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 2009-05-05, 16:15 #1 Kees     Dec 2005 3048 Posts 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.
2009-05-05, 16:26   #2
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

668510 Posts

Quote:
 Originally Posted by Kees 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

 2009-05-05, 16:27 #3 TimSorbet Account Deleted     "Tim Sorbera" Aug 2006 San Antonio, TX USA 11×389 Posts 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".
 2009-05-05, 16:34 #4 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts I can't improve on n
2009-05-05, 16:35   #5
Mr. P-1

Jun 2003

49116 Posts

Quote:
 Originally Posted by retina 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

 2009-05-05, 16:50 #6 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 668510 Posts If the n is large (or the squares are large, or both) and the universe is closed, then only one line is needed.
 2009-05-05, 16:53 #7 10metreh     Nov 2008 2·33·43 Posts 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).
2009-05-05, 16:57   #8
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

1A1D16 Posts

Quote:
 Originally Posted by Mr. P-1 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

 2009-05-05, 17:01 #9 axn     Jun 2003 22×32×151 Posts not true for 3x3 Attached Thumbnails
2009-05-05, 18:20   #10
davieddy

"Lucan"
Dec 2006
England

2×3×13×83 Posts

Quote:
 Originally Posted by axn not true for 3x3
Neat.
Or on second thoughts a bit messy

 2009-05-05, 18:29 #11 S485122     "Jacob" Sep 2006 Brussels, Belgium 3×5×127 Posts 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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