An $(a,b)$-knight moves $a$ units horizontally and $b$ units vertically (or $b$ horizontally and $a$ vertically) for each move. For example, the traditional knight is a $(1,2)$- or $(2,1)$-knight. Does there exist general algorithms to the following problems?
Given $a,b$ and an infinite chessboard, can an $(a,b)$-knight reach every point on the chessboard no matter where it starts?
Given $a,b$ and an $m\times n$ chessboard, can an $(a,b)$-knight reach every point on the chessboard no matter where it starts? Here you can make the assumption that $m,n\gg a,b$ so the space won't be too limited for the knight to move.
In an infinite chessboard, it should be simpler, because the knight can reach every point if and only if it can achieve a single-unit up, down, left and right movement. For $m\times n$ chessboards though, I guess there might still be problems (or requirement for special treatment) with the edge or corner points, even when $m,n\gg a,b$?