4 Let $a_{1}, a_{2}, \cdots, a_{6} ; b_{1}, b_{2}, \cdots, b_{6}$ and $c_{1}, c_{2}, \cdots, c_{6}$ 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
$$
S \geqslant 6 \sqrt[6]{\prod_{i=1}^{6} a_{i} b_{i} c_{i}}=6 \sqrt[6]{(6!)^{3}}=6 \sqrt{6!}=72 \sqrt{5}>160
$$

Next, we prove that $S \geqslant 162$.
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}
$$

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
$$
1,2,3,4,5,6 ; \quad 5,4,3,6,1,2 ; \quad 5,4,3,1,6,2
$$

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

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