# Minimum number of moves to reach a grid point by modified knight in variant chessboard

I apologize if this is not the right board to post this question but I’m cross-posting from the mathematics board. I am dealing with a computational question that extends the question posed in https://math.stackexchange.com/questions/104700/minimum-number-of-moves-to-reach-a-cell-in-a-chessboard-by-a-knight to the variant form of chess posed in https://math.stackexchange.com/questions/710815/knight-move-variant-can-it-move-from-a-to-b. Specifically, what is the number of minimum moves for a modified knight (call it ($$\alpha,\beta$$)-knight) that moves with $$\pm\alpha$$ and $$\pm\beta$$ along the coordinates (instead of the usual $$[\pm 2, \pm 1]$$)) in any direction to reach a point $$(x, y)$$ starting from the origin (0,0)? This would mean moving from $$(x,y)$$ to any of the following: (𝑥±$$\alpha$$,𝑦±$$\beta$$), (𝑥∓$$\alpha$$,𝑦±$$\beta$$), (𝑥±$$\beta$$,𝑦±$$\alpha$$) or (𝑥∓$$\beta$$,𝑦±$$\alpha$$). We can assume without loss of generality that $$x \geq y$$.

This is similar to the question posed in this forum here: Knight on a chessboard. I’d like to know if there is a closed form answer rather than a solution that requires BFS because I would like to work with a chess board (or coordinate grid) that is $$N \times N$$ where $$N$$ is large (e.g. $$10^6$$).

My initial thoughts of how to approach this is as follows:

1. Modify the equation to reach either the $$x$$-axis or the diagonal as well as the number of subsequent moves from the diagonal to reach the origin as solved in https://math.stackexchange.com/questions/104700/minimum-number-of-moves-to-reach-a-cell-in-a-chessboard-by-a-knight.
2. Solve this problem computationally with a recurrence equation that can be solved using dynamic programming (thought I suspect for larger$$N$$ in a $$N\times N$$ chessboard for large $$\alpha$$ and large $$\beta$$, this becomes intractable). Would this be more feasible than the BFS solution posted in: Knight on a chessboard?
3. Use the logic behind the proof posted in https://math.stackexchange.com/questions/710815/knight-move-variant-can-it-move-from-a-to-b to come up with an analytical solution for a recurrence equation rather than try to solve it computationally.
4. Graph approach using Dijkstra’s algorithm akin to solution posted in https://math.stackexchange.com/questions/1585538/chess-knight-move-in-8×8-chessboard. Again, this may become computationally intractable unless this is solved using sparse graphs.

Any suggestions would be appreciated.