Question 62
Arrange $1,2,\ldots,n$ in any order $a_1,\ldots,a_n$. An operation exchanges two disjoint consecutive pairs $(a_j,a_{j+1})$ and $(a_k,a_{k+1})$, where $j\le k-2$, without reversing either pair. Thus the relevant portion changes from $(a_j,a_{j+1},\ldots,a_k,a_{k+1})$ to $(a_k,a_{k+1},\ldots,a_j,a_{j+1})$. Can every permutation be transformed in finitely many operations into either $1,2,\ldots,n$ or $n,n-1,\ldots,1$ when (a) $n=2001$; (b) $n=2011$?