A grid of points with 10 rows and 10 columns is given. Each point is colored red or blue. Whenever two points of the same color are adjacent in the same row or column, they are connected by a segment of the same color as the points. If two points are adjacent but of different colors, they are connected by a green segment. In total, there are 52 red points. Of these red points, 2 are in the corners and another 16 are on the edge of the grid. The other red points are inside the grid.

![](https://cdn.mathpix.com/cropped/2024_05_01_98fe4f82538e82dc6d3ag-5.jpg?height=388&width=394&top_left_y=1311&top_left_x=657)

There are 98 green segments. Determine the number of blue segments.
Initially, observe that there are 9 segments in each row and in each column, so there are $9 \times 10+9 \times 10=180$ segments in total.

Let $A$ be the number of blue segments and $V$ be the number of red segments. Then $A+V+98=180$, so $A+V=82$, since there are 98 green segments.

Notice that from red points, only red or green segments can depart. Let's count the total number of segments that depart from the red points. In this total, the green segments are counted exactly once and the red segments are counted twice, as the red segments connect two red points.

Departing from a corner, there are 2 segments: $\square$.

From a point on the edge, 3 segments depart.

From an interior point, 4 segments depart.
Suggestion: Count the total number of segments determined by the vertices and subtract those that are not spatial diagonals.

Helpful Facts: The number of ways to choose two objects from $n$ distinct objects is $\binom{n}{2}=$ $\frac{n(n-1)}{118}$. See the table on the page.

Suggestion: Count the total number of segments and the total number of segments that depart from red points.

Helpful Facts: No blue segments depart from red points.

Thus, the total number of segments that depart from the red vertices is

$$
2 \times 2+3 \times 16+4 \times 34=188
$$

but since 98 segments that depart from the red points are the green segments, the remaining $188-98=90$ are red and were counted twice, so $V=45$.

Therefore, $A=82-\mathrm{V}=37$.