1. Let $i_{1}, i_{2}, \cdots, i_{10}$ be a permutation of $1,2, \cdots, 10$. Define $S=\left|i_{1}-i_{2}\right|+\left|i_{3}-i_{4}\right|+\cdots+\left|i_{9}-i_{10}\right|$. Find all possible values of $S$.
[2]
Since $S \geqslant 1+1+1+1+1=5$, $S \leqslant 6+7+\cdots+10-(1+2+\cdots+5)=25$,
and $S \equiv \sum_{k=1}^{10} k(\bmod 2) \equiv 1(\bmod 2)$,
therefore, it only needs to be proven that $S$ can take all odd numbers from 5 to 25.