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 $1,2, \cdots, 6$. Find the minimum value of $\sum_{i=1}^{6} a_{i} b_{i} c_{i}$.
(Xiong Bin)
Let $S=\sum_{i=1}^{6} a_{i} b_{i} c_{i}$. By the AM-GM inequality, we have
$$
\begin{array}{l}
S \geqslant 6 \sqrt[6]{\prod_{i=1}^{6} a_{i} b_{i} c_{i}}=6 \sqrt[6]{(6!)^{3}} \\
=6 \sqrt{6!}=72 \sqrt{5}>160 .
\end{array}
$$

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

we have $S \geqslant 162$.
Furthermore, when $a_{1}, a_{2}, \cdots, a_{6} ; b_{1}, b_{2}, \cdots, b_{6} ; c_{1}, c_{2}, \cdots, c_{6}$ are
$$
1,2,3,4,5,6 ; \quad 5,4,3,6,1,2 ; \quad 5,4,3,1,6,2
$$

respectively, we have
$$
\begin{aligned}
S= & 1 \times 5 \times 5+2 \times 4 \times 4+3 \times 3 \times 3+4 \times 6 \times 1+ \\
& 5 \times 1 \times 6+6 \times 2 \times 2 . \\
= & 162 .
\end{aligned}
$$

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