## Solution to: Intriguing Intersections

The first step is that we number the grid lines like in a mathematical grid, to define the coordinates of the five selected points. For the example given in the question, this can be done as shown in the figure below.

Every selected grid point can now be indicated by its coordinates, using its horizontal and vertical line number. For the example given in the question, the coordinates are the following:

 Point: Coordinates: 1 (2, 4) 2 (4, 5) 3 (3, 1) 4 (7, 2) 5 (8, 3)

The center point between two points can be calculated by taking the average of the coordinates. For example, the center point between points 1 (2, 4) and 2 (4, 5) has coordinates (3, 4½), which is not a grid point, and the center point between points 2 (4, 5) and 5 (8, 3) has coordinates (6, 4), which is a grid point.

The center point between two grid point is again a grid point, if both the difference between the first coordinates and the difference between the second coordinates is even.

Therefore, we place the coordinates of each selected point in one of the following four categories:

• (even, even)
• (even, odd)
• (odd, even)
• (odd, odd)

The center point between two grid points is again a grid point, if the coordinates of both points are in the same category. For the example given in the question, the coordinates of the points are in the following categories:

 Point: Coordinates: Category: 1 (2, 4) (even, even) 2 (4, 5) (even, odd) 3 (3, 1) (odd, odd) 4 (7, 2) (odd, even) 5 (8, 3) (even, odd)

As you can see, points 2 and 5 are in the same category (even, odd). The center point between these two grid points is (6, 4), which is a grid point.

Since there are five points and only four categories, we can apply the pigeonhole principle. The pigeonhole principle states that if you place pigeons in pigeonholes, and you have more pigeons than pigeonholes, there must be at least one pigeonhole that contains two or more pigeons.

In our case, there must be at least one category that contains two or more points. This means that whichever five grid points you select on a square grid, there are always at least two of those points of which the center point is also a grid point.

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