Question 40
On a $5 \times 5$ board, two players alternately mark numbers on empty cells. The first player always marks $1$'s, the second $0$'s. One number is marked per turn, until the board is filled. For each of the nine $3 \times 3$ squares the sum of the nine numbers on its cells is computed. Denote by $A$ the maximum of these sums. How large can the first player make $A$, regardless of the responses of the second player?