Let $W_i$ be the group of $n$ consecutive seats starting at seat $i$. All seat indices are interpreted modulo $2n$. Call seats $i$ and $i+n$ opposite.

Mark seat $i$ if the unique complete pair changes when we move from $W_i$ to $W_{i+1}$.

At such a change, the old pair must contain the departing seat $i$, and the new pair must contain the arriving seat $i+n$.

Suppose $s$ is marked. The new pair contains $s+n$ and some seat $t$ strictly between $s$ and $s+n$ clockwise. This pair remains complete until seat $t$ leaves the window. Therefore, $t$ is the next marked seat, and the pair is $\{t,s+n\}$.

This gives two properties of the marked set $S$:

• Every clockwise gap between consecutive marked seats is shorter than $n$.
• At most one seat from each opposite pair is marked.

For the second property, every non-opposite pair appears in some windows and disappears exactly once. Thus, its endpoints occur once among the marked seats and once among their opposite seats. These two sets must be disjoint because each person has only one partner.

Every remaining pair must join opposite seats, since any other pair would appear in some window and produce a mark.