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If we draw a line using a ruler on an $8\times8$ chess board, what line can touch the most squares, how many? Is it the diagonal?

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    $\begingroup$ Do you consider the diagonal line to "touch" only the eight diagonal squares, or also the squares touching the diagonal squares? That is, is it sufficient for the line to intersect a single point of the square's boundary to "touch" it, or must it go into the interior as well? $\endgroup$ – Akiva Weinberger Oct 26 '15 at 19:21
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    $\begingroup$ I would think if you actually did some drawing the answer would present itself rather quickly. $\endgroup$ – Vincent Oct 26 '15 at 19:23
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Maybe this image which presents a line crossing through 15 squares on a chess board might demonstrate or at least help your understanding of the answer.

Chess Board

As you can see, a slightly offset diagonal line fully enters 15 squares in an alternating pattern. And yes, this is the line which touches the most chess squares (or at least one of them).

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    $\begingroup$ A line from one corner to the opposite corner would yield 22. $\endgroup$ – Vincent Oct 31 '15 at 17:11

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