## Task A-4.5.

Given are 100 points arranged in 10 rows and 10 columns. Each point is either red or blue. A point is in a corner if it is at the end of both a row and a column, and on the edge if it is at the end of a row or a column, but not both. Two points are adjacent if they are in the same row or column, with no other points between them. Each pair of adjacent points is connected by a line of red, blue, or green color, such that each red line connects two red points, each blue line connects two blue points, and each green line connects one red and one blue point. It is known that there are 52 red points in total, of which 16 are on the edge and two are in the corners. It is also known that there are a total of 98 green lines.

Determine the total number of blue lines.
## Solution.

Let $P$ be the total number of blue, and $C$ be the total number of red segments.

The total number of points on the boundary is 32, so the number of blue points on the boundary is 16. Exactly two blue points are in the corners, so the total number of blue points in the interior is 30.

Let's count all the segments that come out of the blue points in two ways.

From the points in the corners, two segments come out, from the points on the boundary, three segments come out, and from the points in the interior, four segments come out.

![](https://cdn.mathpix.com/cropped/2024_05_30_11ec2cf7b6def7a4ec70g-22.jpg?height=134&width=509&top_left_y=658&top_left_x=702)

Therefore, the total number of segments that come out of the blue points is

$$
2 \cdot 2 + 3 \cdot 16 + 4 \cdot 30 = 172
$$

From a blue point, only blue and green segments can come out. In the total number of all segments that come out of the blue vertices, green segments were counted once, and blue segments were counted twice.

Thus, $172 = 98 + 2P$,

so $P = 37$.

The total number of blue segments is 37.

Note: A similar method can be used to arrive at the solution by counting all the segments that come out of the red points.

The points a student earns for this method of solving cannot be added to the points from the above solution.

If a student shows that the total number of segments is 180, and concludes that $P + C = 82$, they should receive 2 points.

By counting all the segments that come out of the red points, we get

$$
2 \cdot 2 + 3 \cdot 16 + 4 \cdot 34 = 188 = 98 + 2C
$$

i.e., $C = 45$. If justified, this conclusion is worth 8 points.