Given $n(>2)$ distinct points in space. Consider all lines determined by pairs of these points, and then all angles formed by pairs of these lines. We observe that the smallest (one of the smallest) angles among them is exactly $60^{\circ}$. What can $n$ be?
Any 3 points determine a triangle. It is not possible for 3 points to lie on the same line, because then there would be two lines forming a $0^{\circ}$ angle, which contradicts the fact that the smallest enclosed angle is $60^{\circ}$. Each interior angle of the triangles is also the angle between some pair of lines, and thus they cannot be less than $60^{\circ}$. This can only happen if any selected triangle is equilateral. However, in space, at most 4 points can be arranged such that the distance between any two of them is the same, meaning that any three of them form an equilateral triangle, so $n$ can be at most 4.

For 3 or 4 points, such an arrangement exists if the points are the vertices of an equilateral triangle or a tetrahedron. In this case, intersecting lines form a $60^{\circ}$ angle, while non-intersecting lines are perpendicular to each other. Therefore, $n$ can be either 3 or 4.