In how many ways can the numbers $1, 2, 3, \ldots, 100$ be arranged around a circular table such that any two adjacent numbers differ by at most $2$? In how many ways can the numbers $1, 2, 3, \ldots, 100$ be arranged around a circular table such that any two adjacent numbers differ by at most $2$?

First place ‘1’ at a position. On its either side you can place only two numbers (‘2’ and ‘3’). Let us place ‘2’ to left of ‘1’ and ‘3’ to right of ‘1’. Now to the left of ‘2’ you can place only ‘4’ (‘3’ has been already place at right of ‘1’). On proceeding, all even numbers are next to each other forming a half of net circle and odd numbers are together forming another half. So there is only one arrangement. If clockwise and anti clock wise arrangement re different, then answer is two possible arrangements.
