Imagine we have a $8 \times 8$ chessboard and a person situated on one of the dark squares. The person is allowed to jump diagonally, but only by $1$ square and the person cannot revisit the squares they were on before. Of course, if the player begins in a non-corner square, then as there are two dark squares at the corners they cannot cover all of the dark squares: if they enter a corner square, there is no way of them leaving it without retracing their steps.
Therefore they must begin at a corner; WLOG assume that they start at the bottom left corner. Then, they have a forced move towards the top right square and now they have the same problem: The top right square of the chessboard and the square that is third from the left on the bottom can be entered only by a single way. Therefore the person cannot cover all of the dark squares by jumping diagonally only $1$ diagonals.
If now we allow the player more freedom by giving them the ability to jump $1$ or $2$ diagonal squares, the person now can cover all the dark squares independent of which dark square they begin from. One can check that this is true by showing manually by hand that the person can cover all the squares if they begin at the 10 dark squares that are inside the triangle when we divide the square into 4 triangles from its centre and then all dark squares follow from symmetry.
I was wondering if given a $n\times n $ chessboard and a set of positive integers $S = \{a_1 , \dots , a_k \} \quad k < n$ and $a_k < n \; \; \forall k$ such that gcd$(a_1, \dots a_k) =1$ and all $a_i$ are different, one can cover all the dark squares using jumps with distances $a_1, \dots a_k$.
You should be able to convince yourself that one space plus two space can cover any rectangular board except very small ones. The only problem comes in a corner and there are only two kinds of corner, those with one black square in the corner and those with two black squares next to the corner. You can find a pattern that handles both kinds of corner. Then if you have enough room you can connect the corners using only single space moves. The only problem comes if the corners are too close together and one uses squares that are needed for another.