Question 71
Anna and Baptiste play the following game. At the start of the game, 2022 white cells, numbered from 1 to 2022, are placed in front of them. Then, taking turns and starting with Anna, they choose a white cell and colour it in the colour of their choice: either red or blue. The game stops after 2022 turns, that is, when the last white cell is coloured.
Anna's score is then equal to the number of integers $a \leqslant 2019$ such that the two cells numbered $a$ and $a+3$ have the same colour. Anna wants her score to be as high as possible; Baptiste wants Anna's score to be as low as possible. What is the maximum score that Anna can guarantee, whatever moves Baptiste chooses to play?
Anna's score is then equal to the number of integers $a \leqslant 2019$ such that the two cells numbered $a$ and $a+3$ have the same colour. Anna wants her score to be as high as possible; Baptiste wants Anna's score to be as low as possible. What is the maximum score that Anna can guarantee, whatever moves Baptiste chooses to play?