Question 30
There are 100 cards, each with one of the numbers $1$, $2$, $\ldots$, $100$ written on it, such that each number appears on exactly one card. The cards are stacked in such a way that the numbers $1$, $2$, $\ldots$, $100$ are written from top to bottom in order. Petryk rearranges the cards according to the following rules. If, before his $k$-th move, the numbers on the cards from top to bottom are arranged as follows: $a_1, a_2, \ldots, a_{k-1}, a_k, a_{k+1}, \ldots, a_{100}$, then after his move they are arranged as follows: $a_k, a_{k-1}, \ldots, a_2, a_1, a_{k+1}, \ldots, a_{100}$ (so the order of the cards does not change when $k=1$). Petryk takes turns making moves $1$, $2$, $\ldots$, $100$, and then makes moves $1$, $2$, $\ldots$, $100$ again, and so on. Will the initial arrangement of the cards with the numbers $1$, $2$, $\ldots$, $100$ from top to bottom necessarily be repeated after a finite number of moves?