17.4. Tutorial 6: Part II

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. So, I am denoting the points as (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), and (x_5, y_5). 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 the midpoint of the line joining those 2 points has integer coordinates.

In this chunk, we start with a scenario of choosing distinct points in a 2D plane, each with integer coordinates. We focus on the notion of midpoints, defined by the average of the x-coordinates and y-coordinates of two points. For example, the midpoint formula of two points (x_1, y_1) and (x_2, y_2) yields: $\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}$. The challenge is to demonstrate that at least one pair of chosen points will result in a midpoint whose coordinates are also integers.

Imagine you have five distinct players located on a basketball court, each standing at different integer coordinate positions. If you pair any two players, the location where they meet at the halfway point will always land on the basketball court's grid. It's a simple property of how the average of whole numbers behaves, ensuring the result is also a whole number.

To apply the pigeonhole principle, we need to identify the set of pigeons and the set of holes. We consider the set of 5 arbitrary points and map them based on the nature of their x-coordinates and y-coordinates. Depending on whether the x-coordinates and y-coordinates are odd or even, this gives us 4 possible combinations. For example, if (x_1, y_1) is odd and (y_1, y_2) is even, we denote this as (odd, even). We now have 5 items (the points) mapped to 4 combinations.

The pigeonhole principle is a counting argument that states if you have more items than containers, at least one container must contain at least two items. Here, we categorize our points by examining whether each coordinate is odd or even. Since we have 5 points (the 'pigeons') but only 4 combinations ('holes'), at least two points must belong to the same category (say, both x-coordinates being odd). This implies their midpoints will yield integer results.

Think of this as gathering five friends with 4 different colored shirts. Regardless of their choices, at least two friends will inevitably be wearing the same colored shirt. Similarly, by mapping the coordinates, we assure ourselves that at least two points will fall into the same odd/even category.

Assuming (x_1, y_1) and (x_2, y_2) are the two points mapped to the same ordered pair, we analyze their midpoint. The formula gives us $\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}$. Since x_1 and x_2 (or y_1 and y_2) are either both odd or both even, their sums will be even, thus guaranteeing integer midpoints.

Since both coordinates x_1 and x_2 are either odd or even, when added together, they produce an even sum. When you divide an even number by 2, the result is always an integer. This is crucial in showing that the midpoints of the selected points will have integer coordinates, thus validating our goal.

Imagine you are merging two even groups of candies together. If you have two bags, each containing an even number of candies, their combined total will also be even, making it easy to evenly distribute the candies between two containers. By using this analogy, we realize how the properties of integers ensure that the outcomes we desire—in this case, integer midpoints—are consistently achievable.

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