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Combinatorics Difficulty 5.0 AIME, harder Prove it United Kingdom

The positive integers from 1 to $n$ ( $n \geq 2$ ) inclusive are to be spaced equally around the circumference of a circle so that: (a) no two even numbers are adjacent; (b) no two odd numbers are adjacent; (c) no two numbers differing by 1 are adjacent. What is the smallest value of $n$ for which the above is possible?

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