1. In a magic square, the sum of the numbers in each row, column, and on both diagonals is the same. The figure below shows a $3 \times 3$ square in which four cells have been filled with rational numbers. Determine whether it is possible to fill the remaining empty cells with rational numbers so that the square becomes a magic square.

| 3.6 |  |  |
| :--- | :--- | :--- |
|  |  | $5 \frac{2}{5}$ |
| $4 \frac{4}{5}$ |  | 2.4 |
1. In a magic square, the sum of the numbers in each row, column, and on both diagonals is the same. The image shows a $3 \times 3$ square with four fields filled with rational numbers. Investigate whether it is possible to fill the remaining empty fields with rational numbers so that the square becomes magic.

## Solution.

Let the unknown numbers in the magic square be denoted by $a, b, c, d$, and $e$.

The sum of the numbers in each row/column/diagonal is the same, so from the first column and the second row we get:

$3.6 + c + 4 \frac{4}{5} = c + d + 5 \frac{2}{5}$

$8.4 = d + 5.4$

2 POINTS

$d = 8.4 - 5.4 = 3$

1 POINT

| 3.6 | $a$ | $b$ |
| :---: | :---: | :---: |
| $c$ | $d$ | $5 \frac{2}{5}$ |
| $4 \frac{4}{5}$ | $e$ | 2.4 |

The sum of the numbers on one diagonal is $3.6 + 3 + 2.4 = 9$.

The square will be magic if the sum of the elements in all columns, rows, and both diagonals is 9.

From the second row, we get:

$c = 9 - 3 - 5 \frac{2}{5} = 0.6$

1 POINT

From the third column, we get:

$b = 9 - 3.6 - 2.4 = 1.2$

1 POINT

From the third row, we get:

$e = 9 - 4 \frac{4}{5} - 2.4 = 1.8$

From the first row, we get:

$a = 9 - 3.6 - 1.2 = 4.2$

We need to check if the sum is magic in the first and second columns and on the other diagonal:

$3.6 + 0.6 + 4.8 = 4.2 + 3 + 1.8 = 4.8 + 3 + 1.2 = 9$

Since the sum of the numbers in each row, column, and on both diagonals is 9, the square is magic.

Note 1. If the student arrives at the correct results in a different order, evaluate as follows: determining the first unknown number 3 POINTS, sum by columns/rows/diagonals 1 POINT, each subsequent number 1 POINT, and checking the sum in the remaining rows/columns/diagonals not used for calculating the unknown numbers 2 POINTS.

Note 2. For a complete solution, it is not necessary to show the calculation process with equations; it is sufficient to determine the numbers that are missing (8 POINTS) and verify that the sum in all columns/rows and diagonals is 9 (2 POINTS).

| 3.6 | 4.2 | 1.2 |
| :---: | :---: | :---: |
| 0.6 | 3 | $5 \frac{2}{5}$ |
| $4 \frac{4}{5}$ | 1.8 | 2.4 |