Question 17
$\{a\}_1, \{a\}_2, \ldots, \{a\}_{1982}$ is a permutation of $1, 2, \ldots, 1982$. If $\{a\}_1 \gt \{a\}_2$, we swap $\{a\}_1$ and $\{a\}_2$. Then if$\,$(the new)$\, \{a\}_2 \gt \{a\}_3$ we swap $\{a\}_2$ and $\{a\}_3$. And so on. After $1981$ potential swaps we have a new permutation $\{b\}_1, \{b\}_2, \ldots, \{b\}_{1982}$. We then compare $\{b\}_{1982}$ and $\{b\}_{1981}$. If $\{b\}_{1981} \gt \{b\}_{1982}$, we swap them. We then compare $\{b\}_{1980}$ and (the new) $\{b\}_{1981}$. So we arrive finally at $\{c\}_1, \{c\}_2, \ldots, \{c\}_{1982}$. We find that $\{a\}_{100} = \{c\}_{100}$. What value is $\{a\}_{100}$?