17.3. Lecture No#38

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.

If you have 2 points, a point with coordinates (x1, y1) and another point with coordinates (x2, y2), then the midpoint of the line joining these 2 points is given by the formula (x1 + x2)/2, (y1 + y2)/2.

This chunk introduces the concept of distinct points in a 2D plane with integer coordinates. We want to demonstrate that among any 5 chosen distinct integer coordinate points, there exists at least one pair whose midpoint is also an integer point. The formula for the midpoint is shown, and it shows how we can calculate the average of the coordinates.

Imagine you have 5 unique addresses on a city map that correspond to different houses. If you randomly selected these houses, you can always find at least two houses that, when averaged together as a location, point to a location that also has an exact address – a house or a store, proving that this concept of midpoints applies in real life.

So let us do that. Consider the set of 5 arbitrary points which are all distinct and have integer coordinates. We are trying to map these points based on the nature of their x-coordinates and y-coordinates. Depending upon whether the x-coordinate is even or odd, or whether the y-coordinate is odd or even, I have 4 possible combinations. This function maps these 5 points to the corresponding pair; say if x1 is odd and y1 is even, then I will say that f(x1, y1) is (odd, even) and so on.

In this chunk, we apply the Pigeonhole Principle to our set of points. The principle states that if you have more 'pigeons' (5 points) than 'holes' (4 combinations of even/odd x and y coordinates), some pigeons must share a hole. Essentially, this means at least two of our selected points will have the same combination of even/odd properties. Hence, they will fulfill the condition to ensure that their midpoint is also an integer.

Think of it like having 5 different colored balls (the points) and only 4 available baskets (the combinations of even/odd). No matter how you toss them into the baskets, at least one basket will end up with at least two balls. This illustrates that we must find at least two points that share the same even/odd property, leading to their midpoint being an integer.

Now we want to inspect what happens to the midpoint of the line joining these 2 points, say (x1, y1) and (x2, y2). The midpoint will be ((x1 + x2)/2, (y1 + y2)/2). Since both the points are mapped to the same ordered pair, if both x1 and x2 are odd or both are even, x1 + x2 will always be divisible by 2. Similarly, for y-coordinates, both y1 and y2 need to be either odd or even, leading to y1 + y2 being divisible by 2 as well. Therefore, the midpoint indeed has integer coordinates.

This chunk concludes the theory by confirming that if the x-coordinates are either both odd or both even, their sum will always be divisible by 2, resulting in an integer for the x coordinate of the midpoint. The same applies to y-coordinates. This mathematical principle consolidates our argument that, given any distinct 5 integer-coordinate points, a midpoint with integers must exist.

Imagine measuring distances between two points on a string made of beads representing integers. When you pull the string taut and find the halfway point, regardless of how you measured from those beads, as long as both start points align symmetrically (both odd or both even), you end up exactly on a bead, a whole number on your string.

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