# 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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