You do not need to be able to play chess to solve this problem.

The standard move for a knight on a chess board is $2$ steps in one direction and one step in the other direction. A knight's tour is a sequence of moves in which the knight visits every square on the board once and only once and a circuit is a tour in which the knight returns to the starting point.

Prove that a knight cannot make a tour on a $2$ by $n$ board for any value of $n$.

How many different tours can you find on a $3$ by $4$ rectangular board?

Prove that a knight cannot make a circuit on a $3$ by $4$ rectangular board.