examday

/TMUA

Ava and Ben play a game on a $12\times12$ board. For some positive integer $k$, Ava chooses $k$ squares and writes an $A$ in each of them. Ben then chooses $k+3$ squares and writes a $B$ in each. After Ben is done, Ava wins if there is a sequence of letters forming “$BAA$” or “$AAB$” when read either horizontally (left-to-right) or vertically (top-to-bottom), and Ben wins otherwise. Determine the smallest value of $k$ for which Ava has a winning strategy.