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Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move $n$, one must jump $a_n$ steps in one of the directions north,south,east or west, and mark the ending square as visited?

If $a_n=n$ or if $a_n=n^2$?

Allowing diagonal moves as well, is there a general algorithm, given $a_n$, to check if a path exists?

Note: I am asking if given $a_n$, there exists an infinite sequence of directions, $d_n\in(N,S,W,E)$, such that for all $(x,y)\in Z^2$, there exists a finite integer $k$, k(x,y)$, such that starting at the unit square with center$(0.5,0.5)$, marked as visited, we have after moving sequentially$a_i$steps in direction$d_i$, for$i=1,2,3,...,k$, visited$k+1$different unit squares, including the unit square with center and are situated at$(x+0.5,y+0.5)$. 6 added 40 characters in body Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move$n$, one must jump$a_n$steps in one of the directions north,south,east or west, and mark the ending square as visited? If$a_n=n$or if$a_n=n^2$? Allowing diagonal moves as well, is there a general algorithm, given$a_n$, to check if a path exists? Note: I am asking if given$a_n$, there exists an infinite sequence of directions,$d_n\in(N,S,W,E)$, such that for all$(x,y)\in Z^2$, there exists a finite integer$k$, such that starting at the unit square with center$(0.5,0.5)$, marked as visited, we have after moving sequentially$a_i$steps in direction$d_i$, for$i=1,2,3,...,k$, visited$k+1$different unit squares, including the unit square with center$(x+0.5,y+0.5)$. 5 added 13 characters in body Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move$n$, one must jump$a_n$steps in one of the directions north,south,east or west, and mark the ending square as visited? If$a_n=n$or if$a_n=n^2$? Allowing diagonal moves as well, is there a general algorithm, given$a_n$, to check if a path exists? Note: I am asking if given$a_n$, there exists an infinite sequence of directions,$d_n\in(N,S,W,E)$, such that for all$(x,y)\in Z^2$, there exists a finite integer$k$, such that starting at the unit square with center$(0.5,0.5)$, marked as visited, we have after moving sequentially$a_i$steps in direction$d_i$, for$i=1,2,3,...,k$, visited the unit square with center$(x+0.5,y+0.5)\$.

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