Question 3
Let $n \gt 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that
a. The set of the numbers in each row is a permutation of $0, 1, \ldots, n-1$; the set of the numbers in each column is a permutation of $0, 1, \ldots, n-1$;
b. The board is totally non-symmetric: for $1 \leq i \lt j \leq n$ and $1 \leq i' \lt j' \leq n$, if $(i, j) \neq (i', j')$ then $(a_{i, j}, a_{j, i}) \neq (a_{i', j'}, a_{j', i'})$, where $a_{i, j}$ denotes the entry in the $i^{\text{th}}$ row and $j^{\text{th}}$ column.
a. The set of the numbers in each row is a permutation of $0, 1, \ldots, n-1$; the set of the numbers in each column is a permutation of $0, 1, \ldots, n-1$;
b. The board is totally non-symmetric: for $1 \leq i \lt j \leq n$ and $1 \leq i' \lt j' \leq n$, if $(i, j) \neq (i', j')$ then $(a_{i, j}, a_{j, i}) \neq (a_{i', j'}, a_{j', i'})$, where $a_{i, j}$ denotes the entry in the $i^{\text{th}}$ row and $j^{\text{th}}$ column.