17.1. Discrete Mathematics

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.

This chunk introduces the problem where we have 5 arbitrary distinct points in a 2D plane. Each point has integer coordinates (x, y) which means both x and y can be any integer but must be different for each point. This sets the stage for further exploration into whether we can find a pair of points whose midpoint also has integer coordinates.

Imagine placing 5 distinct pins on a map (the 2D plane) at different integer junctions. Each pin represents a point, and we want to discover if the average position (the midpoint) between any two pins will still land us at an integer junction.

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

The midpoint of two points in a 2D space is calculated by taking the average of their x-coordinates and the average of their y-coordinates. This means that if both coordinates of the points are integers, the result could either be an integer or a half-integer (like 1.5), depending on whether the coordinates are even or odd.

Think of two friends who are standing at different street corners represented by their coordinates. If they meet halfway, where will they end up? If both are on even-numbered street corners, they'll meet at another corner. If one is on an even and the other on an odd corner, they'll end up somewhere that is not a corner, like the middle of the street.

We want to apply the pigeonhole principle. So remember for the pigeonhole principle we have to identify the set of pigeons and the set of holes here and then the mapping which relates the pigeon and the holes.

The pigeonhole principle states that if you have more items (pigeons) than containers (holes), at least one container must hold more than one item. In this case, we are going to map each of the 5 points to a combination of their x and y coordinates being odd or even. There are 4 combinations (odd, odd), (odd, even), (even, odd), and (even, even).

Imagine you have 5 different colored socks (the points) to put into 4 different drawers (the combinations of odd/even coordinates). Because there are more socks than drawers, at least one drawer will end up containing more than one sock.

Without loss of generality, assume that out of those 2 points, say (x₁, y₁) and (x₂, y₂), which are guaranteed to be mapped to the same ordered pair are the first 2 points. Now we want to inspect what happens to the midpoint of the line joining these 2 points.

With our assumption (that out of the 5 points there are at least 2 points that map to the same odd/even combination), we check what this means for their midpoint. If both x-coordinates are odd or both are even, their midpoint (the sum divided by 2) must result in an integer. This shows our original claim that at least one midpoint will also have integer coordinates.

If you think of two cars starting at two corners of a street, if they both start from either even-numbered or odd-numbered intersections, they will meet at another intersection. However, if one starts on an even and the other on an odd, they will not land on another intersection but somewhere between.

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