Problem 10.1. In each cell of a $5 \times 5$ table, a natural number is written in invisible ink. It is known that the sum of all the numbers is 200, and the sum of three numbers located inside any $1 \times 3$ rectangle is 23. What is the central number in the table?

![](https://cdn.mathpix.com/cropped/2024_05_06_9620aefbc3ee8b62d5ddg-39.jpg?height=309&width=311&top_left_y=340&top_left_x=575)
Answer: 16.

Solution. Divide the $5 \times 5$ square without the central cell into four $2 \times 3$ rectangles, and each of these into two $1 \times 3$ rectangles.

![](https://cdn.mathpix.com/cropped/2024_05_06_9620aefbc3ee8b62d5ddg-39.jpg?height=305&width=303&top_left_y=841&top_left_x=575)

We get 8 rectangles of $1 \times 3$, the sum of the numbers in each of which is 23. Since the sum of all the numbers is 200, we find the number in the central cell as $200 - 23 \cdot 8 = 16$.