What does "parity pattern" mean?

Question

Five lattice points are chosen in the plane lattice. Prove that you can always choose two of these points such that the segment joining these points pass through another lattice point. (The lattice consists of all points of the plane with integral coordinates).

The solution reads as follows;

Let us consider the parity pattern of the coordinates of these lattice points. There are only four possible patterns: $(e,e) , (e,o) , (o,e) , (o,o)$. Among the five lattice points, there will be two points, say $a= (a, b)$ and $B = (c,d)$ with the same parity pattern. Consider the midpoint $L$ of $AB$. $L= ( \frac{a+c}{2} , \frac{b+d}{2} )$. $a$ and $c$ as well as $b$ and $d$ have the same parity, and so $L$ is a lattice point.

I do not fully understand the solution provided. How did the midpoint idea come from? How does it work? Would appreciate any insight provided by anyone to this solution.

Answer 1

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If you pick any five points, two of them must be the same letter. The midpoint of any two same letters is another letter.

Answer 2

It means whether the coordinates of the point are "both even", "first even second odd", "first odd and second even" or "both odd". I think the "zeros" are actually o's? The midpoint idea, well I don't know how to come up with it, but it works, since then the sum of coordinates will always be even and you can divide by 2 to get another lattice point.