Any permutation can be achieved, since the cycle $\gamma = (1,2,3,\ldots,n)$ and the transposition $\tau = (1,2)$ together make up a set of generators for the symmetric group $\mathcal{S}_n$. This same classic result has also been used some years ago in a Bulgarian Test. Oh, well ... http://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations.

In each vertex of a regular $n$-gon $A_1A_2...A_n$ there is a unique pawn. In each step it is allowed: 1. to move all pawns one step in the clockwise direction or 2. to swap the pawns at vertices $A_1$ and $A_2$. Prove that by a finite series of such steps it is possible to swap the pawns at vertices: a) $A_i$ and $A_{i+1}$ for any $ 1 \leq i < n$ while leaving all other pawns in their initial place b) $A_i$ and $A_j$ for any $ 1 \leq i < j \leq n$ leaving all other pawns in their initial place. Proposed by Matija Bucic