a) The minimum number of counters that need be placed on a nxn chessboard such that no additional counters can be placed without creating any straight line of 3;

b) The maximum number of counters that can be placed on a nxn chessboard such that no three lie in a straight line.

Remember that positions like A1, B3, and C5 are in a straight line.
(Try to continue the sequences if you can)

Presented below are the n-by-n "chessboards" for both the mininum number and the maximum number of counters. Again, no guarantee these are the actual minimums and maximums. 

I am unaware if there is a sequential pattern. (Before editing, I had stated the maximums appear to be n*2, yet this may be untrue as there was an error in the 7x7 grid example -- now corrected to show the maximum as 13, it may still be true, but I do not have the pattern, if possible, where it is 14).

_(4)_3x3_(6)_
X . X | X X .
. . . | X . X 
X . X | . X X
 
__(4)__4x4__(8)__
X . . X | X . X .
. . . . | X . X .
. . . . | . X . X  
X . . X | . X . X
 
__(6)___ 5x5 __(10)__
. . . . . | X . X . .
. X . X . | . . . X X
X . . . X | . X X . .
. X . X . | X . . . X
. . . . . | . X . X .

____(8)____6x6___(12)____
X X . . . . | X . . X . .
X . . X . . | X . . X . .  
. . . . . . | . X . . X .
. X . . X . | . X . . X .
. . . X X . | . . X . . X
. . . . . . | . . X . . X
 
_____(9)_____7x7____(13)_____
X X . . . . . | X . X . . . .
X . . X . . . | . X . . X . .
. . . . . . X | . . . . X . X
. X . . X . . | X . X . . . .
. . . X X . . | . . . X . X .
. . . . . . . | . X . . . . X
. . . . . . . | . . . X . . .

______(12)_____8x8_____(16)______
. X . . . . X . | X . . . X . . .
X . . . . . . X | X . . . X . . .
. . X . . X . . | . X . . . X . .
. . . . . . . . | . X . . . X . .
. . . . . . . . | . . X . . . X .
. . X . . X . . | . . X . . . X .
X . . . . . . X | . . . X . . . X
. X . . . . X . | . . . X . . . X

Edited on April 27, 2006, 6:11 pm

Posted by Dej Mar on 2006-04-27 05:43:04