Question About An Induction Problem: What Am I Supposed To Prove?

Question

I've encountered this induction problem, and I'm not sure what I'm supposed to prove. I would like an explanation on a few things about the problem; I'm not looking for a hint/solution to the problem. The problem here is:

What I'm confused about is the following:

1. What is "cap"/hemisphere? Right now, I'm imagining something that looks like this:

2. What does choosing a hemisphere mean? And what does removing a hemisphere mean? Right now, I'm imagining something that looks like this:

3. What does "keeping the sphere covered mean"? Right now, I think it means that as long as the sphere has caps on it, it's "covered". But then I doubt this is correct, because the problem would be trivial- since I've chosen 4 caps to remain, it's covered. So I'm wondering what the right interpretation of this is.

Thanks in advance.

Answer 1

I think the important part here is that the hemispheres don't actually stick out of the sphere; they're hemispheres of the actual sphere itself.

Here's a better way to phrase the question: A cap is a hemisphere of the sphere, and we have some set of caps such that their union is the whole sphere. Prove that there is a subset of four caps whose union is still the whole sphere.