This question was presented as an application of the pigeonhole principal, giving the natural answer of 9: Once you distribute 8 points evenly across a sphere split into two hemispheres, the 9th point will ensure that at least one of the hemispheres has five points.

However, a counter argument was made: the first two points on the sphere are always collinear and define an equator, and any point along the sphere's equator are considered to be in both hemispheres simultaneously. As such, equally distibuting four points puts each hemisphere at 4 points, and point number 7 has to bring a hemisphere up to 5.

The final argument takes everything one step further, suggesting that as there are an infinite number of possible equatorial lines, every single arrangements of 5 points must have at least one valid equator which splits the hemispheres into 5 and 0. We tried a whole bunch of arrangements on a tennis ball and couldn't quite find one to disprove this, but we're not exactly convinced either.

Safe to say, the argument had spiralled far past the point of pigeon-holes by the time we got here. Does anyone have anything concrete to nail down the answer with?