, but if there is C in between them, then A can't see B and B can't see A. What is the lowest amount of children, so that any child doesnt see atleast 1 other child? Prove it!

I think the answer is 6- a triangle with one "extended" end on each side by one point. The thing is I have no idea how to prove such thing. I would appreciate any help. Thank you very much!
 
I'm not sure what you mean by "abscissa", which means the x-coordinate of a point.

I think what this is saying is that we want a collection of points such that no four are collinear, but every point is collinear with two others, so that there is a point between it and some other. That is, every point is at one end of some line of three.

Clearly this can't happen with only 3 points, as even if they are collinear, the middle one will "see" both of the others. Now you have to show likewise that it can't happen with 4, or with 5; and then show that your configuration of 6 satisfies the conditions. As I understand it, this is your figure:

Code: 
          F
         /
D---B---A
     \ /
      C
       \
        E
 
Dr.Peterson said:
I'm not sure what you mean by "abscissa", which means the x-coordinate of a point.

I think what this is saying is that we want a collection of points such that no four are collinear, but every point is collinear with two others, so that there is a point between it and some other. That is, every point is at one end of some line of three.

Clearly this can't happen with only 3 points, as even if they are collinear, the middle one will "see" both of the others. Now you have to show likewise that it can't happen with 4, or with 5; and then show that your configuration of 6 satisfies the conditions. As I understand it, this is your figure:

Code: 
          F
         /
D---B---A
     \ /
      C
       \
        E
Click to expand...
Thank you very much for clarifying the solution. I would also like to ask you whether there is a formula for this solution as well, becouse the other part of the problem is to tell how many points are needed for every point to not see atleast 2 (Again I believe the answer is 12-hexagon with extended sides). Also is there a maximum of points that each point cannot see?(I didn't find the solution for 3). I am sorry if this is way too many questions, but I have no idea how to solve this, as these shapes don't have anything in common neither the points in my opinion. Thank you very much for your time Dr.Peterson.
 
I wouldn't expect a "formula"; it is possible that you could prove some fact like this inductively, but I can also imagine that it might not extend beyond 2. I haven't looked into the details, but you have very likely done the same kind of reasoning I would do, in order to come up with your answers.

Can you tell us the entire problem exactly as given to you? What is its source?
 
Let A and B collinear points, then A and B see each other, but if theres a C in between, then A and B can't see each other trough C. There cannot be 4 collinear points.
1.Find the position for the least points, so that not a single point see every point. Prove it
2. Find the position for the least points, so that every point doesnt see atleast 2 other points. Prove it
3. Now decide(and then prove), if this can go on and on. If for every K exists a position for group of points, so that every point doesnt see K other points. OR if there exists a constant C that in any given position for N points(the given rules still aply) SOME of them see atleast N-C other points
 
Thanks for giving the details, so I know your questions are not just your own speculation, but there is reason to consider the general case.

I apologize for just asking questions, but when I asked for the source, my main concern was to know whether this is for a course you are taking, and if so what specific topics you are studying. This could help us see what sort of proof would be required.
 
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