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What are some really interesting math problems you know of?

One I worked on the other day and neither of my math professors had a quick answer to was: Construct 3 disjoint open sets in R with the same boundary, if it is not possible prove why.
An interesting generalization of this is in R^n how many disjoint open sets can share a boundary?
If you message me I can send you the answer(it's also posted on math.SE which is where I got the problem from).

Anyone else have interesting theoretical problems like this?

I'm more interested in topics in topology and analysis(recently I've been looking at the cantor like middle-m sets with my math professor).
  
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