Three, (50 points) In an $8 \times 8$ grid, each cell contains a knight or a knave. Knights always tell the truth, and knaves always lie. It is known that each resident in the grid says: “The number of knaves in my column is (strictly) greater than the number of knaves in my row.” Determine the number of arrangements of residents that satisfy this condition.
Let the number of villains in the $i$-th row and the $j$-th column be denoted as $r(i)$ and $c(j)$, respectively.

Thus, the condition given in the problem is equivalent to: if $c(j) > r(i)$, then the resident in cell $(i, j)$ is a knight; otherwise, it is a villain.
First, we prove that all $c(j)$ are equal.
Assume that columns $j_{1}$ and $j_{2}$ satisfy $c(j_{2}) > c(j_{1})$. If there is a knight in cell $(i, j_{1})$, then $c(j_{1}) > r(i)$, and consequently, $c(j_{2}) > r(i)$. Therefore, there is also a knight in cell $(i, j_{2})$. Thus, the number of knights in column $j_{2}$ is no less than that in column $j_{1}$, which contradicts $c(j_{2}) > c(j_{1})$.

Next, we prove that all residents in the same row are either all knights or all villains (such an arrangement is called a "striped arrangement").

For the resident in cell $(i, j)$ of the $i$-th row, since $r(i)$ is independent of $j$, whether the resident in cell $(i, j)$ is a knight or a villain depends only on $c(j)$. Given the previous argument, we know that all $c(j)$ are equal, so the residents in the $i$-th row are either all knights or all villains.
Finally, we verify that the striped arrangement meets the conditions.
If there is at least one row of villains, then the number of villains in each column containing a villain does not exceed 8 (villains lie), while the number of villains in each row containing a knight is indeed greater than 0 (knights tell the truth). If there are no villains in any row, then every resident is a knight, and the number of villains in their row and column is 0, which contradicts the fact that knights tell the truth.
Therefore, there are $2^{8} - 1 = 255$ arrangements that meet the conditions.