The game of Boxes is played on a rectangular $ab$ grid of points with horizontal and vertical undirected edges. $($There are a total of $2ab+a+b$ edges.$)$ At the start of the game there are no edges. On their move, White or Black can add an edge. If the edge completes one of the squares in the grid, the player who plays that edge gets a point and also gets to make the next move. In Figure $9-22,$ it is Black's turn in the position at the left, and he can complete two boxes before making a move in the upper right to give the position on the right.

$($C$)$ Kendall Hunt Publishing Company

Figure $9-22$: The result of a sequence of Black moves in a game of Boxes.

$($c$)$ The $11$ game is dull because Black always scores the only point on his second move. Analyze the $12$ game and determine who wins and by what margin under optimal play. It may help to exploit symmetries $-$ while White has seven possible first moves, you can restrict attention to only three.