7
$\begingroup$

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 $a_n=n^2$?

  • 1
    The previous edition of this problem has now been posted to MO, http://mathoverflow.net/questions/88659/traversing-the-infinite-square-grid2012-02-16

1 Answers 1