Maximum Permutation

Let \(\sigma\) be a permutation of the set \(\{ 1, 2, \ldots, 999 \}\). What is the maximum value over all permutations, of \[ \displaystyle \min_{1 \leq i \leq 999} |\sigma(i+1)-\sigma(i)|? \]

Details and assumptions

\(\sigma(i)\) denotes the number in the \(i^{th}\) position and we interpret \( \sigma(1000)=\sigma(1) \).

If \(\sigma\) is the identity permutation, then \( |\sigma(i+1) - \sigma(i)|=|(i+1)-i| = 1\), hence \( \displaystyle \min_{1 \leq i \leq 999} |\sigma(i+1)-\sigma(i)| =1\).

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