124. Intersection of Great Circles. It is known that on a sphere, $n$ great circles intersect in $n(n-1)$ points in general. How should the numbers $1,2, \ldots, n(n-1)$ be placed in these points so that the sums of the numbers located on each circle are equal to each other. (Recall that the great circles lie in planes passing through the center of the sphere.)
124. Place the number $b$ at some point, and at the point located at the opposite end of the corresponding diameter, place the "complementary" number, which is $n(n-1)+1-b$. Continue

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-115.jpg?height=588&width=577&top_left_y=1625&top_left_x=748)

this process until all numbers (and points) are exhausted. Since each large circle contains $n-1$ pairs of intersection points, the total sum of all numbers on it is $[n(n-1)+1](n-1)$.

[L. Moser, M. M., 25, 114 (November 1951).]