Solution and rules are found here. \begin{array}{ccc|ccc|ccc} 4 & 8 & 3 & 7 & 2 & 6 & 1 & 5 & 9 \\ 7 & 2 & 6 & 1 & 5 & 9 & 4 & 8 & 3\\ 1 & 5 & 9 & 4 & 8 & 3 & 7 & 2 & 6 \\ \hline 8 & 3 & 7 & 2 & 6 & 1 & 5 & 9 & 4 \\ 2 & 6 & 1 & 5 & 9 & 4 & 8 & 3 & 7 \\ 5 & 9 & 4 & 8 & 3 & 7 & 2 & 6 & 1 \\ \hline 3 & 7 & 2 & 6 & 1 & 5 & 9 & 4 & 8 \\ 6 & 1 & 5 & 9 & 4 & 8 & 3 & 7 & 2 \\ 9 & 4 & 8 & 3 & 7 & 2 & 6 & 1 & 5\\ \end{array}
I noticed that the final solution can be derived from treating the grid as a torus with each digit propagated by generators $(1,3), (3,1).$
To make this a rigorous solution as well as an explanation of how you could come up with the puzzle, two things need to be justified:
- If a digit $d$ is at $(x, y),$ then it is at $(x+3,y+1), (x+1, y+3)$ (taking coordinates $\mod 9$).
- If the central square has $1$ and $2$ at the given positions, then the rest of the square fills in uniquely.
The proof of claim 2 is easy assuming claim 1. Just go through the $\le 5 \cdot 6!$ remaining ways to place $3, 4, \dots, 9$ in the central square, use claim $1$ to fill in the rest of the grid, and see that only one leads to a solution. So I am interested in a proof that the rules of Sudoku and the knight's move, king's move, and orthogonal constraint implies the 1st claim.