Great Circles and Inscribed Cube

Question

A cube is inscribed within a sphere. How many distinct great circles are there that contain at least 2 vertices of the cube along its perimeter?

Intuition tells me any great circle that coincides with two or more vertices of the cube must intersect with 2 vertices that are diagonal from each other. Hence, there are 4 pairs of such diagonals.

However, this was incorrect. I'm not sure where I'm going wrong and any help would greatly be appreciated.

Answer 1

There are infinitely many such great circles. This is because:

1. Any great circle on the sphere must share its center with the center of the sphere.
2. Any two vertices of the cube that are diametrically opposite along a long diagonal of the cube will be collinear with the center; i.e., the long diagonal is a diameter of the sphere.
3. An infinite number of great circles can be drawn through these opposing vertices.