Question 8
It is considered a file with 1000 numbered tokens, ordered in its natural order. The following operation is applied to that file: the first file of the file is placed interspersed between the penultimate and the last of the file, and the second, at the end of all, leaving, therefore, in the first place the one previously occupied by the third. Observing the succession of positions occupied by each of the cards, show that after 1000 analogous operations, applied successively (each to the ordering resulting from the previous operation), the file returns to its natural order. Check that an analog result could not be obtained ($n$ operations for a file of $n$ tabs) if it were a file with an odd number $n$ of tabs.