Module No#08

Today, we are going to explore midpoints between pairs of points with integer coordinates in a 2D plane. Can anyone tell me how to calculate the midpoint between two points?

Is it by averaging the x-coordinates and the y-coordinates?

Exactly! The formula for the midpoint, M, of points (x₁, y₁) and (x₂, y₂) is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Now, what happens if x₁ and x₂ have the same parity?

Then the midpoint will also be an integer!

Right! This leads us to the pigeonhole principle in our next topic. Let’s see how we can apply this to prove a statement.

When we consider five distinct points and their coordinates, how can we relate them to our principle?

We could group them based on their parities into categories, right? For example, odd/even for x and y!

Precisely! This gives us four combinations: (odd, odd), (odd, even), (even, odd), and (even, even). Since we have five points, how does this relate to the pigeonhole principle?

There must be at least two points that fall into the same category!

Exactly! This shows that the midpoint will have integer coordinates as both coordinates share parities. Now, let’s explore a similar idea with a different example.

Let's look at the second problem where we need to select five integers from 1 to 8. Why do we care about sums equaling nine?

Because there are various pairs like (1, 8), (2, 7), (3, 6), and (4, 5) that add up to that!

Correct! Now, can you identify how we can apply the pigeonhole principle here?

By choosing five integers, we will hit one of those pairs, because there are only four pairs that sum to nine!

Exactly! Therefore, irrespective of how we choose our integers, at least one pair will always sum to nine. Let’s summarize what we have learned.

In summary, when faced with five integers chosen from 1 to 8, there will always be a pair of integers that sum to nine. The pigeonhole principle effectively confirms this without extensive enumeration.