# Anna and Boris play the Red Blue game

Anna and Boris play a game on a 9x9 chessboard. Anna goes first and turns alternate thereafter. In each move, Anna puts a red counter on a vacant square while Boris puts a blue counter on a vacant square. When the board is completely filled, a row with more red counters than blue counters is called a red row, and a blue row otherwise. Red and blue columns are similarly defined. The score for Anna is the sum of the numbers of red rows and red columns while that for Boris is the sum of the numbers of blue rows and blue columns. What is the highest possible score for Anna?

Clarification example: After all the squares have been filled, if there are 7 red rows and 3 red columns then Anna’s score is 7+3=10.

2nd clarification: We are not assuming that Boris is playing well. We need to consider all possible games including games in which Boris is trying to help Anna.

Best score for Anna is 16 points. Anna places 41 counters. She must place at least 5 counters in a row to claim it, so she gets at most $$\lfloor41/5\rfloor=8$$ points from rows. Similarly she gains at most 8 points from columns. Therefore her score is at most 16.
$$\begin{matrix} R&R&R&.&.&.&R&R&. \\ R&R&R&R&.&.&.&R&. \\ R&R&R&R&R&.&.&.&. \\ .&R&R&R&R&R&.&.&. \\ .&.&R&R&R&R&R&.&. \\ .&.&.&R&R&R&R&R&. \\ R&.&.&.&R&R&R&R&. \\ R&R&.&.&.&R&R&R&. \\ .&.&.&.&.&.&.&.&R \\ \end{matrix}$$