24.1 ** Select $k$ numbers from $1,2, \cdots, 1000$. If among the selected numbers there are always three that can form the sides of a triangle, find the minimum value of $k$.
Parse and select 15 numbers: $1,2,3,5,8,13,21,34,55,89,144,233,377,610$, 987, such that no three numbers can form the sides of a triangle, so $k \geqslant 16$. Let the 16 selected numbers be $a_{1}<$ $a_{2}<\cdots<a_{16}$, if no three numbers form the sides of a triangle, then $a_{1} \geqslant 1, a_{2} \geqslant 2, a_{3} \geqslant a_{1}+a_{2} \geqslant$ $3, a_{4} \geqslant a_{2}+a_{3} \geqslant 5, \cdots, a_{15} \geqslant a_{13}+a_{14} \geqslant 987, a_{16} \geqslant a_{14}+a_{15} \geqslant 1597$, which contradicts $a_{16} \leqslant$ 1000. Therefore, selecting 16 numbers, there must be three numbers that form the sides of a triangle. In summary, the minimum value of $n$ is 16.