A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of different colours, they are then joined by a green line segment. In total, there are 52 red dots. There are 2 red dots at corners with an additional 16 red dots on the edges of the array. The remainder of the red dots are inside the array. There are 98 green line segments. The number of blue line segments is
(A) 36
(D) 39
(B) 37
(E) 40
(C) 38

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-136.jpg?height=424&width=412&top_left_y=1200&top_left_x=1328)
A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of different colours, they are then joined by a green line segment. In total, there are 52 red dots. There are 2 red dots at corners with an additional 16 red dots on the edges of the array. The remainder of the red dots are inside the array. There are 98 green line segments. The number of blue line segments is
(A) 36
(B) 37
(C) 38
(D) 39
(E) 40

## Solution

First, we note that there are 9 line segments in each row and in each column, so there are $9(10)+9(10)=180$ line segments in total.

Let $B$ be the number of blue segments and $R$ the number of red segments. Then $B+R+98=180$, so $B+R=82$, as there are 98 green line segments.

Coming out of a red dot, there can only be a green line segment or a red line segment. We count the total number of line segments starting from red dots. Note that in this total, the green segments are counted once and the red segments twice, as the red segments have both ends at red dots.

From a corner dot, there are 2 segments

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-271.jpg?height=122&width=141&top_left_y=2275&top_left_x=1247)

From an edge dot (not on corner), there are 3 segments

From an interior dot, there are 4 segments

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-272.jpg?height=398&width=233&top_left_y=262&top_left_x=1217)

So the total number of segments coming from red dots is

$$
2(2)+3(16)+4(34)=188
$$

and so, since 98 segments from red dots are accounted for by green segments the remaining $188-98=90$ segments from red dots are accounted for by red segments, for a total of 45 red segments, or $R=45$.

Therefore $B=82-R=37$.

ANSWER: (B)