number seq?

[Acidic]

OK, this is just something I've been doing for fun, hence, not that important. at www.sporkle.com (jn the multiplayer section), players can play a game called rooms. I'm sure some of you have heard of this. It is a two player game, and you start with a blank grid, lets say 5x5. You can draw a line connecting two dots, the line has to be one unit in length (cannot cross over a dot), and cannot be diagonal.

When a player makes a square (which must be 1x1) the square gets the colour of the player. At the end of the game (once all the squares are coloured) the player with the most squares of his/her colour wins.

I want to work out if there is a sequence for the maximum number of turns, taking into account that if a person gets a pointm they get an extra move that turn.

BTW, the lines on the outside have not been drawn from the start

so, here are my numbers (may be incorrect):
1x1=4 turns
2x2=10 turns
3x3=17
4x4=? i got lots of different answers


edit:
I'm pretty sure 4x4=27