Suppose we start at $(0.5,0.5)$ in an infinite unit square grid, and our goal is to traverse every square on the board.
At move $n$ one must take $a_n$ steps in one of the directions, north,south, east or west. And every square we walk over is marked as visited, we are not allowed to walk over a visited square twice.
Is there a sequence of directions, such that we can visit every square of the board exactly once if $a_n=n$?
Is there such a sequence if we are allowed to walk in diagonal directions aswell?
Is there a general algorithm to check, given $a_n$, if a path exists?
Is there a path in any of the above cases for $a_n=n^2$?