if there are 5 points on a sphere then 4 of them belong to a half-sphere.

If there are 5 points on the surface of a sphere, then there is a closed half sphere, containing at least 4 of them.

It's in a pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension.

Regards

• An answer is given by Calvin Lin here. This is a duplicate in a way, but the topic of that other question is vastly different (a call for trick questions), so I am a bit reluctant to call it a duplicate. Commented Feb 13, 2014 at 22:50
• So if you ever see a headline like "80% of Olympic Sites In The Last 20 Years Have Been In The ___ Hemisphere," it means bupkis. Commented Feb 14, 2014 at 16:36
• @Bob Stein What is the maning of "bupkis" ? I never met this term... Commented Jun 22, 2020 at 6:57
• @JeanMarie "bupkis" means "nothing at all". According to yourdictionary it comes from Yiddish. I think it's especially funny when used in a string of synonyms. Commented Jun 22, 2020 at 10:15
• This was problem A2 from the 2002 Putnam Exam. See the solution here: youtube.com/… Commented Nov 28, 2021 at 17:33

• @zinking: Take 5 points on $\mathbb S^1$ such that the angle of any two neighboring points is $2\pi/5$. Then in every closed hemisphere there lie at most 3 of the points. Commented Feb 14, 2014 at 14:54