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Old 2009-05-05, 16:15   #1
Kees
 
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Default Lines in a Square

Consider an n x n chessboard.
What is the size of the smallest collection of lines such that every square of the chessboard is intersected by at least one line ?

PS: I do not pretend to have a solution, but as this is more puzzle than serious math I decided to put it here.
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Old 2009-05-05, 16:26   #2
retina
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Quote:
Originally Posted by Kees View Post
Consider an n x n chessboard.
What is the size of the smallest collection of lines such that every square of the chessboard is intersected by at least one line ?

PS: I do not pretend to have a solution, but as this is more puzzle than serious math I decided to put it here.
One line only is needed. The puzzle does not specify that the lines have to be straight so I make it a single spiral with pitch of < one square width.

Last fiddled with by retina on 2009-05-05 at 16:27
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Old 2009-05-05, 16:27   #3
TimSorbet
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Assuming straight lines only.
I can't find a way to do it with any less than n lines, no matter the number of squares, and it can always be done with n lines by putting one horizontal line through each row, so my answer is:
n. It takes a minimum of n lines to put a line through every square of a n x n "chessboard".
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Old 2009-05-05, 16:34   #4
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I can't improve on n
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Old 2009-05-05, 16:35   #5
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Quote:
Originally Posted by retina View Post
One line only is needed. The puzzle does not specify that the lines have to be straight so I make it a single spiral with pitch of < one square width.
In mathematics, lines are straight by definition. A spiral is a curve.

Last fiddled with by Mr. P-1 on 2009-05-05 at 16:36
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Old 2009-05-05, 16:50   #6
retina
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If the n is large (or the squares are large, or both) and the universe is closed, then only one line is needed.
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Old 2009-05-05, 16:53   #7
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I am pretty sure the answer is n (although it is possible to miss by one square on smaller boards, not sure about larger ones).
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Old 2009-05-05, 16:57   #8
retina
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Quote:
Originally Posted by Mr. P-1 View Post
In mathematics, lines are straight by definition. A spiral is a curve.
Okay, no argument from me there, but maybe the OP meant this type of line. :p
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Old 2009-05-05, 17:01   #9
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not true for 3x3
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Old 2009-05-05, 18:20   #10
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Quote:
Originally Posted by axn View Post
not true for 3x3
Neat.
Or on second thoughts a bit messy
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Old 2009-05-05, 18:29   #11
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for n even or n=1 one needs n lines, for n uneven one needs n-1 lines

I have no mathematical proof, just a bit of fidling around with a vector drawing program.

Jacob

Last fiddled with by S485122 on 2009-05-05 at 18:31 Reason: stated reasoning (?) behind assertion.
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