CN-TST 2009 Problem 5

Let $m > 1$ be an integer, $n$ is an odd number satisfying $3\le n < 2m,$ number $a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $(1)$ for any $1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $1,2,3,\cdots,m; (2)$ for any $1 < i\le m, 1\le j\le n - 1, |a_{i,j} - a_{i,{j + 1}}|\le 1$ holds. Find the minimal value of $M$ , where $M = max_{1 < i < m}\sum_{j = 1}^n{a_{i,j}}.$