# A drunk knight's tour [duplicate]

Consider an infinite chess board. A knight moves 2 squares forward on one direction, then turn left or right, move 1 square further on. Let's denote this a normal knight, or $$\langle 2,1\rangle$$ knight. And it's known that a knight could reach any square on the board, i.e. denoting the starting square as $$(0,0)$$, a $$\langle 2,1\rangle$$ knight can arrive any square $$(m,n)$$.
Now let's say this knight gets drunk and started to walk $$\langle p,q\rangle$$...
Of course, if both $$p$$ and $$q$$ are even numbers, a $$\langle p,q\rangle$$ knight won't arrive $$(m,n)$$ if both $$m$$ and $$n$$ are odd numbers.
Then brings to the question, in which situation, a drunk $$\langle p,q\rangle$$ knight still could reach any square $$(m,n)$$ ?