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?

I started with a 2x2 grid it turns out that after 2 moves,

the 3rd move lets the fourth score, so the fewest moves is 2.

For a 3x3 grid the fewest is 4. They form a +

For a 4x4 grid the fewest is 10. There are at least two ways of doing this.  I'm sorry Tristan, I still can't manage a picture.  One way is two vertical lines down the middle rows with two spurs off of each side.  The other is an expanded version of the + from a 3x3.

For a 5x5 grid the fewest is 16.  This can be done with three vertical lines and sets of 2 spurs or with an expanded +.

For a 6x6 the fewest is 26.

For a 7x7 the fewest is 36.

I haven't proven any of this but a reasonable formula seems to be for a nxn grid, the fewest is (n-1)^2 or (n-1)^2 + 1 if n is even.

-Jer

Posted by Jer on 2005-01-05 18:57:12