A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square.
Such a board is called [i] interesting[/i] if the following conditions hold:
$\circ$ In all unit squares below the main diagonal, the number $0$ is written;
$\circ$ Positive integers are written in all other unit squares.
$\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers
are actually the numbers $1,2,...,2n$ (not necessarily in that order).

$a)$ Determine the largest number that can appear in a $6 \times 6$ [i]interesting[/i] board.
$b)$ Prove that there is no [i]interesting[/i] board of dimensions $7\times 7$.

### Part (a)
To determine the largest number that can appear in a $6 \times 6$ interesting board, we need to consider the conditions given:

1. In all unit squares below the main diagonal, the number $0$ is written.
2. Positive integers are written in all other unit squares.
3. The sums of all $n$ rows and the sums of all $n$ columns are the numbers $1, 2, \ldots, 2n$.

For a $6 \times 6$ board, $n = 6$, so the sums of the rows and columns must be the numbers $1, 2, \ldots, 12$.

Let's denote the elements of the board as $a_{ij}$ where $i$ is the row index and $j$ is the column index. The main diagonal elements are $a_{ii}$ for $i = 1, 2, \ldots, 6$. Below the main diagonal, $a_{ij} = 0$ for $i > j$.

The sum of all elements in the board is:
\[
\sum_{i=1}^{6} \sum_{j=1}^{6} a_{ij}
\]
Since the sums of the rows and columns are the numbers $1, 2, \ldots, 12$, the total sum of all elements in the board is:
\[
1 + 2 + \ldots + 12 = \frac{12 \cdot 13}{2} = 78
\]

Each number is counted twice (once in the row sum and once in the column sum), so the total sum of the elements in the board is:
\[
\frac{78}{2} = 39
\]

To maximize the largest number on the board, we need to consider the placement of the numbers such that the sums of the rows and columns are distinct and cover the range $1, 2, \ldots, 12$. The largest number will be on the main diagonal or above it.

Since the sum of the elements in the board is $39$, and we need to distribute this sum among the elements above and on the main diagonal, we can place the largest number in the top-right corner (or any other position above the main diagonal).

The largest number that can appear in a $6 \times 6$ interesting board is $\boxed{12}$.

### Part (b)
To prove that there is no interesting board of dimensions $7 \times 7$, we need to consider the sum of the sums of all rows and columns.

For a $7 \times 7$ board, $n = 7$, so the sums of the rows and columns must be the numbers $1, 2, \ldots, 14$.

The sum of these numbers is:
\[
1 + 2 + \ldots + 14 = \frac{14 \cdot 15}{2} = 105
\]

Each number is counted twice (once in the row sum and once in the column sum), so the total sum of the elements in the board is:
\[
\frac{105}{2} = 52.5
\]

Since the sum of the elements in the board must be an integer, and $52.5$ is not an integer, it is impossible to have an interesting board of dimensions $7 \times 7$.

Therefore, there is no interesting board of dimensions $7 \times 7$. $\blacksquare$