TITLE: Sequence of natural numbers
QUESTION [6 upvotes]: Numbers $1,2,...,n$ are written in sequence. It's allowed to exchange any two elements. Is it possible to return to the starting position after an odd number of movements?
I know that is necessarily an even number of movements but I can't explain that!

REPLY [2 votes]: Hint: Define an inversion of a sequence $(a_1,\dots,a_n)$ as a pair $(i,j)$, such that $i<j$ and $a_i>a_j$. Check, that each movement changes the parity of number of inversions.