Example 7 Given that the sum of several positive integers is 2019. Find the maximum value of their product.
Solution: Since there are only a finite number of different integer sets whose sum equals 2019, the maximum value must exist.
Let $x_{1}, x_{2}, \cdots, x_{n}$ all be positive integers,
$$
x_{1}+x_{2}+\cdots+x_{n}=2019 \text {, }
$$

and $P=x_{1} x_{2} \cdots x_{n}$ reaches its maximum.
Obviously, $x_{i} \geqslant 2(i=1,2, \cdots, n)$.
We will prove: $x_{i} \leqslant 4(i=1,2, \cdots, n)$.
If $x_{i}>4$, then
$$
2+\left(x_{i}-2\right)=x_{i}, 2\left(x_{i}-2\right)>x_{i} \text {, }
$$

replacing $x_{i}$ with $2$ and $x_{i}-2$ would increase $P$, leading to a contradiction.
Since $4=2 \times 2$, when $x_{i}=4$, it can be replaced by two 2s, keeping the sum unchanged. Therefore, we can assume
$$
x_{i} \leqslant 3(i=1,2, \cdots, n) \text {. }
$$

Thus, the number of 2s in $\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}$ is at most two. If there are three 2s, then because $2+2+2=6=3+3$, replacing three 2s with two 3s keeps the sum unchanged but increases the product, leading to a contradiction.
In summary, $P=2^{r} \times 3^{s}(r \leqslant 2)$.
Since $2019=3 \times 673$, we have $r=0, s=673, P$'s maximum value is $3^{673}$.

The perturbation method involves assuming an extremum, then transforming and deriving a contradiction to narrow the range of possible changes, thus leading to a conclusion and solving for the extremum.
Finally, here are four practice problems.
1. In a $100 \times 100$ grid, each cell is filled with one of $-1$, $0$, or $1$, such that the absolute value of the sum of all numbers in the grid does not exceed 400. Prove that there exists a $25 \times 25$ subgrid in which the absolute value of the sum of all numbers is no more than 25.
2. Given 100 real numbers $x_{1}, x_{2}, \cdots, x_{100}$ whose sum equals 100, and the absolute difference between any two adjacent numbers is less than 2. Prove that it is possible to select 50 numbers such that the absolute difference between their sum and 50 is less than 1.
3. Let $n \geqslant 3$. On a plane, there are $n$ red points and $n$ blue points, with no three points collinear, and the vertices of the convex hull polygon are of the same color. Prove that it is possible to draw a line that divides the $2n$ points into two sides, each side containing a non-zero number of points, and the number of red and blue points on each side being equal.
4. Find the largest positive integer $n$, such that for any arrangement of the positive integers $1,2, \cdots, 400$ into a $20 \times 20$ grid, there is always a row or a column where the difference between two numbers is not less than $n$.