Let $\sigma : \{1, 2, . . . , n\} \to \{1, 2, . . . , n\}$ be a bijective mapping. Let $S_n$ be the set of all such mappings and let $d_k(\sigma) = |\sigma(k) - \sigma(k + 1)|$, for all $k \in \{1, 2, ..., n\}$, where $\sigma (n + 1) = \sigma (1)$. Also let $d(\sigma) = \min \{d_k(\sigma) | 1 \le k \le n\}$. Find $\max_{\sigma \in S_n} d(\sigma)$.