A few times now I've found myself in a situation where I want to give a precalculus-level undergraduate student something to think about that's kinda fun and that's well outside of their coursework. But at the same time, I want to ask them a question they can figure out just by thinking about it, or pondering over it, rather than having to work out some calculations or necessarily write things down.
So far my go-to question has been this one:
Suppose you take a pen and mark five points on a ball. I claim that no matter where you draw those points, I can find a closed hemisphere of the ball that contains exactly four of those points. Is this true?
I like this question because there's very little you can write down, and the geometric set up is simple enough that a student should be able to play with it in their head (or use an actual ball). But I'm not perfectly happy with this question because I have to explain to them about what I mean by a closed hemisphere, and I usually have to discuss some ideas of spherical geometry with them. But I feel like this gives them the impression that this problem would never have been tractable without my guidance. I want a problem that they evidently can think about and figure out without some necessary guidance, and possibly without having to write anything down or draw out any examples.
Does anyone have better ideas for questions like this I can ask students?
The reason I'm asking for such questions is that I want students to get over their fear of "doing the wrong thing," so I want to give them problems where there is nothing to write down and they have to mentally grapple with the question. This fear seems to spring up more easily if figuring out a question involves writing (maybe because I can see and judge their work?). They'll often sit there with a blank page and just look to me for guidance: they want me to tell them what the "right thing to do" is.