The default chess board has 5 sectors and 2 concentric tracks and the knight moves 3 steps forward in an anticlockwise direction and one step to the right.

You can change the size of the track, the direction and number of steps the knight can move forward, and the number of steps it can move to the side.

When the knight moves over the outside edge of the circular board it re-enters in the same sector on the inside of the track (and vice versa).

Can you make the knight visit every square once and only once and return to its starting point?

Suppose there are p sectors and q concentric tracks and a knight's move is a steps in one direction and b steps in the other direction. Find conditions on the numbers p, q, a and b under which it is possible for the knight to visit every square and return to its starting point.