13.43 A circular stage for a circus performance is illuminated by $n$ searchlights, each of which illuminates a convex shape on the stage. It is known that when any one of the searchlights is turned off, the stage is still fully illuminated; but if any two searchlights are turned off, the stage cannot be fully illuminated. Find all possible values of $n$.
[Solution] For all $n \geqslant 2$, it is possible. $n=1$ clearly does not meet the requirement; when $n=2$, it is only necessary to arrange for each spotlight to illuminate the entire stage.

When $n \geqslant 3$, number these spotlights as $1, 2, \cdots, n$. The different pairs $(i, j)$ formed by $\{1, 2, \cdots, n\}$ total $\frac{1}{2} n(n-1) = m$ pairs. We arrange these pairs in lexicographical order as follows: $(1,2), (1,3), \cdots, (1, n), (2,3), \cdots, (2, n), (3,4), \cdots, (n-1, n)$. Consider a circular stage and its inscribed regular $m$-sided polygon. Each side of this regular $m$-sided polygon, together with the minor arc of the circle, forms a small segment, totaling $m$ segments. We correspond each of the $m$ pairs to one of the $m$ segments. For the $i$-th spotlight, we take the union of the regular $m$-sided polygon and all segments corresponding to pairs containing $i$ (which is of course a convex shape) as its illumination area, which clearly meets the requirements of the problem.

In fact, each segment corresponds to a pair $(i, j)$, and thus is illuminated by two spotlights. When one of the $n$ spotlights is turned off, the entire stage is still illuminated. However, when two spotlights are turned off, the segment corresponding to the pair of their numbers will not be illuminated.