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Keep a record of the set-ups when it is impossible for the knight
to visit every square (that is the four numbers: of sectors, of
tracks and of moves the knight makes in the two directions). What
can you say about these set-ups?

Consider common factors of pairs of these numbers. It may help to consider first the result for one track with p sectors and a knight's move of a steps forward and zero steps to the side for different values of p and a. What joint property of p and a determines whether or not the knight can visit every square?

Try the standard knights move of two steps one way and one step the other way on different boards and see if the knight can visit every square.

Now what happens if the knight moves two steps in each direction?

A convenient notation is to let (x,y) => (x+s, y+t) denote a move of s steps clockwise and t steps outwards. As there are p sectors around the circuit the knight returns to the same sector after p steps clockwise which calls for taking x and x+s modulo p. Similarly as there are q tracks which the knight moves round cyclically, and the knight returns to the same track after taking q steps outwards, so take y and y+t modulo q.

Consider common factors of pairs of these numbers. It may help to consider first the result for one track with p sectors and a knight's move of a steps forward and zero steps to the side for different values of p and a. What joint property of p and a determines whether or not the knight can visit every square?

Try the standard knights move of two steps one way and one step the other way on different boards and see if the knight can visit every square.

Now what happens if the knight moves two steps in each direction?

A convenient notation is to let (x,y) => (x+s, y+t) denote a move of s steps clockwise and t steps outwards. As there are p sectors around the circuit the knight returns to the same sector after p steps clockwise which calls for taking x and x+s modulo p. Similarly as there are q tracks which the knight moves round cyclically, and the knight returns to the same track after taking q steps outwards, so take y and y+t modulo q.