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?

For a 25-dot grid, you can start by alternating step-like patterns. There are a total of 20 segments that can be drawn before the next segment breaks the strategy.

For a 36-dot grid, using the same strategy, you can get 30 segments, and for a 49-dot grid you can get 42 segments.

It looks like it generalizes to: for an n x n grid, you can get n*(n-1) segments.

Posted by Eric on 2005-01-04 19:27:45