n great circles on the sphere are in general position (in other words at most two circles pass through any two points on the sphere). How many regions do they divide the sphere into?
Answer: n2 - n + 2.
We use the well-known formula E + 2 = V + F, where E is the number of edges, V the number of vertices and F the number of faces. It is true for a sphere provided V > 1, so certainly for n > 1.
Each circle intersects every other circle in 2 vertices, so V = n(n - 1). Each vertex has degree 4, so E = 2V. Hence F = V + 2 = n2 - n + 2. It is easy to check that the formula also holds for n = 1.
25th Putnam 1964
© John Scholes
5 Feb 2002