Consider the classic pencil and paper game Dots and Boxes . 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.

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.