Question 93
A gardener and a woodpecker play the following game, in their garden whose shape is a $2022 \times 2022$ grid made of $2022^{2}$ cells. Two cells are considered neighbours if they share a vertex or an edge. Initially, each cell holds a tree of height 0. Then, at each turn of the game,
$\triangleright$ the gardener chooses a cell; the trees of this cell and of the adjacent cells (four to nine cells in all) all see their height increase by 1;
$\triangleright$ the woodpecker then chooses four cells; the trees of these cells all see their height decrease by 1 (or stay equal to 0 if the woodpecker chose a cell with a tree of height 0).
A tree is called splendid if its height is at least $10^{6}$. Find the largest integer $A$ for which the gardener can make sure, in a finite number of turns and whatever the woodpecker's choices, to have grown at least $A$ splendid trees.
$\triangleright$ the gardener chooses a cell; the trees of this cell and of the adjacent cells (four to nine cells in all) all see their height increase by 1;
$\triangleright$ the woodpecker then chooses four cells; the trees of these cells all see their height decrease by 1 (or stay equal to 0 if the woodpecker chose a cell with a tree of height 0).
A tree is called splendid if its height is at least $10^{6}$. Find the largest integer $A$ for which the gardener can make sure, in a finite number of turns and whatever the woodpecker's choices, to have grown at least $A$ splendid trees.