I have been working a bit on the following math problem:
Assume a knight is placed on a random square on a chess board with the coordinates $(x, y)$, where the x-coordinate is the letter of the given square and the y-coordinate is the number. How many moves $N$ would then be required to move the knight to another, arbitrary square with the coordinates $(x_{N}, y_{N})$?
As for my progress so far, there are some things to note that might be of use to solve this problem. Because of how the chess board is structured, squares sharing a corner will always belong to the same color complex. This has a neat consequence, namely that the coordinate-sum of diagonally adjecent squares always share the same parity. Or with other words: $(x+y)\mod{n}=(x_{N}, y_{N})\mod{n}=0$, $n\in\mathbb{Z}$. This practically means we ever only have to consider even or odd $N$s, for any given starting and desired square.
I have also tinkered a bit with a simple BFS-algorithm in Python, to find the largest possible value of $N$ (it happens to be six). $N=6$ only if the sqaures are $a1\longleftrightarrow h8$ or $a8\longleftrightarrow h1$. A similar idea I have had is have a computer measure the absolute distance between every possible square the knight can reach in N moves and see if there is some sort of relationship between the distance $d$ and $N$.
This is about the extent of all the useful conclusions I have come to so far. I would be delighted to hear any ideas on how I could possibly move forward. Ideally, I want to find some sort of piece-wise function that will take an input, consisting of a starting square and a desired square, and compute the minimum amount of moves required to move between them. However, that might be a little ambitious and finding a manual algorithm might be more realistic. Either way, I want the solution to be mathematical and not rely on computers.
After this case is solved, I would like to move on to a $n x n$ chess board to see if a general solution can be found there.
