CZ-SK-MO 2007 Problem 1

A stone is placed in a square of a chessboard with $n$ rows and $n$ columns. We can alternately undertake two operations: (a) move the stone to a square that shares a common side with the square in which it stands;(b) move it to a square sharing only one common vertex with the square in which it stands.

In addition, we are required that the first step must be (b). Find all integers $n$ such that the stone can go through a certain path visiting every square exactly once.