Oh. I thought you meant me since you were quoting my post.
How many moves does a knight need to get somewhere?
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In case anyone is still working on an alternate solution to the shortest knight path problem, here are a few more results given by the formula I found for comparison purposes.
m(7,7)=6
m(13,-11)=8
m(-233,512)=257
m(2317,1449)=1256
For the benefit of anyone who is just coming across this thread, I'll explain what the numbers above mean. They relate to a math problem involving a knight near the middle of a non-standard chessboard which is about to travel to a different square. The board is laid out like a regular chessboard, but instead of being an 8 by 8 board, it has a very large number of ranks and files. For convenience, the starting square of the knight is given coordinates (0,0). A square two files to the right and five ranks below would be at (2,-5), and so on. The problem is to find the length of the shortest path, using standard knight-moves, for the knight to travel from its starting square at (0,0) to a given target square at (x,y), for any particular values of x and y, given that the board is large enough to keep the shortest path from ever reaching the edge of the board. So the answer will be a formula for a function m(x,y) which calculates the shortest path length for any given x and y.
I found a formula for m(x,y) which I believe gives the correct minimum move number for any values of x and y. There can be more than one correct way to solve a math problem, and I hoped a mathematically-minded reader would be inspired to develop another solution. If two solutions found independently give the same path length for the same starting and target squares in every case, it would increase our confidence that our solutions are correct. You could be the first to find an alternate approach to solving the problem.
I reposted my interest in finding an alternate shortest-knight-path formula in the Fun With Chess forum, thinking I might find readers there more interested in working on an interesting math problem that in scoring debating points. As a result, Lugarian, although not able to develop a formula, found an online discussion of this problem in which a shortest-path formula was proposed. It is different from my formula, but gives the same exact shortest path length for every target square I have tested. I've concluded that it satisfies my hope of finding an independently-derived formula which also gives correct results for m(x,y).
Some readers may be interested in seeing the actual formulas, so I'll show them here as a way of wrapping up this thread. They are simplest to express when written for the one-eighth of the board where 0≤y≤x. As I mentioned previously, the results there can easily be extended by symmetry to cover the whole board.
Here is the formula I posted two years ago at another site where I first saw the problem being discussed:
[r] means integer part of r
WOLOG, assume 0≤y≤x.
m(1,0)=3
m(2,2)=4
else if 2y>x, m(x,y)=x-y+2[(2y-x+2)/3]
else m(x,y)=x-y-2[(x-2y)/4]
Here is the simplified form of the formula Lugarian found a link to:
⌈r⌉ means the smallest integer greater than or equal to r
WOLOG, assume 0≤y≤x.
m(1,0)=3
m(2,2)=4
else let z=⌈max(x/2,(x+y)/3)⌉
m(x,y)=z+((x+y+z) mod 2)
I read it once, didn't understand. I'm not going to read it 5 times to try and decipher it.
Sorry, I'll try to be clearer next time.
Not you, lol.