Alex and Bobette are playing on a $20 \times 20$ grid where the cells are square and have a side length of 1. The distance between two cells is the distance between their centers. They take turns playing as follows: Alex places a red stone on a cell, ensuring that the distance between any two cells with red stones is never $\sqrt{5}$, then Bobette places a blue stone on the grid without any restrictions. The game stops when one of them can no longer place a stone. Find the largest $K$ such that Alex can always place at least $K$ stones, regardless of Bobette's responses.
Let's show that $K=100$. Alex can always place 100 stones: it suffices to color the grid in a checkerboard pattern and always play on a white square. Indeed, two white squares are never at a distance of $\sqrt{5}$ from each other. Let's show that Bobette can prevent Alex from placing more than 100 stones: we tile the grid with copies of the rectangle

$A C B D$

$B D A C$

$C A D B$

$D B C A$

If Alex plays on a square, Bobette plays in the same rectangle, on the square of the same color that is not at a distance of $\sqrt{5}$ from the first. Thus, Alex is deprived of the two other squares of the same color in the same rectangle. In the end, Alex cannot cover more than one quarter of the grid, which contains 400 squares.