$A$ writes, at his choice, $8$ ones and $8$ twos on a $4\times 4$ board. Then $B$ covers the board with $8$ dominoes and for each domino she finds the smaller of the two numbers that that domino covers. Finally, $A$ adds these $8$ numbers and the result is her score. What is the highest score $A$ can secure, no matter how $B$ plays? Clarification: A domino is a $1\times 2$ or $2\times 1$ rectangle that covers exactly two squares on the board.