Proving the Existence of a Color-Matching Rectangle

In summary, the pigeon hole principle states that if there are more pigeons than pigeon holes, then there must be at least one pigeon hole with more than one pigeon. In the case of the given problem, the pigeon holes represent the number of colors (blue and red) and the pigeons represent the number of vertices in a rectangle. Therefore, there must exist a rectangle with all vertices of the same color.
  • #1
DaveElliott
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Homework Statement


A plane is colored blue and red in any way . Prove that there exists a rectangle with vertices of the same color


Homework Equations



Its obviously true.

The Attempt at a Solution



I was thinking proofy by pig hole, but haven't quite figured it out.
 
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  • #2
It is called the pigeon hole principle: if there are more pigeons than pigeon holes, then there is at least one pigeon hole containing more than one pigeon.

In the infinite sense (useful for this problem): if there are infinitely many pigeons and finitely many pigeon holes, then there is at least one pigeon hole containing infinitely many pigeons.

In this problem, what are the pigeon holes?
 

FAQ: Proving the Existence of a Color-Matching Rectangle

What is a color-matching rectangle?

A color-matching rectangle is a rectangle in which all four sides have the same color. This means that the opposing sides are the same color, as well as the adjacent sides.

How can you prove the existence of a color-matching rectangle?

One way to prove the existence of a color-matching rectangle is by using mathematical proofs and equations. By using the properties of angles and sides in a rectangle, it can be shown that a color-matching rectangle is possible.

Is it possible to have a color-matching rectangle in real life?

Yes, it is possible to have a color-matching rectangle in real life. It may be difficult to find or create, but it is mathematically possible.

Are there any real-world applications for color-matching rectangles?

Color-matching rectangles can be seen in art and design, where the use of symmetry and color can create visually appealing compositions. They can also be seen in architecture, where buildings may have color-matching rectangles as part of their design.

Can the existence of a color-matching rectangle be disproven?

No, the existence of a color-matching rectangle cannot be disproven. If a rectangle has four sides, it is mathematically possible for all four sides to have the same color. However, it may be difficult to physically create or find a color-matching rectangle in the real world.

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