17.5.2. Application of Pigeonhole Principle

Let us start with question number 8. You are given here arbitrary distinct points in 2 dimensional planes. Each point will have an x-coordinate, y-coordinate and the points are having integer coordinates. So they are arbitrary points except that they are distinct. Our goal is to show that irrespective of the way these 5 points are chosen arbitrarily, there always exists a pair of points such that if you consider the midpoint of the line joining those 2 points, it has integer coordinates.

In this segment, we introduce the problem of finding integer-coordinate midpoints for arbitrary distinct points in a 2D plane. The principle being discussed is based on the Pigeonhole Principle, which in this case helps establish that among five distinct points, at least two must share characteristics that ensure their midpoint is an integer. Specifically, we focus on the x and y coordinates and categorize them into even and odd. The key takeaway here is that no matter how you select those points, a pair will inevitably generate an integer midpoint.

Think of it this way: if you have five people, each wearing either a hat or no hat (let's simplify them to two categories), then at least two people must be wearing the same type of headgear. Similarly, in our problem, when mapping points with even or odd coordinates, two points will share the same 'hat' type, ensuring that their average (midpoint) results in an integer.

We are trying to map these points depending upon the nature of their x-coordinate and y-coordinate. Depending on whether the x-coordinate is even or odd, or whether the y-coordinate is odd or even, we have 4 possible combinations. So, consider the set of 5 arbitrary points which are all distinct.

In this chunk, the focus shifts to how we categorize points based on the parity of their coordinates (even or odd). We recognize there are four distinct combinations for a point's coordinates based on whether they are even or odd. By mapping these points accordingly, we create a relationship between the 5 points and the 4 combinations. The Pigeonhole Principle applies directly here, as we have more items (5 points) than categories (4 combinations), guaranteeing that at least one category must contain multiple points.

Imagine sorting fruits into baskets by color: red, green, yellow, and blue. If you have five fruits, it's unavoidable that at least two will end up in the same basket. In our case, the colors correspond to even and odd combinations of coordinates, ensuring that a repetition occurs.

It follows from the pigeonhole principle that we now have 5 items in the set and 4 items in the set, which means there always exists a pair of points among these 5 points such that both of them are mapped to the same ordered pair. Without loss of generality, assume that among those 2 points which are guaranteed to be mapped to the same ordered pair are the first 2 points.

Here, we conclude that given 5 points and 4 categories for their coordinate combinations, at least two of the points must share a combination (e.g., both odd x and even y). This leads us to examine their midpoint. We denote the first two points and analyze what happens to their midpoint, which is formulated mathematically. Given the shared coordinate characteristics of these two points, we can assure that their average will yield integer results, confirming the original proposition.

If you have two teammates both wearing the same color jersey (like both being in red), when they stand together, they form a group that is easy to identify. Similarly, our two points, sharing coordinate traits, will always produce a midpoint that is predictable and integer-based.

In the end, we established that irrespective of the distinct points chosen, there will always be a pair whose midpoints yield integer coordinates, showcasing the power of the Pigeonhole Principle.

This section wraps up the entire discussion by reiterating the significance of the Pigeonhole Principle in solving the problem. The conclusion is a powerful affirmation that, in any scenario involving five distinct points with integer coordinates, we can guarantee finding pairs whose midpoints conform to integer values, thereby conclusively resolving the posed question.

Think of wrapping gifts; if you have five gifts but only four wrapping paper styles, at least two gifts will end up wrapped in the same style. In our mathematical context, this ensures that pairs of points reliably yield midpoints in integers, further demonstrating the principle's reliability in real-life applications.

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