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7
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edited Feb 19 2012 at 19:17 |
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 $(x+0.5,y+0.5)$.
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6
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edited Feb 18 2012 at 11:05 |
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 $(x+0.5,y+0.5)$.
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5
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edited Feb 18 2012 at 10:37 |
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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4
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edited Feb 18 2012 at 10:25 |
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 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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3
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edited Feb 16 2012 at 20:15 |
Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move n, $n$, 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 aswellas well, is there a general algorithm, given $a_n$, to check if a path exists?
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2
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edited Feb 16 2012 at 19:41 |
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1
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asked Feb 16 2012 at 19:27 |
Traversing the infinite square grid
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 aswell, is there a general algorithm, given $a_n$, to check if a path exists?
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