One possible way to approach these things that maybe has more of the "intrinsic" feel Fred is looking for: Let F : Z^4 -> Z^2 map the four standard basis vectors to four "basis" vectors of the knight's-move lattice; so it's represented by, say, the matrix (1 1 2 2; 2 -2 1 -1). The preimages of any point in Z^2 form a 2-dimensional lattice within Z^4 (offset by some single preimage) whose basis vectors are, if I've not messed up the routine calculations to get them, (1;-1;-2;2) and (-2;-2;1;1). Now questions about distances and shortest paths for a knight on the chessboard become questions about distances and shortest paths from the origin to *any point of that lattice* in Z^4. Or, equivalently, consider only the preimage lattice of 0 and ask about distances from some single preimage to *any point of that lattice*. So, for instance, the question about (2n,n) is the same as: show that of all points (a-2b;-a-2b;n-2a+b;2a+b), the one with the smallest L1-norm is (0;0;n;0). It's not obvious to me that this actually makes anything easier, but I think there's a fair chance that there's some single further bit of technique that makes all these questions, viewed from this perspective, routine. -- g