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Knight Defeated

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.