Consider the classic pencil and paper game [Dots and Boxes][1] . You can assume the 2 by 2 version (that is with 9 dots). 

> Starting with an empty grid of dots, players take turns, adding a
> single horizontal or vertical line between two unjoined adjacent dots.
> A player who completes the fourth side of a 1×1 box earns one point
> and takes another turn. (The points are typically recorded by placing
> in the box an identifying mark of the player, such as an initial). The
> game ends when no more lines can be placed. The winner of the game is
> the player with the most points.

![enter image description here][2]

Note that there are at most 12 moves in the entire game and I think there will be between 9 and 11 turns.


> The puzzle is to determine if the game is a win, lose or draw for the
> first player assuming optimal play by both sides.

Clearly you could brute force it by computer, which is fine and perfectly interesting, but not as interesting as a human understandable proof.


  [1]: http://en.wikipedia.org/wiki/Dots_and_Boxes
  [2]: https://i.sstatic.net/blhGw.png