Using the Pigeon Hole Principle to Solve an Equilateral Triangle Points Problem

[auk411]

Homework Statement

How many points can be placed in an equilateral triangle where each side is of length 2 such that no 2 points are within 1 of each other?

Homework Equations

Need to use pigeon hole principle.

The Attempt at a Solution

I know that there are at least 3. Visually, if you sketch the triangle it looks like there will be a 4. However, I don't know how to prove this.

 

[willem2]

I know that there are at least 3. Visually, if you sketch the triangle it looks like there will be a 4. However, I don't know how to prove this.

divide the triangle in 4 area's that can only contain 1 point each

 

[auk411]

divide the triangle in 4 area's that can only contain 1 point each

Yes. I'm asking how does one show this. I don't know how to show this. Someone who claims to have the right answer told me that we need to add root 3 over 2 plus root 3 over 6 to get about 1.3.

I am completely lost as to how this is an answer.

 

[willem2]

Yes. I'm asking how does one show this. I don't know how to show this. Someone who claims to have the right answer told me that we need to add root 3 over 2 plus root 3 over 6 to get about 1.3.

I am completely lost as to how this is an answer.

Apparently they computed the distance from a corner to a center of the triangle, but this indeed not the answer.

If you can divide the triangle in 4 pieces, such that the maximum distance between 2 points in a single piece is 1, then there can be only a single point in each piece, so the maximum amount of points is 4. You then only have to give a configuration of 4 points to prove that 4 is indeed possible.