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 stopped posting improvements in my solutions, the typing of the actual solution diagram is a pain.
These are the results up till now.
Grid    Moves
2          2
3          4
4          8
5         15
6         23
7         32

This would lead to the formula (n-1)^2-(n-3).
I need help to improve the 5 and 6 diagrams to 14 and 22.  In my best solutions, there is clearly room for  improvement, but
I don't find it and am getting square eyes. 
I am reasonably sure that the others can't be improved.

 

Posted by Hugo on 2005-01-14 18:42:22