a) João found a way to arrange the numbers $\{1,2,3, \ldots, 16\}$ on a $4 \times 4$ board so that the sum of the numbers in each row, each column, and each diagonal is always the same. One of the possibilities is shown in the example below.

| 4 | 6 | 9 | 15 |
| :---: | :---: | :---: | :---: |
| 13 | 11 | 8 | 2 |
| 16 | 10 | 5 | 3 |
| 1 | 7 | 12 | 14 |

Find another example of distributing these 16 numbers that satisfies the same conditions.

b) Verify that in any possible distribution, the sum of the numbers in each row and column is always 34.

c) João has now created a new type of board with other positive numbers. The product of the numbers in each row, each column, and each diagonal is always the same. What is the value of the number $4H$?

| $1/2$ | 32 | $A$ | $B$ |
| :---: | :---: | :---: | :---: |
| $C$ | 2 | 8 | 2 |
| 4 | 1 | $D$ | $E$ |
| $F$ | $G$ | $H$ | 16 |
Solution

a) Other possible distributions would be:

| 10 | 1 | 12 | 11 |
| :---: | :---: | :---: | :---: |
| 2 | 15 | 6 | 16 |
| 13 | 8 | 5 | 3 |
| 9 | 7 | 4 | 14 |


| 1 | 14 | 15 | 4 |
| :---: | :---: | :---: | :---: |
| 10 | 8 | 5 | 11 |
| 7 | 9 | 12 | 6 |
| 16 | 3 | 2 | 13 |

b) Let $l$ be the sum of the numbers written in a column. Adding the numbers of the four columns we have:

$$
\begin{aligned}
4 l & =1+2+3+\ldots+16 \\
& =136
\end{aligned}
$$

Therefore, $l=136 / 4=34$. The same argument can be applied to the rows.

c) Performing cancellations in the product of some rows, columns, and diagonals, we obtain:

$$
\begin{aligned}
(A \cdot 8 \cdot D \cdot H)(F \cdot 1 \cdot 8 \cdot B)(C \cdot 2 \cdot 8 \cdot 2) & =(1 / 2 \cdot 2 \cdot D \cdot 16)(1 / 2 \cdot 32 \cdot A \cdot B)(1 / 2 \cdot C \cdot 4 \cdot F) \\
4 H & =1
\end{aligned}
$$