Sequence of natural numbers 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!

Basically, if you make an odd number of switches, then at least one of the numbers has only been moved once (unless you repeat the same switch over and over which is an easy case to explain). But if you start in some configuration and move a number only once and want to return to the start, you must move again. 
Try induction -- an easy base case is $n=2$.
