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?

Just so it's clear, you're looking for the fewest number of moves before someone has to let the other player score.  That means you're looking for an ideal setup with as few lines drawn as possible where the next move must allow a score.

The solution is harder than it first appears.  It is less than (n-1)^2.  I don't have any idea how to generalize the solution, so I suggest doing each part separately.

Posted by Tristan on 2005-01-05 01:18:17