A grid with nine cells, as shown in the image, is filled with nine consecutive natural numbers. These numbers are arranged in ascending order from left to right and from top to bottom (i.e., the smallest number is in the top left, and the largest is in the bottom right). The thick broken line divides the grid into two parts. The sum of the numbers in the left part is 100 less than the sum of the numbers in the right part.

![](https://cdn.mathpix.com/cropped/2024_04_17_f53d56ddc45593eccca2g-2.jpg?height=377&width=376&top_left_y=477&top_left_x=840)

Which number is in the middle cell?

(L. Šimůnek)
On each line, the number in its rightmost field is 2 greater than the number in its leftmost field. The sum of the numbers in the rightmost column is therefore $3 \cdot 2=6$ greater than the sum of the numbers in the leftmost column. The number in the lower field of the middle column is 3 greater than the number in its middle field. These differences are highlighted in different shades of gray in the grid.

![](https://cdn.mathpix.com/cropped/2024_04_17_f53d56ddc45593eccca2g-2.jpg?height=371&width=371&top_left_y=1231&top_left_x=845)

The sum of the numbers in all gray fields on the right side of the grid is therefore $6+3=9$ greater than the sum of the numbers in all gray fields on its left side. However, the sum of all numbers on the right side should be 100 greater than the sum of all numbers on the left side. Therefore, the number in the only unhighlighted field must be $100-9=91$. In the middle field of the grid, the number must then be $91+3=94$.

Scoring suggestion. 3 points for determining suitable subsets in the two separate parts of the grid and determining the difference in the sums of these subsets; 2 points for calculating the number in the unmarked field (or another auxiliary number); 1 point for determining the number in the middle field.

Alternative solution. If we denote the number in the middle field by $x$, then the numbers in the other fields are

| $x-4$ | $x-3$ | $x-2$ |
| :---: | :---: | :---: |
| $x-1$ | $x$ | $x+1$ |
| $x+2$ | $x+3$ | $x+4$ |

The sum of all numbers in the left part of the grid is $4x-3$, and the sum of all numbers in the right part is $5x+3$. The second expression should be 100 greater than the first. This gives us the equation

$$
4x-3+100=5x+3,
$$

whose solution is $x=94$. The number in the middle field is 94.

Scoring suggestion. 3 points for setting up the equation where the unknown is one of the numbers in the grid; 3 points for calculating the number in the middle field.

Note. With a different choice of the unknown, we arrive at a different equation with a similar solution. If, for example, $y$ denotes the number in the top-left field, then we get

$$
\begin{aligned}
y+(y+3)+(y+4)+(y+6)+100 & =(y+1)+(y+2)+(y+5)+(y+7)+(y+8), \\
4y+113 & =5y+23, \\
y & =90.
\end{aligned}
$$

The number in the middle field is then determined as $y+4=94$.