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.

[QUOTE=Kees;172428]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.[/QUOTE][spoiler]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.[/spoiler]

Assuming straight lines only.
[spoiler]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".[/spoiler]

[spoiler] I can't improve on n [/spoiler]

[QUOTE=retina;172430][spoiler]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.[/spoiler][/QUOTE]

[spoiler]In mathematics, lines are straight by definition. A spiral is a curve.[/spoiler]

[spoiler]If the n is large (or the squares are large, or both) and the universe is closed, then only one line is needed.[/spoiler]

[spoiler]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).[/spoiler]

[QUOTE=Mr. P-1;172435][spoiler]In mathematics, lines are straight by definition. A spiral is a curve.[/spoiler][/QUOTE]Okay, no argument from me there, but maybe the OP meant [url=http://en.wikipedia.org/wiki/Line_(electrical_engineering)]this type of line[/url]. :p

not true for 3x3

[quote=axn;172440]not true for 3x3[/quote]
Neat.
Or on second thoughts a bit messy:smile:

[spoiler]for n even or n=1 one needs n lines, for n uneven one needs n-1 lines[/spoiler]

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

Jacob