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

In summary, the question is asking for the maximum number of points that can be placed in an equilateral triangle with side length 2, where no two points are within 1 unit of each other. By using the pigeon hole principle, it can be shown that there are at least 3 points that can be placed. Visually, it appears that there can be 4 points, but it needs to be proven. The solution involves dividing the triangle into 4 pieces, each with a maximum distance of 1 between points, resulting in a maximum of 4 points. A configuration of 4 points can then be provided to prove that 4 is indeed possible. The suggested solution of adding root 3 over 2 and root
  • #1
auk411
57
0

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.
 
Physics news on Phys.org
  • #2
auk411 said:
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
 
  • #3
willem2 said:
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.
 
  • #4
auk411 said:
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.
 

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

1. What is the Pigeon Hole Principle?

The Pigeon Hole Principle is a mathematical concept that states that if there are more pigeons than pigeon holes, at least one pigeon hole must contain more than one pigeon.

2. How is the Pigeon Hole Principle used to solve problems?

The Pigeon Hole Principle is used to prove the existence of a solution to a problem. It works by considering the number of "pigeons" (items or elements) and the number of "pigeon holes" (categories or options), and showing that there must be at least one category with more than one item.

3. What is the "Equilateral Triangle Points" problem?

The "Equilateral Triangle Points" problem involves finding the minimum number of points needed to form an equilateral triangle when each point is connected to every other point. It is a classical example of applying the Pigeon Hole Principle.

4. How can the Pigeon Hole Principle be applied to solve the "Equilateral Triangle Points" problem?

The Pigeon Hole Principle can be applied by considering the vertices of the equilateral triangle as "pigeon holes" and the points as "pigeons". By showing that there must be at least one vertex with more than one point, we can prove that the minimum number of points needed is three.

5. Are there any other applications of the Pigeon Hole Principle?

Yes, the Pigeon Hole Principle has many other applications in various fields such as computer science, cryptography, and graph theory. It can also be used to solve problems in real-life situations, such as scheduling conflicts or distributing resources among a group of people.

Similar threads

Back
Top