Problem: A \emph{site} is any point $(x,y)$ in the plane
for which $x,y \in \{1, \dots, 20\}$.
Initially, each of the $400$ sites is unoccupied.
Amy and Ben take turns placing stones on unoccupied sites,
with Amy going first;
Amy has the additional restriction that no two of her stones
may be at a distance equal to $\sqrt5$.
They stop once either player cannot move.
Find the greatest $K$ such that Amy can ensure that
she places at least $K$ stones.

Solution: The answer is $K = 100$.

First, we show Amy can always place at least $100$ stones.
Indeed, treat the problem as a grid with checkerboard coloring.
Then Amy can choose to always play on one of the $200$ black squares.
In this way, she can guarantee half the black squares,
i.e.\ she can get $\half \cdot 200 = 100$ stones.

Second, we show Ben can prevent Amy from placing more than $100$ stones.
Divide into several $4 \times 4$ squares and then further partition
each $4 \times 4$ squares as shown in the grid below.
\[
  \left[
  \begin{array}{cccc}
    1 & 2 & 3 & 4 \\
    3 & 4 & 1 & 2 \\
    2 & 1 & 4 & 3 \\
    4 & 3 & 2 & 1
  \end{array}
  \right]
\]
The squares with each label form $4$-cycles by knight jumps.
For each such cycle, whenever Amy plays in the cycle,
Ben plays in the opposite point of the cycle,
preventing Amy from playing any more stones in that original cycle.
Hence Amy can play at most in $1/4$ of the stones, as desired.