Let $x, y, z, w \in [0,1]$. Find the maximum value of $S=x^{2} y+y^{2} z+z^{2} w+w^{2} x-x y^{2}-$ $y z^{2}-z w^{2}-w x^{2}$.
Suppose $x=\max \{x, y, z, w\}$, then
$$
\begin{aligned}
S & =y\left(x^{2}-z^{2}+y z-x y\right)+w\left(z^{2}-x^{2}+x w-z w\right) \\
& =y(x-z)(x+z-y)+w(z-x)(z+x-w) \\
& \leqslant y(x-z)(x+z-y) \leqslant\left(\frac{1}{3}(y+x-z+x+z-y)\right)^{3} \\
& =\frac{8}{27} x^{3} \leqslant \frac{8}{27} .
\end{aligned}
$$

Furthermore, when $x=1, y=\frac{2}{3}, z=\frac{1}{3}, w=0$, we have $S=\frac{8}{27}$. Therefore, the maximum value of $S$ is $\frac{8}{27}$.