Let $n$ be a nonzero natural number and let $S = \{1, 2, . . . , n\}$. A $3 \times n$ board is called beautiful if it can be completed with numbers from the set $S$ like this as long as the following conditions are met: $\bullet$ on each line, each number from the set S appears exactly once, $\bullet$ on each column the sum of the products of two numbers on that column is divisible by $n$ (that is, if the numbers $a, b, c$ are written on a column, it must be $ab + bc + ca$ be divisible by $n$). For which values of the natural number $n$ are there beautiful tables ¸and for which values do not exist? Justify your answer.