$$
\text { Three. (25 points) Among the } 1,2, \cdots, 2008 \text { numbers, }
$$
add a negative sign in front of $n$ numbers so that the algebraic sum of these 2008 numbers equals zero. Find the maximum and minimum values of $n$.

Three, find the minimum value of $n$.
To make $n$ as small as possible, the numbers to which negative signs are added should be as large as possible. Let these numbers be:
$$
2008, 2007, \cdots, 2010-n, k
$$

where, $00, \\
\frac{1}{2} n^{2}-\frac{4019}{2} n+10105270 \text{. }
$$

Since $n$ is a positive integer less than 2008, we have $n \leqslant 589$.
Similarly, from equation (3), we get $n>588$.
Therefore, $n=589$.
Substituting $n=589$ into equation (1) gives $k=392$.
When $n=589$, the 589 numbers $392, 1421, 1422, \cdots, 2008$ with negative signs in front of them meet the requirements.
Obviously, the maximum value of $n$ is $2008-589=1419$.
Apart from these 589 numbers, the other 1419 numbers with negative signs in front of them also meet the requirements.
In summary, the maximum value of $n$ is 1419, and the minimum value is
589.