CGMO 2017 Problem 8

Let $n$ be a fixed positive integer. Let $$A=\begin{bmatrix} a_{11} & a_{12} & \cdots &a_{1n} a_{21} & a_{22} & \cdots &a_{2n} \vdots & \vdots & \cdots & \vdots a_{n1} & a_{n2} & \cdots &a_{nn} \end{bmatrix} \text{and} B=\begin{bmatrix} b_{11} & b_{12} & \cdots &b_{1n} b_{21} & b_{22} & \cdots &b_{2n} \vdots & \vdots & \cdots & \vdots b_{n1} & b_{n2} & \cdots &b_{nn} \end{bmatrix}$$ be two $n\times n$ tables such that $\{a_{ij}|1\le i,j\le n\}=\{b_{ij}|1\le i,j\le n\}=\{k\in N^*|1\le k\le n^2\}$ . One can perform the following operation on table $A$ : Choose $2$ numbers in the same row or in the same column of $A$ , interchange these $2$ numbers, and leave the remaining $n^2-2$ numbers unchanged. This operation is called a transposition of $A$ . Find, with proof, the smallest positive integer $m$ such that for any tables $A$ and $B$ , one can perform at most $m$ transpositions such that the resulting table of $A$ is $B$ .