17.2. Module No#08

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.

In this section, we begin by analyzing a problem involving distinct points on a two-dimensional plane, where each point is described by integer coordinates. The problem emphasizes that these points are unique, meaning no two points share the same coordinates. This is a key aspect because it ensures that our approach to finding pairs of midpoints becomes straightforward, as each point can be distinctly analyzed without overlap.

Imagine you have 5 different cities on a map, each represented by a unique location. Each city has a specific latitude and longitude (akin to the x and y coordinates) and we're examining ways to connect these cities and find a midpoint between any two. Since each city is unique, our connections will also yield unique paths, making our analysis clearer.

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.

The objective is to prove that among any given 5 distinct points with integer coordinates, there will always be at least one pair whose midpoint also has integer coordinates. The formula for finding a midpoint between two points (x1, y1) and (x2, y2) is ((x1 + x2) / 2, (y1 + y2) / 2). For the midpoint to have integer coordinates, both x1 + x2 and y1 + y2 must be even, meaning either both coordinates are even or both are odd.

Think of it like finding the average height of pairs of people. If you take two people who are both shorter than average or both taller than average, their average height will also be average. However, if one is short and the other is tall, their average height may not be clear-cut. Hence, to achieve integer coordinates in the midpoint, we must stick to pairs that share the same 'height category'.

So let us do that. Now it follows from pigeonhole principle that we have now 5 items here in the set and 4 items in the set then there always exist a pair of points among these 5 points.

The Pigeonhole Principle helps us conclude that since there are more items (5 points) than categories (4 potential combinations of even/odd coordinates), at least two points must fall into the same category. This means that at least one pair of our selected points will have the same parity (both odd or both even) for both their x and y coordinates. This ensures that the midpoint of this pair will also have integer coordinates.

Imagine you have 5 socks, and only 4 different colors available. If you randomly pick 5, one color must repeat. The same reasoning applies to our points: with odd/even categories for x and y, at least one category of coordinate parity must repeat among the 5 points, ensuring a valid midpoint.

So now we want to inspect what happens to the midpoint of the line joining these 2 points.

Once we know that two points share the same coordinate parities, we apply the midpoint formula. If both points have odd x-values, their sum will be even, ensuring that the midpoint's x-coordinate will be an integer. Similarly, for the y-values, if both are either odd or even, the calculations follow suit and yield an integer for the y-coordinate, confirming our hypotheses.

Imagine you're coordinating dance moves for two dancers who always move in sync. If both dancers make their moves from even numbered steps, their average (midpoint) will also fall into an even number step. This predictable outcome reflects how their actions lead to harmonious results, analogous to our points meeting at an integer midpoint.

--