My friend and I used to play a simple game. An abitrarily large array of dots was drawn on paper, and we took turns connecting adjacent dots vertically or horizontally. Whenever a box connecting four adjacent dots was made, the player who finished it got an extra turn and a point. When all possible lines were drawn, the game ended and the one with the most points won.

My friend and I were both horrible at this game; we both used the same ineffective strategy. On each of our turns, when possible, we would always make a move that would not allow the other player to make a box the next turn.

Using this strategy and 25 dots in a 5x5 grid, what is the fewest number of moves possible before someone has to let the other player score? What if we use 36 dots in a 6x6 grid? And 49 dots in a 7x7 grid?

It seems like everyone else is writing a conclusion statement so I will too, even though I am taking up 3 comments to do so!

For an odd number of dots (and even number of line segments on each line across and even number of boxes created by a completed game) the best you can do is half the current number of line segments. Because there are 2(n)(n-1) dots, the best you can do here is (n)(n-1) if n is the number of dots across

However, for an even number of dots, you can have the entire perimeter comprised of empty segments. So this means you only use (n-1)(n-1) segments because half the outside isn't used up with segments.

Posted by Gamer on 2005-01-04 21:10:15