Question 95
Sahand and Gholam play on a $1403 \times 1403$ grid, initially with all cells white. For each row and each column, there is a button (total $2 \times 1403 = 2806$ buttons). Starting with Sahand, each player, in his turn, presses a button that has not yet been pressed. Then it's the other player's turn, until all buttons are pressed. When Sahand presses a button for a row or a column, all cells in that row or column turn to black, regardless of their color before pressing the button. When Gholam presses a button for a row or a column, all cells in that row or column turn to red, regardless of their color before pressing the button.
At the end, after all buttons have been pressed, Gholam's score is the number of red cells minus the number of black cells. Sahand's score is the number of black cells minus the number of red cells. If Gholam and Sahand both play their best, what would be the minimum score of Gholam? (In other words, find the least score Gholam can guarantee for himself, regardless of Sahand's moves.)
At the end, after all buttons have been pressed, Gholam's score is the number of red cells minus the number of black cells. Sahand's score is the number of black cells minus the number of red cells. If Gholam and Sahand both play their best, what would be the minimum score of Gholam? (In other words, find the least score Gholam can guarantee for himself, regardless of Sahand's moves.)