, 2011

Note that for a given $i$, $a_i$ and $b_i$ are never both in one of $\{1, 2, ..., n\}$ or $\{n+1, ..., 2n\}$. Suppose, without loss of generality, that $a_i, b_i \le n$. Then $a_1, a_2, ..., a_i \le n$, and $b_i, b_{i+1}, ..., b_n \le n$, giving $n+1$ elements less than or equal to $n$, which is impossible. Then every $|a_i - b_i|$ is a difference between an element in $\{n+1, ..., 2n\}$ and an element in $\{1, 2, ..., n\}$. Therefore:
$$\sum_{i=1}^{n} |a_i - b_i| = \sum_{i=1}^{n} (n + i) - \sum_{i=1}^{n} i = n^2.$$