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?

There are better solutions out there, for all three cases.  Just keep at it, and I'll post the solution when you get there.





HINT: This solution has already been propsed for the 6x6 grid:

_| | | |_
_ _| |_ _
_ _   _ _
_  | |  _
 | | | |

You might notice that this solution has 24 lines, which is one less than the (n-1)^2 that might be expected.  Thinking about this may reveal ways to get even lower, not only in the 6x6 grid, but in the other two grids as well.

Posted by Tristan on 2005-01-11 05:07:51