color each point in the coordinate plane either red or blue. prove that no matter how the coloring is done there must exist two points of distance one that are the same color.
To prove that no matter how the coloring is done, there must exist two points of distance one that are the same color on the coordinate plane, we can use a mathematical technique called the Pigeonhole Principle.
The Pigeonhole Principle states that if you have more pigeons than available pigeonholes, at least one pigeonhole must contain two or more pigeons. In this case, the pigeons represent the points in the coordinate plane, the pigeonholes represent the two colors (red and blue), and the distance between points represents the "distance" between pigeonholes.
To apply the Pigeonhole Principle to this problem, let's consider a square on the coordinate plane with side length 2. The four corners of this square represent four points. Since we have two colors (red and blue), there are only two pigeonholes.
No matter how we color these four points, at least two of them must have the same color. This is because we are trying to fit four points into two pigeonholes, which means at least one pigeonhole must contain two or more points.
Now, if two points have the same color and their distance is exactly one, we have proven that such points exist no matter how we color the rest of the coordinate plane. This is because if two points have a distance of one, they are either horizontally or vertically adjacent. Therefore, it is guaranteed that there must exist two points of distance one that are the same color.
Thus, using the Pigeonhole Principle, we can conclude that no matter how we color the points in the coordinate plane with two colors, there must always be two points of distance one that have the same color.