10. Given
$$
S_{n}=|n-1|+2|n-2|+\cdots+10|n-10| \text {, }
$$

where, $n \in \mathbf{Z}_{4}$. Then the minimum value of $S_{n}$ is $\qquad$ .
10.112.

From the problem, we know
$$
\begin{array}{l} 
S_{n+1}-S_{n} \\
=|n|+2|n-1|+\cdots+10|n-9|- \\
{[|n-1|+2|n-2|+\cdots+10|n-10|] } \\
=|n|+|n-1|+\cdots+|n-9|-10|n-10| .
\end{array}
$$

When $n \geqslant 10$, $S_{n+1}-S_{n}>0$, thus, $S_{n}$ is monotonically increasing;
When $n<10$, $S_{n+1}-S_{n}<0$;
When $n=6$, $S_{7}-S_{6}<0$.
Therefore, $S_{n}$ is monotonically decreasing in the interval $[1,7]$, and monotonically increasing in the interval $[7,+\infty)$.
Hence, the minimum value of $S_{n}$ is $S_{7}=112$.