28. Let $a_{1}, a_{2}, \cdots, a_{6} ; b_{1}, b_{2}, \cdots, b_{6} ; c_{1}, c_{2}, \cdots, c_{6}$ all be permutations of $1,2,3,4,5,6$. Find the minimum value of $\sum_{i=1}^{6} a_{i} b_{i} c_{i}$. (2005 China National Training Team Selection Exam Problem)
28- Let $S=\sum_{i=1}^{6} a_{i} b_{i} c_{i}$. By the AM-GM inequality, we have
$$\left.S \geqslant 6 \sqrt[6]{\prod_{i=1}^{6} a_{i} b_{i} c_{i}}=6 \sqrt[6]{\prod_{i=1}^{6}(6!}\right)^{3}=6 \sqrt{6!}=72 \sqrt{5} \geq 160$$

Next, we prove that $S>161$.
Since the geometric mean of $a_{1} b_{1} c_{1}, a_{2} b_{2} c_{2}, a_{3} b_{3} c_{3}, a_{4} b_{4} c_{4}, a_{5} b_{5} c_{5}, a_{6} b_{6} c_{6}$ is $12 \sqrt{5}$, and $26161
\end{aligned}$$

Therefore, $S \geqslant 162$.
Furthermore, when $a_{1}, a_{2}, \cdots, a_{6} ; b_{1}, b_{2}, \cdots, b_{6} ; c_{1}, c_{2}, \cdots, c_{6}$ are $1,2,3,4,5,6 ; 5,4,3,6,1,2 ; 5,4,3,1,6,2$ respectively, we have
$$\begin{aligned}
S= & 1 \times 5 \times 5+2 \times 4 \times 4+3 \times 3 \times 3+4 \times \\
& 6 \times 1+5 \times 1 \times 6+6 \times 2 \times 2=162
\end{aligned}$$

Thus, the minimum value of $S$ is 162.