Looking for a challenging task or variant related to the knight's tour problem

Question

I recently took it upon myself to investigate the knight's tour problem for a math assessment.
I decided to investigate how the problem differs with a general knight $(m, n)$ that moves $m$ squares along one axis and $n$ squares along the other. I feel like I have dug myself into a hole as I have only been able to prove fairly obvious instances where the knight's tour is impossible. I am struggling to find a specific task that is mathematically challenging yet not impossible. To clarify, I am not asking for any answers or work regarding my assignment, but rather some feasible ideas to explore related to a general knight $(m, n)$. I have been considering looking at permutations of the knight vs. the general knight within the first few moves but I feel this task has no real significance about how a general knight differs from a traditional knight in the knight's tour problem.
Are there smaller tasks I can tackle to build up to more complex ideas or is the Hamiltonian path problem mostly trial and error?
Any exploration ideas related to the knight's tour problem would be greatly appreciated.

Answer 1

Late reply, but hoping it can still be helpful.
Now, if I am not getting wrong, you are simply describing the $(m,n)$ leapers from fairy chess on a $2$D chessboard (see Leapers $(m, n)$). If this is not the case, let me introduce my idea for a $k$-dimensional $x_1 \times x_2 \times \cdots \times x_k \subseteq \mathbb{Z}^k$ chessboard as $k$ becomes sufficiently large for the given leaper.

Let $\sqrt{m^2+n^2} \notin \mathbb{N}$, then the given $(m,n)$ would be uniquely defined by the (fixed) Euclidean distance covered at any move and thus we can go beyond the usual limit of the planar moves.
As an example, the FIDE definition of the knight stated in the FIDE Handbook points to this kind of solution and I recently proved the counterintuitive result that such (Euclidean) knight's tour exists even on each $2 \times 2 \times \cdots \times 2$ chessboard with at least $2^6$ cells (Reference here: https://arxiv.org/abs/2309.09639).