Age 14 to 16 Challenge Level:
For the $2$ by $n$ board, if there is a tour then it must pass
through the corner square. Is this possible?
It might help to think of the squares as vertices of a graph. Then
there is an edge joining two vertices if and only if there is a
knight's move between the corresponding squares.
Eight of the vertices are of degree two (only one path in and one
out of that square). To construct a tour you are forced to visit
these vertices in a particular order.