For the 2 by 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> board, if there is a tour then
it must pass through the corner square. Is this
possible?
<p/>
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. 
<p/>
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.

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