Question 145
Let $m$ be an integer greater than $1$, and let $n$ be an odd number with $3 \le n \lt 2m$. Numbers $a_{i,j}$ ($i, j \in \mathbb{N}$, $1 \le i \le m$, $1 \le j \le n$) satisfy:
(1) For every $1 \le j \le n$, $a_{1,j}$, $a_{2,j}$, ..., $a_{m,j}$ is a permutation of $1, 2, ..., m$;
(2) $|a_{i,j} - a_{i,j+1}| \le 1$ for every $1 \le i \le m$, $1 \le j \le n-1$.
Find the minimal possible value of $M = \max_{1 \le i \le m} \sum_{j=1}^{n} a_{i,j}$.
(1) For every $1 \le j \le n$, $a_{1,j}$, $a_{2,j}$, ..., $a_{m,j}$ is a permutation of $1, 2, ..., m$;
(2) $|a_{i,j} - a_{i,j+1}| \le 1$ for every $1 \le i \le m$, $1 \le j \le n-1$.
Find the minimal possible value of $M = \max_{1 \le i \le m} \sum_{j=1}^{n} a_{i,j}$.