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Given a grid of $n \times m$ points, on a sheet of paper, what's the minimum amount of lines (which can extend infinitely) required to pass through each point, that you can draw without lifting a pencil?

Sample grid and answer: Connect the dots

Note: Lines are of less width then points are radius, all lines must be straight.

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    $\begingroup$ Could we just say that the points are of zero radius and the lines are of zero width? Otherwise the answer is "technically" 1 because you can always have the dots large enough that they overlap and you can just put a line through their intersection. Also, must the lines be end to end as stated in the puzzle you linked? (if not it's trivial) $\endgroup$ – Ben Frankel Sep 23 '15 at 3:39
  • $\begingroup$ @Ben See the note $\endgroup$ – warspyking Sep 23 '15 at 9:32
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    $\begingroup$ @warspyking Well, your note is (a) grammatically incorrect (b) difficult to understand. Could you please state the question in a more elaborate form, so we can atleast understand it (and the edit it for you)? $\endgroup$ – ghosts_in_the_code Sep 23 '15 at 12:43
  • $\begingroup$ Basically I give you a grid of dots, and without lifting your pencil you have to draw over every dot with nothing but straight line segments. (The grid is $n \times m$) $\endgroup$ – warspyking Sep 24 '15 at 2:13
  • $\begingroup$ I have voted to close this question as unclear. If you address @ghosts_in_the_code's questions and modify the question, I will retract it. $\endgroup$ – Rohcana Sep 29 '15 at 18:25
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I hope the lines are not required to intersect.

We want maximum number of points on each line. We can have a maximum of $m$ points on it (for $m\geq n$). Hence we draw $n$ parallel lines, each with $m$ dots.

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    $\begingroup$ Even if the lines are required to intersect, because the points are not restricted to zero radius, you can always tilt the lines such that the $n$ lines are in a zigzag formation. $\endgroup$ – Hackiisan Sep 23 '15 at 7:17
  • $\begingroup$ I think @warspyking meant that (a) the points are of zero-radius (b) the lines must form a continuous curve. We'll have to wait and see. $\endgroup$ – ghosts_in_the_code Sep 23 '15 at 7:23
  • $\begingroup$ See the Note ${}$ $\endgroup$ – warspyking Sep 23 '15 at 9:32
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This is my hypothesis, but I haven't figured out how to prove it yet:

For $m = n$, the answer is

$2(n-1)$. This employs the solution given in the linked question, of basically using the well-known four-line solution for a $3\times3$ grid, and then extending a square spiral outward to encompass the other dots:

enter image description here

Any other solution (e.g. using a triangle spiral instead of a square) doesn't seem to improve on this number.

For $m > n$, the solution is

$2n-1$. There doesn't appear to be any more efficient solution than just drawing a spiral or zigzagging across the rows:

enter image description here or enter image description here

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  • $\begingroup$ By the time you're reading this, you've already counted to nine in your head. Twice. $\endgroup$ – user1717828 Oct 1 '15 at 4:34
  • $\begingroup$ @user1717828 I've counted 3 times :P $\endgroup$ – warspyking Oct 1 '15 at 13:05

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