Let us consider the problem where Amy and Ben take turns placing stones on a 20x20 grid consisting of sites $(x, y)$ where $x$ and $y$ are integers between 1 and 20 inclusive. Amy's condition for placing a red stone is that the distance between any two red stones is not equal to $\sqrt{5}$. This occurs specifically when the coordinates of two stones differ by 2 in one coordinate and 1 in the other, which are equivalent to the vector differences $(\pm 2, \pm 1)$ or $(\pm 1, \pm 2)$.

The goal is to find the maximum number $K$ such that Amy can ensure placing at least $K$ red stones regardless of Ben's move choices. Our task requires an arrangement that avoids any pair of red stones being placed at a distance of $\sqrt{5}$.

### Strategy

An efficient strategy is to place stones such that any two stones are more than $\sqrt{5}$ units apart. We focus on constructing a checkerboard pattern, exploiting the grid nature.

1. To prevent placing red stones at a distance of $\sqrt{5}$, consider only placing red stones on sites where both coordinates, $x$ and $y$, are either both odd or both even.

2. This choice ensures that the difference $(\pm 2, \pm 1)$ or $(\pm 1, \pm 2)$ cannot align two red stones with the distance of $\sqrt{5}$, since these would require parity mismatches in both coordinates.

3. Constructing the grid in this way, one subset of the grid (say, all sites with both coordinates odd) will contain no nearby red stones to violate the distance rule.

### Calculation

- Each 20x20 grid contains 400 total sites.
- The number of odd-odd sites is equal to the number of even-even sites since both are $\frac{20}{2} \times \frac{20}{2} = 10 \times 10 = 100$.

Amy can guarantee placing at least 100 red stones because whenever she uses only odd-odd or even-even sites, each site choice naturally keeps all pairs in the valid configuration.

Thus, the greatest $K$ such that Amy can guarantee at least $K$ stones is:

$$\boxed{100}$$

This solution represents a strategic positioning allowing Amy to always utilize the entire set of sites based on parity distinctions, ensuring Ben’s choices will not interfere with her minimum $K$ target.